{"topic":"navier-stokes","tier":"stated","total":420,"limit":50,"offset":0,"claims":[{"claim_id":2601119,"arxiv_id":"2608.12969","paper_version":1,"claim_text":"Theorem 2.1 states that for α, β, κ, T, μ > 0, η ≥ 0, d ∈ {2,3}, Ω a bounded $C^{{2,ν}}$ domain, an admissible pressure p with growth γ ∈ (1,∞), and initial data ρ_0 ∈ $L^{{γ̃}}$(Ω), (ρu)_0 ∈ $L^{{2γ̃/(γ̃+1)}}$(Ω), c_0 ∈ $W^{{1,2}}$(Ω) satisfying the compatibility conditions (2.16), there is a finite energy weak solution (ρ,u,c) on Ω_T. The continuity equation holds weakly, the momentum equation with artificial pressure p_α(ρ) = p(ρ) + (α/2)ρ² and forcing αρ∇c holds weakly, the parabolic equation β∂_t c − κΔc + α(c−ρ) = 0 holds a.e., and the energy inequality with E[ρ,ρu,c] = ∫ ( |ρu|²/(2ρ) + W(ρ) + (α/2)|ρ−c|² + (κ/2)|∇c|² ) dx holds. The solution has ρ ∈ C_w([0,T];$L^{{γ̃}}$), u ∈ L²(0,T;$W_0^{{1,2}}$), c ∈ C([0,T];$W^{{1,2}}$) ∩ L²(0,T;$W^{{2,2}}$), and ∂_t c ∈ L². No monotonicity of p_α is required, and the result covers all γ>1, improving on the range γ>d/2 known for the compressible Navier–Stokes equations, because the quadratic coupling and capillary terms in the energy supply extra control of the density.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Global-in-time existence of finite energy weak solutions to a relaxed Navier-Stokes-Korteweg model","paper_submitted_at":"2026-08-13T08:49:56+00:00"},{"claim_id":2666708,"arxiv_id":"2608.12815","paper_version":1,"claim_text":"The main discovery is that the cascade structure of the linearized Stokes–heat system makes coupled controllability possible even though the Stokes operator and the Dirichlet Laplacian have incompatible spectral decompositions. Theorem 1.1 states that for each $N\\in\\{2,3\\}$ there exist $T_0\\in(0,1)$ and $C>0$ such that for every $T\\in(0,T_0)$ there is $\\delta_T>0$ with the following property: for every $(y_0,\\theta_0)\\in V\\times H^1_0(\\Omega)$ with $\\|y_0\\|_V+\\|\\theta_0\\|_{H^1_0(\\Omega)}\\le\\delta_T$, controls $v_0$ and $v$ satisfying $v_{N-1}\\equiv0$, $v_N\\equiv0$ drive the controlled Boussinesq system to $y(T)=0$, $\\theta(T)=0$, with cost bounded by $C\\exp(C/T)$ times the size of the initial datum. In dimension two this means no velocity control at all; in dimension three only the first velocity component is controlled. The proof proceeds through a mixed observability inequality for the adjoint system, a Lebeau–Robbiano iteration using only Stokes frequency cut-offs, and a time-iteration argument that transfers the linear estimate to the nonlinear system without changing the exponential order of the cost.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A Lebeau--Robbiano approach to the controllability of the Boussinesq system with a reduced number of controls","paper_submitted_at":"2026-08-13T04:46:15+00:00"},{"claim_id":2703782,"arxiv_id":"2608.11553","paper_version":1,"claim_text":"The paper's central claim is that terminal loss of boundedness is compatible with every standard regularity guarantee short of the sup norm. Concretely, for any $T>0$ and any $1\\le p,q<\\infty$ with $1/p+1/q>1$, Theorem 1.2 provides a force $f\\in L^q(0,T;L^p(D))\\cap L^1(0,T;L^2(D))$ and a classical solution $(v,P)$ of (1)–(2), smooth on $D\\times[0,T)$, such that $\\|v(t)\\|_{L^\\infty(D)}\\to\\infty$ as $t\\uparrow T$ while $v\\in L^\\infty(0,T;L^2(D))\\cap L^2(0,T;H^1(D))$. The solution extends strongly in $L^2$ to the unique Leray–Hopf solution on $[0,T]$ and satisfies the energy equality. The construction gives two-sided rates, e.g. $\\|v(t)\\|_{L^\\infty(D)}\\asymp(T-t)^{-(1-\\alpha)/2}$ and $\\|f(t)\\|_{L^p(D)}\\asymp(T-t)^{-(3-\\alpha)/2+1/p}$ for a parameter $\\alpha$ chosen in (8). It also classifies the mixed norms exactly: within this family $f\\in L^q(0,T;L^p(D))$ holds precisely for $\\alpha>3-2/p-2/q$, and $v\\in L^\\sigma(0,T;L^m(D))$ precisely for $\\alpha>1-2/m-2/\\sigma$.