claims depot shelf
Navier-Stokes Regularity
Formal claims (Lean)
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On the paper's own terms, the central discovery is Theorem 1.2: there exists a uniform exponent $2<q\ll3$ such that for any nonnegative smooth function $e(t):[0,T]\to[0,\infty)$ there is a weak solution $u\in C_t([0,T];L^q(\mathbb{T}^3))$ of the periodic Navier-Stokes system with $\int_{\mathbb{T}^3}|u(x,t)|^2\,dx=e(t)$ for every $t$. Taking $e_k(t)=1-\cos(kt)$ for integers $k\ge1$ yields infinite
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The central claim is that the linearized coupled operator, a $2\times 2$ system with a heat-type equation for the porous pressure and a Stokes operator for the fluid velocity whose domain encodes the interface conditions as non-diagonal boundary traces, generates an exponentially stable, compact, analytic semigroup with maximal $L^r$-regularity. Under the Beaver-Joseph-Saffman condition this holds
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Corollary 2.10: For finite-energy initial data, the same framework yields a local weak solution even when the thick solid is purely elastic (delta = 0).
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Theorem 2.6: If P_in/out is time-periodic with sufficiently small L2 norm, then the coupled 3D fluid / 2D shell / 3D viscoelastic solid system (1.16) has at least one time-periodic weak solution (u, eta, d) in the energy space, and sup E + int D <= C0.
Stated claims
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The central claim is Theorem 1.1: there exist positive constants $\varepsilon_0$ and $\delta_0$ such that for any $\varepsilon\in(0,\varepsilon_0)$ and wave strengths $\delta_l\sim\delta_r\le\delta_0$, the Cauchy problem for the compressible Navier-Stokes equations has a family of global smooth solutions $(v^\varepsilon,u^\varepsilon)$ converging as $\varepsilon\to0^+$ to the entropy solution $(V,U)$ of the Euler equations. The convergence holds in $L^p(\mathbb{R})$ for every $p\in[2,+\infty)$ with the explicit bounds $\|(v^\varepsilon-V,u^\varepsilon-U)(t,\cdot)\|_{L^p}\le C(\delta_l+\delta_r)^{1/2}\varepsilon^{1/p}$ for $0\le t\le t_0$, plus $C(\delta_l+\delta_r)(t-t_0)^{1/(2p)}\varepsilon^{1/(2p)}$ for $t\ge t_0$. The proof treats the pre-collision phase, the collision point, and the post-collision shock-rarefaction composite as one connected picture, using an approximate collision time to close uniform energy estimates before merging into a shifted composite wave after the collision.
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The paper's core assertion is that a D-solution is trivial if, for r >= 1, the supremum of velocity or vorticity on cylinders |x'|=r decays as |u| <= C r^{-2/3} [log(e+r)]^{-gamma} or |omega| <= C r^{-5/3} [log(e+r)]^{-gamma} with gamma > 1/3, with no symmetry hypothesis needed for this Liouville statement (abstract, result (ii)). It also claims the improved decay rates in result (i).
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The central claim is that the matched asymptotic expansion is not merely formal: the leading outer profiles satisfy the chemotaxis-Euler system (2.4), which is locally well-posed for initial data in $H^m_{xy}\times H^m_{xy}\times H^{m+1}_{xy}$ satisfying compatibility and curl-free conditions; and the inner profiles $v_{B,0}^2, n_{B,1}, v_{B,1}^1, v_{B,1}^2, n_{B,2}, u_{B,1}^1, u_{B,2}^2, v_{B,2}^1, u_{B,2}^1, u_{B,3}^2, p_{B,2}$ defined by (2.5)-(2.15) each admit a unique solution with weighted anisotropic Sobolev regularity, provided the outer solution has sufficiently high tangential regularity. The main difficulties overcome are the loss of diffusion in the Euler limit, handled through the curl-free structure of $v$ and elliptic div-curl estimates, and the unbounded normal transport terms $z\,a(t,x)\partial_z f$ in the inner equations, handled through polynomial weights and boundary homogenization.
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The central claim is that the S1-S3 loop (deduce a priori bounds, verify sharpness by variational maximization, extract mechanisms) yields closed solutions to two model problems. In the Burgers problem, the instantaneous bound $dE/dt \le C\nu^{-1/3}E^{5/3}$ is sharp in its exponent, and the finite-time numerical maximizers found by Ayala & Protas (2011) grow like $E_0^{3/2}$; Albritton & Nitti (2023) then proved the matching upper bound, so the problem is mathematically closed. In the 2D Navier-Stokes problem, Matharu et al. (2022) showed that the combined estimate (34)-(35) is saturated by six branches of extreme initial conditions that maximize enstrophy dissipation, so the estimate is declared sharp and offers no room for improvement other than, perhaps, a logarithmic correction. For 3D Euler flows, maximizing the $\dot{H}^3$ seminorm over Gevrey-class initial data yields a flow whose norm growth is consistent with finite-time singularity formation, with the near-singular structure being two colliding jets forming a flattened vortex-ring gap.
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The central claim is Theorem 1: for divergence-free u0 in L^p_w(R^n_+), p>n, there exists a time T(u0)>0 and a smooth solution (u,π) of the Navier-Stokes IBVP, expressed by the representation formulas (8)-(9), satisfying the four-term estimate (10) with constants controlled by K(t,rho) and the weighted norm of u0, the limit (11) in the chosen metric, the dual-space convergence (12), and the pressure estimate (13). Theorem 2 asserts uniqueness of this solution in the class detected by Theorem 1. For suitably small weighted norm, the same results hold for all positive times.
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The paper's central claim is Theorem 5.1: for any global Leray–Hopf weak solution of the 3D Boussinesq system with Navier boundary conditions, if the friction coefficient α ∈ L∞(∂Ω) is nonnegative and positive on a boundary subset of positive surface measure, then the total L2 energy decays exponentially: ‖u(t)‖² + ‖ρ(t)‖² ≤ D(‖u0‖² + ‖ρ0‖²)$e^{{-Kt}}$ for all t≥0, with K,D depending only on ν, κ, Ω, and α. The proof does not require the rigid-motion kernel to be trivial nor any geometric restriction on the domain; the weighted boundary term in the Korn–Poincaré inequality accounts for the kernel component. When α≡0, the scalar ρ and the velocity component orthogonal to the rigid-motion kernel K
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The central claim, Proposition 1, is that the BDF2 incremental Helmholtz-Leray projection method with residual-based VMS stabilization on the predicted velocity satisfies a discrete velocity error of order dt^2 + h^q. The multiscale decomposition is applied only to the predicted velocity, whose unresolved part is modeled by the momentum residual; neither pressure nor corrected velocity is decomposed. The modeled fine scale enters all three substeps, producing SUPG-like stabilization in the predictor, a PSPG-like term in the pressure Poisson equation, and no pressure-fine-scale/grad-div term. On equal-order elements the scheme matches reference data for manufactured solutions, lid-driven cavi
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Theorem 1 states that for initial data satisfying local bounds on the density, a velocity gradient strictly below one, and finite total energy, there exist functions (ρ,u) and a Radon measure τ on (0,T)×R such that |∂xu|≤1, τ=π∂xu, π≥0, and π(1−|∂xu|)=0 almost everywhere, and (ρ,u) solves the continuity and momentum equations in the distributional sense. The density is locally bounded away from zero and infinity on every compact set, with constants depending only on the data and the compact set. The construction first solves the truncated problem on Ωk=(−2k−2,2k+2) with homogeneous Dirichlet boundary conditions, obtains estimates independent of p and k, passes p→∞ for fixed k to get a satura
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Theorem 1.1 states: if phi belongs to \dot B^{1/2}_{2,1} cap \dot B^{9/2}_{2,1} and the initial perturbation has \|(rho_0-rho_*, u_0)\|_{\dot H^{1/2-delta} cap \dot H^3} <= epsilon_0, then (1.5) has a unique global strong solution satisfying (1.8), with \|(rho_0-rho_*, u_0)\|_{L2} arbitrarily large. Theorem 1.2 asserts \|\nabla^k(rho-rho_*, omega)(t)\|_{L2} <= C(1+t)^{-(k-s)/2} for k=0,1 whenever the initial data are finite in \dot B^s_{2,\infty}, s in [-3/2,-1), and Theorem 1.4 gives matching lower bounds under a decay-character condition.
