Hirota-Satsuma gains sharp local well-posedness off-diagonal
Existence holds for unequal regularities when dispersion coefficients meet a ratio condition.
· “Sharp local well-posedness for the Hirota-Satsuma system”
math.AP
Analysis of PDEs
Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics
sort pith recommended most recent
Ten-lobed domain breaks Pompeiu and Schiffer conjectures
Certified interval arithmetic proves a noncircular real-analytic domain with a boundary-constant Neumann eigenfunction.
· “A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures”
φ-DeepONet modifies the DeepONet architecture to handle discontinuities by using multiple…
φ-DeepONet learns mappings with discontinuities in inputs and outputs by combining multiple branch networks with a nonlinear interface…
Explicit optimal Hardy weight found for fractional Laplacian
The weight is null-critical and fixes the sharp constant for n^{-2σ}.
· “An optimal fractional Hardy inequality on the discrete half-line”
Infinite family of non-circular domains disproves Schiffer conjecture
A bifurcation branch on a real-symmetry relaxation reaches integer symmetries, refuting Schiffer and Pompeiu together.
Measure-driven nonlocal equation on Heisenberg group has a weak solution
Existence is obtained by weak convergence inside a new function space the authors call the Walker space.
One-period bubble truncation yields Lane-Emden ground states at every mass
The gradient tail drop in the periodic direction restores compactness without lower-order perturbations
· “On Dancer-type solutions for the Lane--Emden equation via semivirial-vanishing geometry”
Global solutions exist for power-rate fragmentation-coagulation in any dimension
Power rates give classical global solutions without data-size limits once fragmentation dominates diffusion and coagulation.
Sobolev deficit equals distance to affine extremals
Near-optimal functions sit, up to linear maps, at a squared distance controlled from both sides by the energy gap.
· “Sharp Stability for the Affine Fractional Sobolev Inequality”
Global weak solutions exist for chemotaxis fluid model of vessel formation
Existence holds on the 3D torus for pressure exponents above 4/3 and finite-energy data, grounding long-time descriptions of early blood-v.
Nitsche method gives optimal errors for Stokes-Poisson-Boltzmann with slip
Proves well-posedness, optimal convergence, and reliable residual-based error estimators for the coupled system.
· “Nitsche method for the Stokes-Poisson-Boltzmann equation with Navier slip boundary condition”
Fractional-log Laplacian eigenvalues mix scaling with log growth
Differentiating the fractional Laplacian in the order parameter produces a new operator whose Dirichlet spectrum obeys a Weyl law with an n/
· “The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues”
Couette flow stable for compressible NS up to Re^{-1} perturbations
Global solutions exist and remain close to base flow for initial data within ε Re^{-1} in H^4 when ε is small and Re-independent.
Sharp Nitsche bound proved for all higher dimensions
Every harmonic homeomorphism A(r,1)→A(R,1) in n≥3 has R≤nr/(n−1+r^n), and equality is radial up to rotation.
· “The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity”
Riesz transform weak-type bound is 2 in every dimension
A fractional obstacle decomposition plus a Lewy-Stampacchia estimate settles a 1986 problem from the International Congress of…
· “A dimension-free weak-type (1,1) bound for the vector Riesz transform on mathbb{R}^n”
Balls minimize the first Neumann–Poincaré eigenvalue in all dimensions
The three-dimensional case settles the 1/6 conjecture for the Neumann–Poincaré operator.
· “A sharp isoperimetric inequality for the Neumann--Poincar\'e operator in every dimension”
Boundary mass becomes a Dirac peak as diffusivity vanishes
Rigorous rate proves left-moving swimmers leave a wall layer that feeds a new boundary ODE.
· “Boundary layers and vanishing diffusivity in run-and-tumble models”
Fractional ground states are strictly superharmonic on every open set
A new proof covers every exponent s in (0,1) and yields the quantitative bound −Δu > λ^{1/s}u pointwise in the domain.
Complete spectrum computed for exponential biharmonic Choquard bubbles
Morse index one, five kernel modes, uniform coercivity, and an explicit sharp stability constant.
Stirring random flows stop blow-up in reaction–diffusion systems
Tuned transport noise yields global classical solutions with probability as close to 1 as desired.
Asymptotic stability proved for the degree-one vortex
Small equivariant perturbations radiate away; the internal mode drains on the epsilon^-2 time scale and the vortex returns to rest.