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Pure-swirl loss of boundedness under $L^1_tL^2_x$ forcing: exact mixed-norm ranges","paper_submitted_at":"2026-08-12T01:40:31+00:00"},{"claim_id":2276654,"arxiv_id":"2608.10951","paper_version":1,"claim_text":"The paper's central claim is that, in the SGSRRHR regime with $\\mu < \\mu_c$, small-amplitude solutions of the limit Navier-Stokes system near the critical Taylor number $T_c(\\mu)$ are described at leading order by the amplitude equations (5)-(6), namely two coupled complex Ginzburg-Landau equations for the amplitudes $A$ and $B$ of the two non-axisymmetric critical modes. All coefficients of this system—the linear dispersion terms and the cubic coupling coefficients $b$ and $c$—are obtained from the eigenfunctions $\\zeta_1, \\zeta_2$ and the adjoint eigenfunction via explicit formulas (40)-(41), evaluated by Chebyshev collocation. The reduced system admits helicoidal-wave and ribbon-wave solutions, and their existence and stability are governed by the signs of the real parts $b_r, c_r$, $b_r + c_r$, and $b_r - c_r$, with thresholds at $\\mu_1 \\approx -0.814$, $\\mu_2 \\approx -0.848$, and $\\mu_3 \\approx -0.8$. The paper also reduces the search for stationary spatial modulations to a third-order ODE system (18). If these claims hold, they give a systematic weakly nonlinear description of the non-axisymmetric onset in the small-gap limit.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Couette-Taylor instabilities in the small gap regime: the very counter-rotating case","paper_submitted_at":"2026-08-11T14:20:17+00:00"},{"claim_id":2355086,"arxiv_id":"2608.08841","paper_version":1,"claim_text":"On the paper's own terms, the central claim is that the initial-boundary problem (1.2) is well posed in two regimes: a weak one without any size restriction on the data, and a strong one under the smallness inequality (3.2). The weak solution lies in $L^\\infty(0,T;H)\\cap L^2(0,T;V)$, has time derivative in $L^{4/3}(0,T;V')$, and satisfies the strong energy inequality (3.1); the pressure is recovered as a distribution with the time-regularity split given in Theorem 4.1. The strong solution is unique, belongs to $L^\\infty(0,T;V)\\cap L^2(0,T;H^{3/2}(\\Omega)^3)$, has $v_t$ and $Av$ in $L^2(0,T;\\mathcal{L}^2)$ (where $\\mathcal{L}^2=L^2(\\Omega)^3\\times L^2(\\Gamma_O)^3$), and carries a pressure $p\\in L^2(0,T;W^{1,3/2}(\\Omega))$. The loss of one half of a derivative compared with the Dirichlet case is inherent: an $L^2(\\Gamma_O)$ outlet datum gives $H^{3/2}(\\Omega)$ rather than $H^2(\\Omega)$ interior regularity.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On the evolutionary Navier-Stokes equations in distorted pipes under dynamic and energy-stable outflow boundary conditions","paper_submitted_at":"2026-08-09T18:05:42+00:00"},{"claim_id":2223490,"arxiv_id":"2608.08140","paper_version":1,"claim_text":"Theorem 3.1 is the paper's central claim. For a semidiscrete CutFEM with ghost-penalty stabilization, a Lagrange multiplier on the reference interface, and finite elements of degree $k\\geq 3$, it establishes the error bound $$\\max_{t\\in[0,T]}\\big(\\|u_h-u\\|^2_{$L^{2}$(\\Omega_f[X_H])}+\\|d_H-d\\circ\\Phi_H\\|^2_{$H^{1}$(\\hat\\$\\Omega$^s_H)}+\\|\\partial_t d_H-\\partial_t