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Under the one-signed decomposition ω_0^ν = μ_0^ν + f_0^ν with μ_0^ν ≥ 0 and f_0^ν bounded in L^p (p>1), plus uniform kinetic energy and total vorticity variation, the author proves Dν(δ,T) = ν∫_δ^T ‖ω^ν(t)‖_2^2 dt ≤ C νT + C log( log(1/(νδ))/log(1/(νT)) ) ≲_{δ,T} 1/|log ν|. Because Dν is exactly the kinetic energy removed by viscosity, this is a quantitative no-anomalous-dissipation statement with an explicit rate. The same estimate disproves a published conjecture predicting that the rate 1/√|log ν| can be achieved by some viscosity-independent datum in this class. When initial velocities are relatively compact in L^2, the bound extends to growing observation times Tν with log Tν = o(|log ν
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The central claim, Theorem 1.1, is that the mixture-theory system is locally strongly well-posed: initial data v_j,0 in H^1, phi_1,0 in H^2 with 0 < phi_1,0 < 1 and the compatibility condition div(phi_1,0 v_1,0 + (1 - phi_1,0) v_2,0) = 0 produce a unique solution on some time interval with v_j in L^2(H^2) intersected with H^1(L^2), phi_1 in L^2(H^3) intersected with H^1(H^1) and H^2(H^{-1}), and p in L^2(H^1_{(0)}). In the authors' terms, the linearized principal operator is invertible with a uniform bound independent of the time horizon, and the remaining nonlinear terms are locally Lipschitz with Lipschitz constants that vanish as the time interval shrinks; the contraction-mapping theorem
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The central claim is that global mass conservation in a closed low-Mach-number system can be enforced exactly and efficiently for any single-phase equation of state by solving the nonlinear constraint M0 = ∫ρ(p0,T)dV at each stage. The paper derives a Newton-Raphson update for p0 whose residual derivative is the volume integral of ρ times isothermal compressibility, a quantity available from analytic equations of state or property tables; in the ideal-gas limit the update reduces to the standard explicit formula. The algorithm separates thermodynamic from hydrodynamic updates: temperature is advanced first, p0 is corrected, density and properties come from the equation of state, and only the
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The central claim is that the hydrostatic Helmholtz projection P_h v = P_H v + \bar v splits off the surface pressure and enforces the hydrostatic divergence constraint, reducing the primitive equations to the quasilinear evolution ∂_t v + A v = F(v), where A = −P_h(Δ_N + Ric) is the hydrostatic Stokes operator on the product manifold N = Σ×(−h,0). The paper proves that A admits a bounded H∞-calculus on L^q_{σ}(N;TΣ), which supplies maximal L_q-regularity and local well-posedness for critical initial data. A long chain of a priori estimates, carried out at p=q=2, controls the H^2 norm of solutions in terms of the initial H^2 norm and time; combined with the smoothing property of the semiflow
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Theorem 1.1 states that, for small initial data in H^s (s ≥ 4), the rescaled perturbation g^ε satisfies (I−P)g^ε → 0 strongly and the hydrodynamic moments converge, up to a subsequence, to (u, ϑ) solving ∂_t u + (1/m_2) u·∇u + ∇p = ν_ℏ Δu, ∇·u = 0, ∂_t ϑ + (1/m_2) u·∇ϑ = κ_ℏ Δϑ, with ϱ = −ϑ. The limiting distribution is the infinitesimal quantum equilibrium associated with the fixed global equilibrium, not the classical Maxwellian. The positive transport coefficients ν_ℏ and κ_ℏ are defined through the microscopic auxiliary equations L eA = A and L eB = B, and the quantum-adapted thermal mode q^ε = (ϑ^ε − βϱ^ε)/(1+β) together with the solenoidal velocity converge strongly in local Sobolev sp
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The central claim is Theorem 3.1 and Theorem 3.2: for the eddy-viscosity system (1.1), under the symmetry (SYM) and the weak-stratification hypothesis ∂_z g, ∂_zz g, ∂_zzz g = O(ε), initial data with bounded energy functional (3.4) yield unique strong solutions on an ε-independent time interval with the uniform bound (3.6); and as ε→0 the slow part of the solution converges to a strong solution of the incompressible primitive equations (1.2). The mechanism is a three-wave decomposition of the linearized dynamics—a slow/mean wave, a fast horizontal acoustic wave at frequency O(1/ε), and a very fast vertical acoustic wave at frequency O(1/ε²)—whose projection operators are non-orthogonal and d
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Theorem 1.1 (Global exact Lagrangian realization): for every p in T^3, T>0, F* in SL(3,R) and ν≥0, every sufficiently large odd integer N admits a real-analytic curl eigenfield W_N with curl W_N = N W_N such that the explicit amplitude u_N = c_N e^{-ν N^{2} t} W_N is an exact unforced Euler (ν=0) or Navier–Stokes (ν>0) solution whose Lagrangian flow satisfies abla_a X_N(p,T)=F*, with the lifted trajectory an embedded analytic arc of nowhere-vanishing velocity. Consequently the set of one-particle deformation gradients generated by this class is exactly SL(3,R).
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For weak solutions of the three-dimensional degenerate compressible Navier-Stokes equations with density-dependent viscosity on a family of expanding domains, the simultaneous inviscid and low-Mach limits yield strong local convergence of density to 1 and of momentum to a smooth incompressible Euler velocity, even from ill-prepared compactly supported initial data, on any time interval short of the Euler lifespan.
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For the incompressible Navier-Stokes system driven by a dual-scale admissible memory kernel, the Cauchy problem is globally well-posed in the Hadamard sense for small divergence-free data in the critical Besov space Ẋ^{-κ}_{∞,∞}(ℝ^N) that satisfy the high-frequency adherence condition lim_{j o+∞} 2^{-jκ}‖Δ_j u_0‖_{L^∞}=0, where κ=(1-α_∞)/(1+α_∞). This regime properly contains the little-Besov closure. At the same time, for every Lebesgue exponent 1<p<p_c with p_c=N(1+α_∞)/(1-α_∞) the data-to-solution map fails to be uniformly continuous at the origin by instantaneous norm inflation of the second Picard iterate.
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The central discovery is that the vectorial LBM defined by (5) with the equilibria of Section 2, under parabolic scaling (4), is formally second-order consistent with the incompressible Navier-Stokes system (1) provided the viscosity parameters satisfy 2μα_qx(1/ω_qx−1/2)=2μα_qy(1/ω_qy−1/2)=ν. The proof uses the moment equations and the pressure ansatz (7) to convert the density equation into a divergence-free constraint at leading order and to identify the momentum equations with the Navier-Stokes momentum balance. Numerical tests confirm second-order L2 convergence for the Taylor-Green vortex (Table 1) and for Poiseuille flow once the boundary conditions are corrected for non-equilibrium re
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If (v, q, a, Ψ, h) is a smooth solution of the Lagrangian fluid-plate system on [0, T) and the combined higher-order norm Y(0) is smaller than a fixed ε > 0, then Y(t) ≤ C Y(0) e^{-t/C} for every t in [0, T). Here Y controls the fluid velocity in H^3 together with its first two time derivatives and the plate displacement in H^4 together with its first three time derivatives. The smallness condition is independent of T, and the initial height may differ from the flat equilibrium.