d\\circ\\Phi_H\\|^2_{$L^{2}$(\\hat\\$\\Omega$^s_H)}\\big)+\\int_0^T\\|u_h-u\\|^2_{$H^{1}$(\\Omega_f[X_H])}\\,dt+$H^{2}$\\int_0^T\\big(\\|p_h-p\\|^2_{$L^{2}$(\\Omega_f[X_H])}+\\|\\hat\\sigma_H-\\hat\\$\\sigma$\\circ\\Phi_H\\|^2_{$H^{{-1/2}}$(\\hat\\Gamma_H)}\\big)\\,dt\\le C($h^{{2k}}$+$H^{{2k}}$),$$ provided the exact solution is smooth and the moving-domain Stokes problem has uniform $H^2\\times H^1$ regularity. The proof treats the fluid equations on the numerically deformed domain, so the discrete interface, its normal, and the pulled-back traction all depend on the solid displacement error; conversely, the solid error is driven by the fluid traction and kinematic mismatch. The authors close this loop by introducing an interpolated configuration, establishing uniform norm equivalences and trace/inverse estimates on the moving cut domains, and using a continuation-in-time argument to upgrade a priori smallness assumptions into unconditional convergence on the whole smoothness interval.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Convergence of a CutFEM for fluid--structure interaction with a deforming interface","paper_submitted_at":"2026-08-08T13:57:28+00:00"},{"claim_id":1920567,"arxiv_id":"2608.06533","paper_version":1,"claim_text":"The paper's central claim is that, for smooth strictly positive Gaussian-tailed initial vorticity, the random batch vortex blob particle system is quantitatively close at the level of laws to the solution of the regularized 2D Navier-Stokes vorticity equation, uniformly on finite time intervals. The normalized relative entropy $H_N(\\tilde F^N_\\varepsilon(t)|\\omega_\\varepsilon(t)^{\\otimes N})$ stays below $C_T(h_0^N+\\varepsilon^{-4}\\tau^2(1+h_0^N)+N^{-1})$. This is achieved by separating the two error sources: the random-batch error is controlled through a locally coupled auxiliary partition, a symmetric law comparison, and Fisher-information dissipation, while the mean-field fluctuation is controlled by exploiting the oddness and divergence-free structure of the Biot-Savart kernel together with score-difference cancellation. The estimate implies $L^1$ propagation of chaos for fixed-particle marginals and, after letting $\\varepsilon\\to 0$, convergence of the particle approximation to the true vorticity solution of the Navier-Stokes equations.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Mean field error estimate of the random batch method for vortex blob dynamics for the 2D Navier--Stokes Equation","paper_submitted_at":"2026-08-06T19:34:54+00:00"},{"claim_id":1734068,"arxiv_id":"2608.06173","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Vanishing viscosity limit to two interacting shocks from the same family for the compressible Navier-Stokes equations","paper_submitted_at":"2026-08-06T15:38:40+00:00"},{"claim_id":1619709,"arxiv_id":"2608.06040","paper_version":1,"claim_text":"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).","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations","paper_submitted_at":"2026-08-06T13:49:30+00:00"},{"claim_id":1799977,"arxiv_id":"2608.05940","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Boundary layer analysis for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part I: Well-posedness","paper_submitted_at":"2026-08-06T12:08:06+00:00"},{"claim_id":1428619,"arxiv_id":"2608.04859","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Extreme flows: where physics meets mathematically rigorous bounds","paper_submitted_at":"2026-08-05T13:51:07+00:00"},{"claim_id":1538368,"arxiv_id":"2608.04621","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"The IBVP for the Navier-Stokes equations in the half-space in a class of weighted Lebesgue spaces","paper_submitted_at":"2026-08-05T09:27:10+00:00"},{"claim_id":1813273,"arxiv_id":"2608.04191","paper_version":1,"claim_text":"The