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Augmenting the Huang–Shen BDF–IMEX consistent splitting scheme with a directional Maday–Kaber–Tadmor spectral vanishing viscosity operator yields a scheme that remains stable and optimally accurate at high Reynolds number: SVV contributes a viscosity-independent coercive term on the high modes in the energy identity, the design temporal orders k=2,3,4 are retained, and the bare scheme’s breakdown at Re=10^4 is eliminated in manufactured, Kovasznay, and Kelvin–Helmholtz tests.
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Sufficiently strong, sufficiently resolved tangential boundary feedback on a nonempty open subset Γ generates a coercive L2 spectral gap for the assimilation error; the limiting gap equals that of the mixed-boundary problem with homogeneous Dirichlet data on Γ and is of order ν. Whenever this gap dominates the long-time averaged symmetric-gradient energy of the reference solution, the assimilated velocity converges exponentially to the reference velocity.
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Among eight surrogates compared on a shared pipeline, no single architecture wins both regimes. On the CMP film a one-shot full-field DeepONet reaches 3.2% relative error on cumulative wall shear stress; on the Kármán wake a latent autoregressive DeepONet retains about 96% of the shedding power that direct and one-shot models collapse to nearly zero. The axis that flips the winner is the treatment of time—autoregressive feedback for the self-sustained limit cycle, a direct map for the boundary-driven Stokes film—while representation only changes the margin. Pointwise RMSE ranks the wrong model in both regimes, so failure-mode-resolved metrics are required; neither the winning architecture no
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For the fully discrete IMEX-BDFk Taylor-Hood scheme with k = 1,...,6 on the 3D incompressible Navier-Stokes equations with no-slip boundaries, the numerical solution is uniformly bounded in the energy norm and the errors satisfy optimal bounds of the form O(h^{l+1} + τ^k) in L2 velocity, O(h^l + τ^k) in H1 velocity, and a matching L2-in-time pressure bound, with the time-step restriction independent of the spatial mesh size.
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The paper's central claim is that the very weak formulation of the Navier-Stokes system with nonhomogeneous Dirichlet boundary data is well-posed on domains of Sobolev-multiplier class M_W^{1+α,ρ}(ε) for sufficiently small ε, with a unique solution in L^s_t L^q_x. The proof reduces the problem to an equivalent integral equation u = U - Q(u,u), built from the Stokes semigroup, and defines the inverse Stokes operator on the rough nonlinear term by duality; the argument closes by a contraction. Local-in-time existence follows from the decay of the semigroup term as t→0, and global-in-time existence from an explicit smallness condition on the data.
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On its own terms, the paper's central claim is Theorem 1.1: for an exterior domain with locally Lipschitz boundary and a prescribed boundary velocity v* in the trace space W^{1/2,2}, if Re C0(Ω)|Φ| < 1 where Φ is the net flux of v* through the body surface, then problem (1.1) has at least one weak solution. Previously the same problem was solvable only under the stronger conditions Φ = 0 and small ||v*||. The engine is Lemma 3.1, which constructs a solenoidal extension ev* of v* satisfying 2 Re |∫(u−V)·W(u)·ev*| ≤ (γ + C(Ω)Re|Φ|)||u||², allowing the nonlinear term to be absorbed. The proof then runs a fixed-point argument on bounded subdomains and passes to the exterior limit. The theorem al
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The central claim is that velocity and vorticity inherit different spatial localization from Gaussian initial data. For d=2,3, a strong vorticity solution keeps a Gaussian bound up to the maximal lifespan. For d≥2, a strong velocity solution with Gaussian-localized L∞ data satisfies an explicit asymptotic expansion: u(x,t) = −Σ_{|α|=0}^{d-1} (−1)^{|α|}/α! ∇∂^α K^{i,j}(x) ∫_0^t M^{i,j}_α(s) ds + R(x,t), with sup_{t∈[0,T]}|R(x,t)| = O(|x|^{−2d−1}) as |x|→∞, uniformly on any compact time interval inside the maximal lifespan. Here K^{i,j}=∂²_{i,j}Γ is the second derivative of the fundamental solution of the Laplacian and M^{i,j}_α are moments of u^i u^j. The expansion holds up to maximal lifespa
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The paper establishes that the classical MWI face flux inherits a spurious dependence on solver parameters because the mobility coefficient b_F is proportional to the inverse of the interpolated diagonal momentum coefficient A_P, which in practical solvers contains 1/omega and 1/Delta-t contributions. By writing the relaxed, time-integrated momentum balance in a divided form and applying the same elimination of the non-pressure term as in the classical derivation, the author derives a modified flux expression in which the pressure-correction mobility is scaled only by the spatial coefficient, while the temporal and relaxation effects reappear as coefficients multiplying stored flux-correctio
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The central claim is Theorem 1.2: if an open C^3 filament evolves by the binormal flow, admits a regular tubular neighborhood, and satisfies a uniform chord-arc condition, then for Γ/ν and νT small enough there is an exact Navier–Stokes solution whose vorticity is the Lamb–Oseen profile ω0 = Γ/(4πνt) e^{−r²/(4νt)} T(s,t) plus a remainder ω̃. The remainder obeys sup_{0<t<T} (νt)^{1/2−3/(2p)}∥ω̃∥_p ≤ C C_F(χ,Γ,νT) for all p in [3/2,∞], and the velocity it induces is bounded uniformly in space. Thus the actual vorticity is a moving Gaussian tube around the binormal-flow filament, with a controlled perturbation that vanishes as t→0 for p<3.
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Theorem 1.2 asserts that for any divergence-free initial datum in L² there exists a weak solution whose final regularity interval begins at θ0, with θ0 in (s, s + π/4[(1−η)(α+β)]^{-1}); here s = 0 when R(α+β)||v0||² ≤ π/2, and s = R/(2γ²) log[4/π (α+β)||v0||²] otherwise. Because α grows like R³ and β like R, the interval's length shrinks like R^{-3} while s grows like R log R, so at large Reynolds numbers the waiting time is dramatically shorter than Leray's bound, which diverges like a high power of R. The estimates are established for the Leray approximating sequence and transferred to the weak solution via the uniqueness of the weak limit.