central claim is that the artificial-compressibility relaxation with a single scalar $\\varepsilon$ turns the global divergence constraint into a local residual, at a modelling-error price controlled by $\\varepsilon$ and quantified by the scaling $\\partial_x u + \\partial_y v = -\\varepsilon\\,\\partial_t p$. The paper shows that both divergence and velocity error decrease monotonically and saturate as $\\varepsilon\\to 0$, with no stiffness-induced degradation down to $\\varepsilon=10^{-4}$ on a smooth benchmark. On the $Re=100$ cylinder wake the data-free forward AC-PINN relaxes to the steady symmetric wake, a valid low-residual solution; this failure is not a training artifact but a structural property, so sparse-sensor data assimilation is required to recover shedding. The paper also claims that assimilation fidelity is bounded by the reference fidelity, and that viscosity is identifiable only when sensors are placed in the boundary layer where the viscous term is significant.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"An Artificial-Compressibility Physics-Informed Neural Network for the Unsteady Incompressible Navier--Stokes Equations","paper_submitted_at":"2026-08-04T19:49:53+00:00"},{"claim_id":2512100,"arxiv_id":"2608.04138","paper_version":1,"claim_text":"On the paper's own terms, the discovery is Theorem 2.3: under the full-time endpoint measure $\\mu_*$ of Lemma 2.1, if $\\mu_*(\\{a\\})=m>0$, then one can select times $\\tau_j\\uparrow T_*$, nested balls $B_{j+1}\\Subset B_j\\downarrow\\{a\\}$, orthonormal divergence-free packets $q_j$ supported in $B_j$ with $q_j$ a harmonic gradient in $B_{j+1}$, and a single backward adjoint $A$ solving the adjoint Oseen equation such that $\\langle u(t),A(t)\\rangle=\\sqrt{m}$ for all $t<T_*$, $|A(t)|^2\\,dx \\rightharpoonup^* \\delta_a$, $A(\\tau_j)-q_j\\to 0$, and the late triangle is uniformly saturated: $\\sup_{k>j\\ge J}\\|U(\\tau_k,\\tau_j)^*q_k-q_j\\|_2\\to 0$, with the same uniformity for the forward propagator and both dissipation integrals. The Cauchy–Schwarz saturation identifies $a_\\infty=\\sqrt{m}$ and $d_A=1$, upgrading weak compactness to a unit adjoint. Corollary 2.6 then removes the family from the obstruction: for every sufficiently late root, $\\int_{\\tau_{J+1}}^{T_*}\\|\\Delta U(t,\\tau_J)q_J\\|_2^2\\,dt=\\infty$, so the budget $R_u$ is infinite.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Full-Tail Dynamical Rigidity Forced by Atomic Navier-Stokes Energy Concentration","paper_submitted_at":"2026-08-04T18:46:28+00:00"},{"claim_id":1254948,"arxiv_id":"2608.03672","paper_version":1,"claim_text":"The central discovery is that in the critical regime (particle diameter $\\varepsilon^3$, spacing $\\varepsilon$), the steady Navier-Stokes equations with prescribed flux rate or prescribed pressure drop homogenize to the Brinkman-type system $$-\\$\\Delta$ u + (\\nabla\\times u)\\times u + \\nabla\\Phi + G u - J = f,\\qquad \\operatorname{div} u=0,$$ in the original unperforated admissible pipe domain, with the same mixed boundary conditions on the inlet, outlet, and walls, and with the same flux or pressure-drop prescription (Theorems 2.4 and 2.5). Here $G\\in L^2(\\Omega;\\mathbb{R}^{3\\times 3})$ is the limiting density of Stokes resistance matrices of the particles and $J\\in L^2(\\Omega;\\mathbb{R}^3)$ encodes their prescribed velocities; both arise as limits of $\\varepsilon$-scaled sums over the particles, as in (2.12)-(2.14). The result holds without any smallness condition on the data, and the pressure-drop version needs only the kinetic-energy bound (2.11), while the flux version additionally assumes the inlet and outlet strips stay particle-free (condition (4.3)).","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes","paper_submitted_at":"2026-08-04T13:49:09+00:00"},{"claim_id":1362561,"arxiv_id":"2608.00972","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Exponential decay for the 3D Boussinesq equations with Navier boundary conditions","paper_submitted_at":"2026-08-02T03:52:49+00:00"},{"claim_id":1205258,"arxiv_id":"2608.00843","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A Helmholtz-Leray projection method with variational multiscale stabilization for the Navier-Stokes equations","paper_submitted_at":"2026-08-01T19:50:27+00:00"},{"claim_id":1206058,"arxiv_id":"2608.00689","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Existence of Weak Solutions to a Power-Law Model for Compressible Non-Newtonian Fluids on 1D Unbounded Domain","paper_submitted_at":"2026-08-01T14:29:28+00:00"},{"claim_id":1206950,"arxiv_id":"2608.00465","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Compressible Navier--Stokes equations with a potential force: global well-posedness and optimal time-decay rates for arbitrarily large $L^2$ initial data","paper_submitted_at":"2026-08-01T06:12:26+00:00"},{"claim_id":1127798,"arxiv_id":"2608.00234","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:15.618407+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Delayed Dissipation for Two-Dimensional Vortex Sheets","paper_submitted_at":"2026-07-31T19:29:04+00:00"},{"claim_id":1071163,"arxiv_id":"2607.29298","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Local Well-Posedness for a Diffuse Interface Model for Two-Phase Flows from Mixture Theory","paper_submitted_at":"2026-07-31T11:15:07+00:00"},{"claim_id":1077761,"arxiv_id":"2607.29224","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A global mass-preserving numerical method for low-Mach-number real-gas flows in closed systems","paper_submitted_at":"2026-07-31T09:55:17+00:00"},{"claim_id":870735,"arxiv_id":"2607.27724","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"The primitive equations on curved surfaces","paper_submitted_at":"2026-07-30T06:03:14+00:00"},{"claim_id":881911,"arxiv_id":"2607.27583","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation","paper_submitted_at":"2026-07-30T01:59:51+00:00"},{"claim_id":889837,"arxiv_id":"2607.27477","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Rigorous justification of the hydrostatic-incompressible approximation for weakly stratified isothermal flow","paper_submitted_at":"2026-07-29T21:37:24+00:00"},{"claim_id":835418,"arxiv_id":"2607.26895","paper_version":1,"claim_text":"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 \nabla_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).","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Exact Lagrangian Realization and Robust Strain Sensing in Incompressible Flow","paper_submitted_at":"2026-07-29T13:26:40+00:00"},{"claim_id":842776,"arxiv_id":"2607.25028","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Inviscid Incompressible Limit for Degenerate Compressible Navier-Stokes Equations on Expanding Domains","paper_submitted_at":"2026-07-27T19:40:12+00:00"},{"claim_id":843048,"arxiv_id":"2607.25009","paper_version":1,"claim_text":"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\to+∞} 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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces","paper_submitted_at":"2026-07-27T19:07:16+00:00"},{"claim_id":847466,"arxiv_id":"2607.24635","paper_version":1,"claim_text":"On its own terms, the central claim is Theorem 1.1: for every $0<\\alpha<1$, $2\\le p<\\infty$, and $0<s<1-2\\alpha+3/p$, with $X=W^{s,p}$ or $X=B^s_{p,q}$ and $1\\le q\\le\\infty$, equation (1.1) has the norm-inflation property. Precisely, for every $\\varepsilon>0$ there exists a divergence-free $u_0\\in C_c^\\infty(\\mathbb{R}^3)$ with $\\|u_0\\|_X\\le\\varepsilon$ and a time $0<t_\\varepsilon\\le\\varepsilon$ such that the unique smooth solution exists on $[0,t_\\varepsilon]$ and satisfies $\\|u(t_\\varepsilon)\\|_X\\ge\\varepsilon^{-1}$. The same norm measures the datum and the inflated solution, so the statement is not a cross-space transfer. The paper argues that the vortex-ring mixing mechanism, previously used for the classical Navier–Stokes and Euler equations, survives fractional dissipation across the whole range $0<\\alpha<1$, and that the strict scaling-supercritical gap is precisely the condition that makes the full-space fractional residual negligible relative to the geometric error. Along the way it also proves the Besov version for every fixed summability index $q$ and the Sobolev version by homogeneous interpolation.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Positive-regularity norm inflation for the 3D hypodissipative Navier--Stokes equations","paper_submitted_at":"2026-07-27T16:32:56+00:00"},{"claim_id":857652,"arxiv_id":"2607.24059","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes equations","paper_submitted_at":"2026-07-27T06:59:40+00:00"},{"claim_id":832032,"arxiv_id":"2607.23756","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Exponential Decay of Solutions to a Fluid-Plate Model with Small Initial Data","paper_submitted_at":"2026-07-26T17:04:13+00:00"},{"claim_id":833055,"arxiv_id":"2607.23720","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations","paper_submitted_at":"2026-07-26T15:32:50+00:00"},{"claim_id":833676,"arxiv_id":"2607.23707","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Continuous Data Assimilation for the 2D Navier-Stokes Equations from Partial Tangential Boundary Observations","paper_submitted_at":"2026-07-26T15:06:18+00:00"},{"claim_id":834429,"arxiv_id":"2607.23667","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study","paper_submitted_at":"2026-07-26T14:02:46+00:00"},{"claim_id":835060,"arxiv_id":"2607.23635","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system","paper_submitted_at":"2026-07-26T12:50:26+00:00"},{"claim_id":861365,"arxiv_id":"2607.23234","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On well-posedness theory of very weak solutions to Navier-Stokes equations on irregular domains with nonhomogeneous Dirichlet boundary data","paper_submitted_at":"2026-07-25T14:40:46+00:00"},{"claim_id":879072,"arxiv_id":"2607.22492","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Steady Motion of a Self-Propelled Body in a Viscous Fluid: Dirichlet Boundary Conditions with Nonzero Flux","paper_submitted_at":"2026-07-24T17:03:21+00:00"},{"claim_id":883130,"arxiv_id":"2607.22146","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"High Order Asymptotic Expansion at Infinity for Strong Solutions to Incompressible Navier-Stokes Equations","paper_submitted_at":"2026-07-24T09:45:38+00:00"},{"claim_id":886308,"arxiv_id":"2607.21926","paper_version":1,"claim_text":"The central claim is Theorem 1.1. For initial data satisfying $\\rho_0\\in L^\\gamma\\cap H^1\\cap W^{1,q}$ with $q\\in(3,6)$, $u_0\\in D^1$, and $\\sqrt{\\rho_0}u_0\\in L^2$, let $\\bar\\rho=\\|\\rho_0\\|_{L^\\infty}$. If the quantity\n\\[\\left[\\bar\\$rho^{3}$\\left(\\frac12\\|\\sqrt{\\rho_0}u_0\\|_{$L^{2}$}^2+\\frac{1}{\\gamma-1}\\|P(\\rho_0)\\|_{$L^{1}$}\\right)+\\bar\\$rho^{{\\frac{3-\\gamma}}${2}}\\right]\\left(\\|\\nabla