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The central claim is Theorem 1: for p∈(2,3) and u0∈L^p(Ω), where Ω is an exterior domain, the whole space, or a half-space, there exists a weak solution u to the Navier–Stokes initial-boundary value problem, and this solution satisfies a structure theorem of the classical kind. Specifically, there are a time θ≥0 and a sequence of open intervals (θ_l,T_l) such that the complement of their union together with [θ,∞) in (0,∞) has zero Lebesgue measure, and u is regular on [θ,∞) and on every (θ_l,T_l). The proof splits the datum u0=v0+w0 with v0 small in L^3 and w0 in L^2, solves a globally regular problem for v, and solves for w a perturbed Navier–Stokes problem with extra linear terms. The key
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The paper establishes, for each sufficiently small wavenumber α, a unique eigenvalue curve (c_r(ν), c_i(ν)) for the even modes in the Tollmien–Schlichting region, with neutral points at ν = J_{ν,−}(α)|α|^7 and ν = J_{ν,+}(α)|α|^{11}; at these points the eigenvalue is simple and c_i crosses zero with nonzero speed (∂ν c_i ~ −|α|^{-5} lower, ~ |α|^{-7} upper). The same result holds in the fixed-ν formulation: for each small viscosity there is a unique neutral wavenumber pair with α² ~ ν^{2/7} and α² ~ ν^{2/11}. Because the simplicity and transversal-crossing conditions are verified, the classical Hopf bifurcation framework applies, yielding traveling-wave solutions (ν_s, Φ_s) with ν_s = ν^{[0]
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The central assertion, Theorem 1.4, is that a one-parameter local family of nontrivial periodic travelling-wave solutions bifurcating from the Nusselt profile at a parameter pair (θ*, γ*) can be continued as a continuous curve of solutions, real-analytic away from a discrete set, that is global. The continuation is unconditional except for an explicit dichotomy: as the curve parameter goes to infinity, the quantity measuring conformal-map degeneracy, second derivatives, velocity, stress, and—when surface tension is zero—the reciprocal surface-stagnation margin must blow up, or the curve closes by returning to the Nusselt solution. The key structural insight is to recast the problem, includin
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On its own terms, the discovery is that the corner-vortex cascade in the triangular cavity is a discrete fractal. At Re=1, the solver resolves seven nested counter-rotating eddies whose centers lie on the cavity midline, with successive size ratios of about 0.49 and intensity ratios of about 0.0012, matching the classical corner-vortex predictions. Modeling each eddy's bounding streamline as a semi-ellipse with major axis twice the minor axis, the paper obtains perimeters and areas that halve and quarter from one vortex to the next. Substituting these into the area-perimeter relation D≈2logP/logA gives fractal dimensions 1.229, 1.260, 1.299, 1.354, 1.432, 1.555, and 1.776 for vortices V1 thr
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On its own terms, the paper's central claim is that the posterior formed from a Gaussian likelihood for noisy velocity observations and a maximum-entropy prior constrained by boundedness, smoothness, incompressibility, and momentum balance has a MAP estimate that accurately reconstructs steady velocity and pressure in patient-specific aneurysms and coarctation, and a Laplace covariance that is conservative (roughly 99% of nodal values inside nominal 95% intervals). The authors demonstrate this on synthetic data for three geometries across SNR 2.5–10 and resolutions 0.5–2.5 mm, reporting region-of-interest WSS errors near or below a few percent for the proposed method, substantially lower tha
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On its own terms, the central claim is Theorem 2.5: for the exterior domain Ω=R^3\B and every 1<q<∞, the fluid–structure operator A_{α,q} generates a bounded analytic semigroup on the subspace X_q of states satisfying the no-penetration condition. Together with Theorems 2.3 and 2.6, the paper claims the linearized system (1.7) with Navier-slip boundary conditions (C2)–(C3) has maximal L^q regularity on finite intervals (with constants independent of T for q<3/2), and that its semigroup satisfies sharp L^r–L^q decay estimates for 1<r≤q<∞, including derivative and pressure bounds. The proof route is: self-adjointness and accretivity of the L^2 operator via a coercive bilinear form; a localizat
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The central claim is that the incompressible Navier-Stokes-Fourier limit holds for any kinetic equation satisfying a set of structural assumptions on the linearized collision operator L, the bilinear part Q, and the fully nonlinear remainder R^ε. Specifically, the paper proves that for small initial data, the kinetic solution f^ε decomposes as f_NS + f_kin^ε + f_disp^ε + f_err^ε, where f_NS is the Navier-Stokes-Fourier fluid limit, f_kin^ε decays exponentially in time with rate λ/ε², f_disp^ε vanishes in averaged L^p norms and uniformly away from t=0, and f_err^ε vanishes uniformly in time. The proof uses a sum space X^s = H^s + K^s_ε that captures both hydrodynamic and kinetic regimes, and
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The central claim is Theorem 3.7 and Theorem 3.9. For n≥3, consider the Navier–Stokes system ∂_t u − Δu + ∇π + (u·∇)u = 0 with div u = 0, and an initial velocity u_0 that is divergence-free and belongs to the critical space H^{n/2-1}. Theorem 3.7 asserts that there is T* > 0 and a mild solution u in C([0,T*); H^{n/2-1}) obtained as a fixed point of the integral equation u(t) = e^{tΔ}u_0 − ∫_0^t e^{(t−s)Δ} P ∇·(u⊗u)(s) ds; if the critical norm of u_0 is small, one may take T* = ∞. Theorem 3.9 asserts uniqueness: two mild solutions in C([0,T); H^{n/2-1}) with the same initial data are equal on all of [0,T). The proof uses maximal regularity of the heat semigroup to close the uniqueness argumen
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The central claim is that the bad subset for nonlinear elimination should be selected from a data-driven reconstruction G(F(X)) of the residual rather than from F(X) itself. The reconstruction is obtained by training an extractor on residual snapshots gathered during the stagnation phase. The reconstruction preserves the dominant 'slow' components responsible for Newton stagnation and filters out 'fast' oscillatory components; the selection criterion then becomes robust to the threshold and to the inner-solve tolerance. With local models trained independently per subdomain, the method converges for Re=10,000 cavity flow where the baseline NE preconditioner stagnates, and the selected bad sub
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The central claim is Theorem 1.1: for any fixed friction factor α=(1−κ)/κ>0 there exist ε, ε0∈(0,1/4) and a ν-threshold ν0 such that for all 0<ν≤ν0 and all H^6 initial data satisfying the Robin consistency conditions and ∥v_in∥_{H^6}≤ε ν^{1/3}, the perturbation remains in an O(ν^{1/3}) neighborhood of Couette, the non-zero modes decay like e^{−ε0ν^{1/3}t}, and both inviscid damping and enhanced dissipation hold. In particular, the critical stability exponent is β=1/3 for every fixed finite friction coefficient, exactly as in the free-slip and boundaryless settings, rather than the β=1/2 of the non-slip case.
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The central discovery is that the macroscopic velocity difference b-u (particle bulk velocity minus fluid velocity) and the temperature difference sqrt(2)omega-sqrt(3)theta (particle temperature variable minus fluid temperature) act as genuine damped modes. Even when mu=lambda=kappa=0, the linearized system has no Laplacian terms, yet these modes, together with the microscopic component of the distribution function, produce coercive dissipation for the fluid velocity and temperature. The paper proves this by establishing uniform a priori estimates independent of mu, lambda, kappa, passing to the limit, and then using low-high frequency decomposition to obtain the decay rates. The pure-fluid
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The core claim is Theorem 1: for the ETD-mr-SAV-MS2-L scheme, if the forcing f lies in L∞(0,∞;H), then the discrete enstrophy obeys ||ω^{n+1}||^2 + |q^{n+1}|^2 ≤ e^{-θΣτ}(||ω^1||^2+|q^1|^2) + (1/θ)(||f||^2/(λ1ν)+γ) with θ = min{νλ1,γ} > 0, giving a bound in L∞(0,∞;L2) independent of Reynolds number and step size. The argument tests the vorticity equation against ω and the auxiliary-variable equation against q; the author asserts that the coupling terms cancel, leaving a differential inequality that is then iterated over arbitrarily many steps. The mean-reverting parameter γ is what supplies the extra dissipation when the discrete advection term fails to vanish exactly, and a cumulative dissi
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The central claim is a pair of synchronization theorems. Theorem 5.2: if a unit field m in W^{2,∞} admits a controlled adapted covering and its shear defect is viscously absorbable, then every strong solution v of the nudged system with gain μ ≥ B*/2 satisfies |v(t)−u(t)|² ≤ exp(−(νλ₁/2)(t−t₀))|v(t₀)−u(t₀)|². Theorem 6.2: if the solenoidal kernel K_m = {0}, then for every μ ≥ Λ*C_η, with no upper gain restriction, the same decay holds with rate νλ₁. Type-I coarse observations synchronize under the gain–resolution condition μh² ≲ ν, and Theorem 8.1 upgrades all four L2 results to H1. The proof uses the exact error identity and decomposes the nonlinearity into observable logarithmic terms and
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Theorem 1.2 states that for the regularized kinetic Lamb–Oseen initial data (1.12), the unique solution f^ε of the rescaled Boltzmann equation (1.3) satisfies ||(f^ε−f^ε_LO)(t)||_{X^{1,k}} ≥ C t/ε³ for all t ≤ T_ε = δ ε²/(C⋆|ln ε|⁶). In the core annulus a₀ε ≤ |x| ≤ b₀ε the macroscopic velocity, divergence, density, and temperature obey the pointwise lower bounds (1.15)–(1.17): the tangential velocity gap grows at least like t/ε³, the radial velocity like t²/ε⁵, the divergence like t²/ε⁶, and the density plus temperature like t³/ε⁷. The kinetic evolution therefore does not lock onto the heat-evolved incompressible Navier–Stokes state in the initial layer; instead a genuinely compressible kine
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The central claim is that the cloud of balls is asymptotically invisible: the fluid velocity converges to the incompressible Navier–Stokes solution, and the ball trajectories converge to integral curves of that velocity field, d/dt h_n = u(t,h_n). The critical number N(r) for which this holds is improved from a logarithmic bound to a polynomial bound in 1/r. The proof constructs approximate divergence-free test functions that are rigid on the balls by replacing each ball's center by its cluster projection; nearby centers merge into one projection, so the boundary condition acts as a single rigid body on each cluster.
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The central claim is Theorem 1.3: for d = 2 and ε = o(δ^{1+ι}) for some ι > 0, after subtracting deterministic macroscopic corrections v^{ε,δ} that solve a Navier–Stokes-type system with full quadratic self-interaction, the rescaled fluctuation δ^{-d/2}(u^{ε,δ} − v^{ε,δ} − u) converges in probability in L²(0,T;H^{−β}) for every β > 0 to a Gaussian field z solving dz + [z·∇u + u·∇z + ∇p_z]dt = (1+ν)Δz dt + χ dW·∇u, with enhanced viscosity ν = 1/16‖K‖²_L² and noise intensity χ = (F_{R²}K)(0), where W is a space-time white noise on the divergence-free mean-zero subspace. The companion law of large numbers (Theorem 1.1) identifies the deterministic limit in d = 2,3 as a Navier–Stokes system with
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The paper's central claim is that the triadic convolution structure of the Navier-Stokes nonlinearity, together with a scale-invariant energy flux and a power-law ansatz for Fourier coefficients, imposes a self-consistency condition on the scaling exponent. For local interactions, each triad contributes roughly K^(1-3α) when the coefficients scale as K^(-α); summing over the ~K^3 triads at scale K gives a flux scaling K^(4-3α). Requiring the flux to be independent of K in the inertial range forces α=4/3, which corresponds to the Kolmogorov energy spectrum E(k) ~ k^(-5/3). The author stresses this is a conditional, formal consistency result, not a proof of turbulence.
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The velocity solution u is shown to equal ω + ϑ(q), where αω − µΔω = ϖ (with ϖ the divergence-free rotational part of the force) and αϑ − µΔϑ = −∇q (with q harmonic). Both ω and ϑ(q) are divergence-free because of their Helmholtz-like structure and carefully chosen boundary conditions; the sum satisfies the tangential part of the Dirichlet condition for any q. The harmonic 'solid pressure' q is then chosen to satisfy the boundary equation Aq = (ω − g)·n, where Aq = −ϑ(q)·n is a self-adjoint, positive-definite isomorphism from H^{−1/2}_0(∂Ω) to H^{1/2}_0(∂Ω). With that q, u = ω + ϑ(q) and p = π + q solve the full generalized Stokes problem, and q is unique.
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The central claim is that the GOC-LBM, built on the D2Q9 lattice with raw moments, central moments, or a single relaxation time, solves the 2D Navier-Stokes equations on orthogonally clustered or curved grids. The key technical step is a Chapman-Enskog analysis that identifies deviation terms E3 and E4 in the second-order non-equilibrium moments; these are then eliminated by introducing corrections to the moment equilibria—Eqs. (76)-(78)—so that the recovered viscous stress tensor matches the GOC constitutive relations. The analysis also shows that shear stresses emerge exactly from a non-equilibrium moment without extra corrections, and that the formulation is Galilean invariant to third or
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The central claim is Theorem 1.1: given any initial velocity v0 ∈ H¹(Ω) that is periodic in x and satisfies v0|z=1 = 0, and any forcing K ∈ L²_loc([0,∞);L²(Ω)), the initial-boundary-value problem for the morning glory model has a unique global strong solution. The solution belongs to C([0,T];H¹) ∩ L²(0,T;H²) with ∂t v ∈ L²(0,T;L²) for every finite T. This extends the earlier small-data theory to large data, and the central new step lies in the a priori H¹ estimate: instead of trying to bound ∂zv directly as in the primitive equations, the authors derive an evolution equation for the vertical velocity w and use its energy estimate to control the boundary term that had blocked previous attempt
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Under suitable smallness assumptions on the data, a residual-based a-posteriori estimator constructed from a regularized Taylor-Hood P2P1 discretization is both reliable and efficient: it supplies computable upper and lower bounds relating the estimator to the error between the exact solution of the original Navier-Stokes problem (with L2 Dirichlet data) and its finite-element approximation.
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For the three-dimensional Navier–Stokes–Cahn–Hilliard system with phase-dependent viscosity and mobility, perforated by holes of diameter order ε^α with α>3, the homogenization limit is the original NSCH system when capillary strength λ_ε converges to a positive constant, and a Stokes–Cahn–Hilliard system when λ_ε tends to zero.
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In three space dimensions, sufficiently regular initial data admit no nontrivial homothetic forward self-similar solutions of the incompressible Navier-Stokes equations. In two dimensions the only decaying homothetic solution is the Oseen vortex. The linearized Euler operator about that vortex is stable, yet another homothetic solution carries an unstable approximate eigenvalue.
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For a Lipschitz density patch and L2 divergence-free initial velocity there is at most one immediately strong solution of the 2D inhomogeneous Navier-Stokes system with vacuum, and that velocity obeys the log-Lipschitz bound that forces the flow into L^∞_t C^{1-ε}_x for every ε∈(0,1), so the Hausdorff dimension of the patch boundary stays equal to 1 for all positive times.
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For the penalized hyperviscous system (1.3), arbitrary L2 data give global weak solutions, small H2 data give unique global strong solutions with bounds independent of the penalty parameter ε, and small L1 ∩ H2 data give the optimal heat-equation decay rates ||∇^k u(t)||_2 ≤ C(1+t)^{-3/4-k/2} for k=0,1,2, likewise uniform in ε.
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Under the abstract assumptions of Section 5 (coercivity of the transport noise relative to viscosity, linear growth of the multiplicative coefficients, Carathéodory regularity, and the Landau potential), the stochastic Navier–Stokes–Cahn–Hilliard system possesses at least one global weak martingale solution on any finite time horizon; when the spatial dimension is two the solution is pathwise unique.
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In dimension d>2, for a stationary ergodic δ-hardcore point process whose two-point correlation decays slightly faster than |x|⁻², the infinite-volume mean settling speed exists, is independent of container geometry, and equals the Stokes velocity of a single particle plus Batchelor’s two-particle correction plus a remainder of higher order in the density.