u_0\\|_{$L^{2}$}^2+\\|P(\\rho_0)\\|_{$L^{2}$}^2\\right)\\]\nis sufficiently small, then the initial-boundary value problem (1.1)-(1.4) admits a unique global strong solution in the upper half-space. The density may contain vacuum regions and the initial data may have large oscillations; the smallness condition is independent of the size of the initial data and is invariant under the natural scaling (1.6) of the system. The theorem therefore provides a half-space counterpart of the scaling-invariant global theory for the Cauchy problem and demonstrates that the slip boundary condition is compatible with the scaling.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Global well-posedness of isentropic compressible Navier--Stokes equations with smallness on scaling-invariant quantity in a half-space","paper_submitted_at":"2026-07-24T02:58:22+00:00"},{"claim_id":887474,"arxiv_id":"2607.21736","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On the Removal of Solver-Induced Dependencies in Momentum-Weighted Interpolation for Primal and Continuous-Adjoint Flow Solvers","paper_submitted_at":"2026-07-23T18:35:15+00:00"},{"claim_id":890403,"arxiv_id":"2607.21439","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Evolution of viscous vortex filaments and soliton-type propagation","paper_submitted_at":"2026-07-23T15:47:48+00:00"},{"claim_id":890785,"arxiv_id":"2607.21380","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On Leray's Theoreme de Structure for a weak solution to the Navier-Stokes equations: an improvement of the result","paper_submitted_at":"2026-07-23T14:43:48+00:00"},{"claim_id":891557,"arxiv_id":"2607.21282","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"An $L^p$-theory for global weak solutions to the Navier-Stokes equations in exterior domains","paper_submitted_at":"2026-07-23T13:03:09+00:00"},{"claim_id":891625,"arxiv_id":"2607.21265","paper_version":1,"claim_text":"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]","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Neutral curves and traveling waves in plane Poiseuille flow","paper_submitted_at":"2026-07-23T12:38:33+00:00"},{"claim_id":894072,"arxiv_id":"2607.21142","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Global bifurcation for steady viscous roll waves on an incline","paper_submitted_at":"2026-07-23T10:19:43+00:00"},{"claim_id":895132,"arxiv_id":"2607.20976","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Fractal Scaling of Moffatt Vortices in Triangular Cavity Flow","paper_submitted_at":"2026-07-23T06:54:55+00:00"},{"claim_id":902426,"arxiv_id":"2607.20224","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Bayesian finite element regression for vascular flow reconstruction with quantified uncertainty","paper_submitted_at":"2026-07-22T14:46:26+00:00"},{"claim_id":907725,"arxiv_id":"2607.19736","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Regularity and Decay for Navier-Slip Fluid-Structure Interaction in Exterior Domains","paper_submitted_at":"2026-07-22T04:05:46+00:00"},{"claim_id":915327,"arxiv_id":"2607.18939","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Incompressible Navier-Stokes limit of non-bilinear kinetic equations and application to the BGK, nonlinear Fokker-Planck and Boltzmann-Fermi-Dirac equations","paper_submitted_at":"2026-07-21T10:22:49+00:00"},{"claim_id":916946,"arxiv_id":"2607.18841","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-17T03:36:13.391603+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A lecture on Navier-Stokes equations","paper_submitted_at":"2026-07-21T08:26:46+00:00"}]}