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Any smooth Type-I solution of 3D Navier-Stokes that is rotated self-similar (or rotated discretely self-similar with scaling factor near 1) must vanish identically once the rotation parameter lies outside a compact interval that depends only on the Type-I constant.
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The admissible two-forms that embed the reversible field form an affine space. For any positive-definite metric A the unique minimum-Hilbert–Schmidt gauge is the weighted wedge (A X)^♭ ∧ r / ⟨A X, X⟩; the identity metric recovers the original skew-gradient embedding, while the same least-squares principle produces regularized, residual-correcting and invariant-preserving gauges that still obey the entropy or free-energy law.
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For viscosities of the form μ = a₁ ρ^δ, λ = 2a₁(δ-1)ρ^δ with δ ∈ (13/18,1) and adiabatic exponents γ ∈ (4/3,6δ-3), the free-boundary problem for the spherically symmetric barotropic Navier–Stokes–Poisson system admits a unique global classical solution for arbitrary large initial data that satisfy the physical-vacuum condition. The solution stays smooth up to the free boundary and realises the same vacuum behaviour as the stationary Lane–Emden configuration.
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If a mild solution of the 3D Navier-Stokes equations develops a critical-point singularity (vorticity of order |x|^{-2} in L^{3/2,∞}) while its direction field remains bounded in the space bmo_{1/|log r|}, then the first possible singular time cannot actually be singular. The logarithmic weight forces the stretching eigenvalue to vanish on the super-level sets, improves the distribution function of vorticity, and ultimately drives the geometric sparseness of the velocity below the analyticity radius, contradicting blow-up via the harmonic-measure maximum principle.
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Under a random Lyapunov structure and a generalized asymptotic coupling with controlled Girsanov cost, a Markov cocycle over an ergodic base flow admits a unique stationary family that is exponentially mixing in both pullback and forward time. Under additional tracking, escaping-energy, and tightness assumptions, the small-noise stationary measures satisfy the Freidlin–Wentzell upper large-deviation bound with good pullback-quasipotential rate function; the full LDP holds when the deterministic pullback attractor is a random point.
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The core mechanism is a unified weak formulation built on three interlocking ideas. First, the bulk mesh velocity and the interface tangential velocity are freed from the fluid velocity via an ALE framework, so the mesh can move independently while geometric consistency is maintained. Second, the bending force — a nonlinear fourth-order geometric quantity — is handled through an evolution equation for the mean curvature, where the ALE technique absorbs the tangential convection naturally. Third, skew-symmetric forms for the convective terms in both bulk and surface Navier–Stokes equations ensure that these terms contribute zero net energy in the discrete test, allowing the energy dissipation
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On the paper's own terms, the central claim is that under Hypotheses (SF), (RZ), (LCR), (GD) in finite dimensions — and (SF), (RZ), (DLP), (ALC), (GD) in infinite dimensions — system (1.1) is exponentially mixing: for initial states v,v' in X, ∥D(u_k(v)) − D(u_k(v'))∥_var ≤ C e^{−γk} in finite dimensions, and the dual-Lipschitz analogue in infinite dimensions. Moreover, there is a unique-in-law two-sided process {û_k} extending the dynamics in the sense D(û_k) = S_*(D(û_{k−1}, η̂_k)) with η̂ distributed as η. If the noise is stationary, {û_k} is the unique stationary process and its one-time marginals give the unique stationary measure. The proof works by lifting (1.1) to a Markov process on
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For every Riesz exponent s∈(0,1) and every viscosity ε∈(0,1], sufficiently small initial density and irrotational-velocity perturbations, together with a solenoidal velocity of size O(ε), produce a unique global smooth solution of the compressible Navier-Stokes-Riesz system that decays in L^{2} and L∞ and converges globally in time to the irrotational Euler-Riesz solution at the explicit rate O(ε^λs) in W^{k,p}.
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Under the parameter range 3/2 < p < 3, 1 ≤ σ < ∞ and suitable q, r, θ, any divergence-free initial velocity whose horizontal part satisfies a smallness condition of the form ∥a_h∥ exp(C ∥a_3∥^{r/θ}) ≤ η admits a unique global solution in the corresponding Chemin-Lerner spaces, with the vertical velocity allowed to be large in ḊB^{-1}_{∞,σ}.
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Under the sole hypothesis that a reference solution of the modified Oberbeck–Boussinesq system remains uniformly bounded, any weak martingale solution of the associated stochastically nudged system converges to it in expectation: for every γ>0 there exist a sufficiently large nudging parameter Λ and a sufficiently fine interpolant scale δ such that the expected L2 distance satisfies the exponential-decay-plus-error bound (4.1) on the prediction interval.
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A very weak suitable solution (u,P) of the Navier–Stokes equations on [0,T]×ℝ³ becomes a Leray solution whenever the initial velocity lies in L², the map t ↦ u(t,·) is weakly continuous in L²_loc for t>0 and strongly continuous at t=0, and u belongs to the local Morrey space M^{p,γ}_{t,x} for parameters satisfying 0<γ<3≤p<∞ and γ/p − 3/p + 2/3 <0.
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For the D2Q9 model the second-order equivalent momentum equations contain non-vanishing cubic parasitic terms Sx and Sy. The scheme therefore cannot recover fully Galilean-invariant second-order Navier–Stokes unless those terms are neglected or reduced by a non-standard heat-flux equilibrium; the highest-moment equilibrium is invisible at second order yet controls higher-order fidelity, as demonstrated by the oblique dipole test in which truncating it destroys the coherent structures while a carefully tuned free-rate package best preserves trajectories.
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Numerical maximisers of the instantaneous rate of growth of the Lq norm of velocity, obtained for several q>3 and for norms spanning several orders of magnitude, saturate the power-law upper bound d/dt ||u||_q^q ≤ C ||u||_q^{q(q-1)/(q-3)} as the norm tends to infinity. The bound is therefore sharp up to a numerical prefactor and cannot be fundamentally improved.
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The central claim is a finite-scale coarse-grained decomposition near the CKN local regularity framework: the local resolution lemma bounds the full Psi by resolved and residual parts at any ell, while the depletion theorem supplies a constructive active-work extraction and weighted telescoping inequality for the signed work density G^ell equals Pi^ell plus div(P^ell U^ell) that accounts for all forward combined work and resolved dissipation using only initial energy, leakage, and backscatter.
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The paper's central discovery is that modular nudging is not limited to the backward Euler time discretization used in earlier work. With BDF2 in the forecast step and the analysis step written as v^{n+2} = \tilde v^{n+2} + (2Δtχ/(3+2Δtχ)) I_H(u(t^{n+2}) - \tilde v^{n+2}), the scheme is stable (Theorem 4.2), and under either of two sets of conditions—one with no time-step restriction but a Reynolds-number bound, and one with a small-time-step plus observation-resolution condition—the velocity error is O(Δt²) and remains bounded over time (Corollaries 5.4.2 and 5.5.1). The explicit analysis step holds when I_H is an L² projection with I_H=I_H²; if not, the formula carries a consistency-error
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The central claim is that for any T > 0 and any pair (δ, γ) in a specific open triangle A (or B), there are smooth solenoidal initial data V0 and a force f in L2_loc(R+; L6/5) (respectively L5/4_loc(R+; L2)) such that the Cauchy problem has exactly one global Leray-Hopf solution, this solution satisfies the energy equality, and both the L3 norm of V and the L2 norm of ∇V tend to infinity as t → T; away from the single point (0, T), the solution is smooth. The construction is explicit: the velocity is a time-shrinking bubble (T−t)^{2δ}(2yz,−xz,−xy)/[(T−t)^{2γ}+|ξ|2]^{5/2}, the pressure is obtained by solving −ΔP = ∇·((V·∇)V), and the force is then read off as Vt − ΔV + (V·∇)V + ∇P.
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Under the structural hypotheses, a potential singular point that stays outside the CKN regime at all dyadic scales must either exhibit non-effective moving-window observability or admit an NS-realizable invisible scale-critical defect cascade. When effective observability holds and such cascades are excluded in controlled window classes with dyadic defect extraction, observable depletion, and moving-window growth control, the CKN smallness criterion is satisfied, implying local regularity.
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The central claim is a theorem-driven reduction: under the explicitly listed structural inputs of prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate, one obtains r_reg(0,0) greater than or equal to c sub M,theta times the absolute value of log C_3(1) to the power minus sigma over 3. The comparison is performed in the harmonic-pressure quotient, the Reynolds commutator is treated as positive covariance stress absorbed by an unresolved-variance buffer, and the the
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Under the fixed scale-invariant local bound Phi(1) <= M, smallness of the critical vertical-component quantity C_3(1) = integral over Q_1 of |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The argument converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli-Kohn-Nirenberg smallness at a smaller scale, with the pressure approximation measured in a quotient by spatially harmonic functions. This pressure topology accounts for the obstruction that time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradient
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The author proves global in time regularity of solutions of the Navier-Stokes equations defined in the complex space. This means the solutions exist and stay smooth for every positive time with no singularities forming at finite times.
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The central claim is Theorem 1.3: on a cone-like domain with aperture α≤π/6, under the Navier total-slip boundary condition, if sup r|v_{0,θ}| ≤ C_* and ∫ r v_{0,θ} = 0, then global bounded strong solutions exist for all T>0, with v in L∞_tx ∩ H1_t L2_x ∩ L2_t H2_x and P in L2_t H1_x; the weighted angular momentum ∫ r v_θ is conserved and the energy inequality holds. Uniqueness holds among strong solutions. The genuinely new content is that the total-slip (β=0) boundary is treated directly: unlike the NHL case, boundary terms from integration by parts have bad signs, and the proof absorbs them through the new unknowns K, F, O, a pressure estimate, a De Giorgi argument for L∞ control of Γ=rv_
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On any smooth hypersurface M^n embedded in R^(n+1), the ambient Bochner Laplacian restricted to tangential fields decomposes as the deformation Laplacian Δ_B + Ric plus a radial boundary-shear term built from normal derivatives and the shape operator. The Gauss equation Ric = nH S − S² turns the zero-order extrinsic combination into the intrinsic Ricci tensor. Boundary conditions then decide the normal profile: Navier slip forces ∂_rU = 0, killing the radial term and leaving Δ_B + Ric; the Hodge condition forces ∂_rU = 2SU and ∂²_rU = 6S²U, making the radial term equal −2Ric and leaving Δ_B − Ric. The interpolating condition ∂_rU = 2αSU gives Δ_Def − 2αRic − 4α(1−α)S². All extrinsic quantiti
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We prove that no first threshold time occurs for axisymmetric swirl solutions by converting bounded score and source size into smooth continuation, extracting smaller descendants from every leakage or fragmentation channel, and showing in the remaining coherent case that the strict full-Dirichlet bridge inequality together with coefficient-calibrated local balance contracts the selected packet.
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On its own terms, the paper proves—by direct construction and by side-by-side numerical validation—that isothermal compressible Navier–Stokes is exactly equivalent to a coupled system of amplitude equations. In 2D the closed system is ∂_t Θ = µ_c ∆Θ + V_Θ Θ, ∂_t Ξ = ν ∆Ξ + V_Ξ Ξ, and ∂_t Ψ_α = D_α(∇+A_α)² Ψ_α + V_α^(v) Ψ_α, with Ψ_α = ρ^α Θ^{1−2α} and α ≠ 0, 1/2. The 3D analogue keeps the same structure for the compressive and density-carrying sectors and adds component-wise heat-type equations for the solenoidal vector streamfunction amplitudes. All nonlinearities are encoded in self-consistent scalar and vector potentials built from reconstructed fields, and the original density and veloci
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The paper proves that, under its Fourier spectral discretization, numerical blowup of the 3D Navier-Stokes solution can be rigorously linked to loss of regularity of the true PDE solution. This is achieved through an energy-based conditional regularity argument: if certain energy-type quantities remain bounded, the numerical solution retains regularity; conversely, if the numerical solution diverges, the analytical solution must have lost regularity. The paper also proves exponential convergence in space and algebraic convergence in time for the chosen discretization.
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For the systems of viscoelasticity of Kelvin–Voigt type in one space dimension (Lagrangian coordinates) and for barotropic compressible Navier–Stokes, solutions that retain persistent oscillations can be constructed, and ideas from the kinetic formulation of conservation laws yield effective equations consisting of a kinetic equation coupled to the macroscopic flow.
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There exists a family of weak solutions (u,b) of the incompressible flow with passive tracer such that both ||u(t)||_L^∞ and ||b(t)||_L^∞ blow up as t approaches a finite time T_*, while the solutions remain classical away from T_*. An infinite family of instantaneous blow-up solutions of the two-dimensional MHD system is also obtained, with critical velocity blow-up rate, and the non-uniqueness is sharp relative to the endpoint Ladyzhenskaya–Prodi–Serrin space L^{2}_t L^∞_x.
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In the hyperdissipative class defined by a sharp Fourier-symbol criterion, the Lions exponent α=5/4 remains the critical energy-growth threshold: if α≥5/4 every Hs solution is global, while for every α>1 one has global strong solvability for sufficiently small Hs data; moreover, if a classical Navier-Stokes flow blows up at a first singular time T* in a continuation norm X, the corresponding vanishing-hyperdissipation family cannot remain uniformly bounded in X on any interval approaching T*.
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There exists a small constant ε₀ > 0 such that whenever the initial data (Q₀, u₀) satisfy ‖Q₀‖_{L∞} + ‖u₀‖_{BMO^{-1}} ≤ ε₀, the simplified active system admits a unique local solution (Q, u) belonging to the Banach spaces X_T × Y_T on a positive time interval [0, T], with the solution norm controlled by a multiple of ε₀.
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The divergence-free BDM subspace of degree k on a triangulated surface admits the L2-orthogonal splitting J^k_BDM = rot(S^{k+1}_0) ⊕ H^k_BDM, where dim(H^k_BDM) equals the first Betti number of the surface. Consequently every incompressible flow discretized in that subspace can be reformulated with a scalar streamfunction and finitely many harmonic coefficients as the only unknowns, eliminating pressure while retaining exact tangentiality, pointwise divergence-freeness and pressure-robustness.
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We establish the global existence of weak solutions to the isentropic compressible Navier-Stokes equations in three-dimensional annular cylinders with Navier-slip boundary conditions, allowing large axisymmetric initial data and vacuum states, provided that the bulk viscosity is sufficiently large. We identify a regime in which compressible and incompressible effects coexist. The compressible component interacts with pressure and density to produce an effective dissipation mechanism, while the divergence-free component enjoys improved regularity. This shows that large bulk viscosity strongly suppresses the compressible effect, thereby relaxing restrictions on the size of the initial data. In
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The paper's central claims are stated as Theorem 2.1 and Theorem 2.2. Theorem 2.1 says that any global weak solution of the diffuse-interface system in a bounded three-dimensional domain that satisfies ∇v∈L^q_loc L^r_loc and v∈L^{2q/(q-2)}_loc L^{2r/(r-2)}_loc, q,r≥2, obeys the energy equality E(t)+2∫_0^t∫ν(φ)|Dv|²+∫_0^t∫|∇µ|²=E(0) for all t; the singular double-well potential is allowed, needing no continuity of its second derivative. Theorem 2.2 treats the general case with a non-constant gradient-energy coefficient and non-degenerate mobility: if the initial velocity is L²-small and the initial phase field is H²-close to a local minimizer φ* of the free energy, there is a unique global st
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For every alpha in (1/2,1) and every divergence-free (1-2alpha)-homogeneous initial datum that is locally Lipschitz, the self-similar profile equation admits a weak solution U = U0 + V with V in H^alpha; when alpha exceeds 2/3 every such weak solution is smooth and satisfies the sharp far-field bounds |nabla^k (U - U0)(x)| less than or equal to C (1+|x|)^{1-4alpha} (or with a logarithm when the data are only Lipschitz).
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For initial perturbations in H^{3} sufficiently close to equilibrium, the compressible barotropic Navier–Stokes–Vlasov–Fokker–Planck system with density-dependent friction admits global classical solutions whose regularity bounds are independent of the viscosity coefficient. These bounds imply a global inviscid limit with convergence rate linear in the viscosity, and therefore the first global classical solutions of the compressible Euler–Vlasov–Fokker–Planck system. Under a mild extra assumption on the data, the solutions and their spatial derivatives decay at optimal rates, with dissipative and microscopic components decaying half an order faster than the macroscopic solution.
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For the initial-boundary value problem of the Navier-Stokes-Cahn-Hilliard-heat system with variable coefficients and a singular potential, global weak solutions exist in two and three dimensions; when the densities match and the spatial dimension is two, those weak solutions are unique under suitable assumptions on the initial temperature, mobility and thermal diffusivity.
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The central claim is Theorem 1.1: for every dimension d≥2, every θ>0, and every q,r∈[1,∞], there exists divergence-free initial data u_in ∈ B^{-θ}_{q,r} with arbitrarily small norm such that two mild solutions u and v, both in C([0,T];B^{-θ}_{q,r}), satisfy u(0)=v(0)=u_in but u(t)≠v(t) for every t∈(0,T]. The proof constructs non-trivial 'singular solutions' to the stationary fractional Navier-Stokes equations via convex integration. These are distributions u with zero mean and zero divergence that satisfy the stationary equation with u⊗u defined as a paraproduct in a negative Sobolev space H^{-s}; such a u automatically gives a time-independent mild solution. Starting from a nonzero smooth s
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On its own terms, the paper's discovery is Theorem 2: for a fixed phase field ψ_h whose mobility is bounded below by δ, the generalized trilinear form b(M(ψ_h), v_h, v_h) is coercive with respect to a DG seminorm whenever η_e ≥ max_{p∈{p_{K_e^-}, p_{K_e^+}}} p(p+d−1) and Λ_e(M(ψ_h)) ≥ tilde M_e^{2α}(ψ_h) λ*. Here λ* is the maximum over elements of a ratio of edge mobility averages to the minimum of the mobility inside the element, and α ∈ [0,1/2] interpolates between the existing α=0 fluxes and the new, more diffusive α=1/2 fluxes. From this coercivity the paper derives optimal convergence, structure preservation, and a discrete maximum principle, and it validates these properties numericall
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The central claim is Theorem 3.8: if a stochastic process u satisfies an energy bound E∫_0^T ||u||_Z^ℓ dt<∞, the abstract SPDE is locally well-posed in a critical setting X with Z embedded in its trace space, and u has the strong weak-strong uniqueness property, then whenever the excess Exc_X of Z over critical regularity satisfies 0<Exc_X<1/ℓ, the ε-singular times satisfy dim_M(T^ε_Sin)≤1−ℓExc_X with M^{1−ℓExc_X}(T^ε_Sin)=0, and the singular times satisfy dim_H(T_Sin)≤1−ℓExc_X with H^{1−ℓExc_X}(T_Sin)=0. For quenched strong Leray–Hopf-type solutions of stochastic 3D Navier–Stokes with multiplicative noise, this yields dim_M(T^ε_Sin)≤1/2, M^{1/2}(T^ε_Sin)=0 and the same Hausdorff bounds; und
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The central discovery is that the boundary of uniqueness for Navier–Stokes mild solutions in the critical Besov scale is exactly the line p<n (any q) together with the point p=n, q≤2; beyond that line — p=n with q>2, or p>n with any q — uniqueness collapses. The mechanism is a new multi-scale construction of stationary building blocks: each block is spread over many frequency shells within a dyadic band, with amplitudes decreasing like 1/√ℓ, and the blocks are arranged so that the low-frequency part of the self-interaction cancels the linear diffusion of the previous block. The remaining forcing is smooth enough and small enough to be absorbed by a contractive fixed point in a better-regular
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The central discovery is that the functions q_l(u,v)—the Gaussian conditional expectations of the squared mode amplitudes given energy and enstrophy—are regular, Lipschitz, and satisfy a monotonicity property (Theorem 2.3): the increments (1−μ_i^{-1})q̂_i are non-decreasing in mode index i. This monotonicity forces low-index modes to have conditional expectations of order 1/N, which is exactly what makes the stationary measure of the diffusion condense. The paper then defines an elliptic diffusion on the open cone 0<v<u<λ_N v with coefficients built from these q_l, proves it has a unique stationary law via a Lyapunov-Foster condition, and derives the condensation inequality (5.1) by combinin
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The central claim is Theorem 5.1: as epsilon -> 0, the stationary laws of (|X_t|^2, |X_t|^2_{-1})_{t>=0} under the generator L^eps = L + (1/eps)(B + kappa D) converge weakly to the law P of the stationary diffusion in the open cone C = {(u,v): 0 <= v <= u <= lambda_N v} with generator A in (2.10), regardless of kappa. The proof is an averaging over the fast motion B + kappa D on the level set X_{u,v}: this motion equilibrates exponentially fast away from the singular rays u = lambda_l v, and the 'untamed' set near those rays has uniformly vanishing mass. Theorem 6.1 then yields the quantitative inviscid condensation bound 2 lim E[U_0 - V_0] <= [(B_1 - B_0)/(lambda_{l0}-1) + (lambda_3/(lambda
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The central result is Theorem 1.1 with Corollary 1.1: for every ε∈(0,ε0], under smallness of prepared initial data (1.19), the scaled VFP-CNS system (1.4) admits a unique global solution of the expansion form (1.17), and the pointwise bound (1.23) holds: |fε(t,x,v)-(1+m0(t,x))M(v)| + |uε(t,x)-u0(t,x)| + |ρε(t,x)-(1+h0(t,x))| ≲ ε, uniformly for all times and positions. The leading particle distribution is the Maxwellian with amplitude 1+m0, and the first-order kinetic correction is explicitly (v·u0(1+m0)-v·∇m0)M. The proof reduces convergence to a uniform energy estimate for the remainder system (3.2), closing with the dissipation structure built from macro-micro decomposition.
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The central claim is that the ETD-mr-SAV-MS2o scheme is unconditionally long-time stable: for any variable time-step sequence, the quantity ||u^{n+1}||^2 + |r^{n+1}+1|^2 decays like e^{-θ∑τ} toward a data-dependent constant, where θ = min{νλ1, γ} and the constant involves only the forcing size and γ. This bound is independent of the Reynolds number and of all step sizes, meaning the numerical solution cannot blow up even with arbitrarily large or wildly varying time steps. The proof cancels the nonlinear advection exactly via the skew-symmetry relation ⟨B(u,u),u⟩=0, leaving only viscous dissipation and the mean-reverting damping of the auxiliary variable r.