· “Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model”
Exact heat-control cost rate: 0.6966 L^2/T
Closes the gap between the 1/2 lower bound and the series upper bound for 1D fast controls.
· “Optimal cost of fast boundary controls for the one-dimensional heat equation”
Three shears make every magnetic field grow exponentially
A time-periodic divergence-free velocity on the 3-torus is proved to be a universal ideal dynamo at large shear amplitude.
· “Exponential growth and decay in the ideal induction equation”
Square-wave Benjamin–Ono revivals reduce to finite log-jump sums
Up to a continuous remainder, square-wave revivals are log and jump kernels; most irrational times stay continuous.
C∞ harmonic functions can vanish flat on large boundary sets
A nowhere-dense boundary set of positive measure can carry the entire jet of a nonzero harmonic function.
· “Smooth Failure of Boundary Unique Continuation for Harmonic Functions”
One integral decides blow-up for damped wave equations
Shifted Glassey range and loss-free lifespan bounds now hold even in the mass-dominant regime.
This paper gives precise conditions under which the deterministic probability-flow ODE…
The probability-flow ODE yields a unique regular Lagrangian flow transporting p0 onto the diffusion marginals under Sobolev/BV score…
· “Mimicking diffusion processes with differential equations”
One Lipschitz flow turns the flat torus into a fast dynamo
For every small diffusivity, one fixed velocity field produces an exponentially growing magnetic mode at a uniform rate.
Pathwise solutions for stochastic wave equations with white noise
A forty-year open problem resolved: constructing sample-by-sample solutions to nonlinear wave equations forced by multiplicative white-in-2.
· “Fourier restriction norm method adapted to controlled paths: stochastic wave equations”
KdV cnoidal waves stay stable under localized hits
First nonlinear proof for multi-mode periodic waves in a Hamiltonian system, via space-time modulation
Shifted variables close uniform chaos propagation for second-order CBO
Translation-mode separation yields integrable coupling and Monte Carlo rates that hold for all time.
· “Uniform-in-time propagation of chaos for Second-Order Consensus-Based Optimization”
Analyticity sets exact uniqueness threshold in Calderón problems
Uniqueness holds for analytic metrics but fails densely in every non-analytic Gevrey class for both fixed-potential and fixed-frequency data
· “A Sharp Regularity Threshold for Uniqueness in Riemannian Calder\'on-type Problems”
Time-periodic weak solutions exist for nonlinear fluid-plate system
Nonlinear Koiter energy forces single Leray-Schauder argument on the coupled Galerkin system in periodic channels.
· “Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates”
Naked singularities stable under small Hölder perturbations
Result for Christodoulou's family shows that weak cosmic censorship depends on the regularity class chosen for perturbations
L^{3,1} endpoint is sharp for axisymmetric Euler well-posedness
Multi-ring data with ω0/r in L^{3,q} for any q>1 produce L^∞ vorticity inflation, showing the space is necessary.
· “Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces”
Exact blowup profiles exist for 3D Euler below critical regularity
For every alpha less than 1/3, self-similar C^alpha vorticity profiles are constructed and drive finite-time singularity from compactly held
Kuramoto game equilibria are unique above critical strength
The synchronized branch is the only smooth family emerging from the uniform state and converges smoothly back to it.
· “Uniqueness of synchronized stationary equilibria in the Kuramoto mean field game”
Invasion-wave speed falls strictly as diffusion ratio rises
A long-observed numerical monotonicity in the Lotka-Volterra strong-competition model now has an analytic proof
Positive and negative Tzitzéica solutions found on lattice graphs
A discrete proof yields exponentially decaying solutions of both signs for the full two-term vortex equation on Z^n.
· “Sign-preserving solutions to the Tzitz\'eica equation on lattice graphs”
Hamiltonian stationary graphs: singular set has dimension at most n-5
Every weak solution is smooth off a small singular set, and a new 5-D cone shows the bound is optimal.
· “Sharp partial regularity of Hamiltonian stationary and special Lagrangian graphs”
Balls minimize a key divergence constant
The paper proves every Lipschitz domain has this constant at least n, with balls attaining it—in all orders.
· “A unified approach to the divergence equation and related functional inequalities”
Non-shear smooth states exist where analytic states must shear
The same far-field shear profile allows smooth flexibility and analytic rigidity, and the dichotomy is sharp.
Vortex surfaces as coordinates tame slender-vortex motion
The framework recovers the classic ring-speed formula with axial flow and solves for the deformed vorticity surfaces.
· “Vortex filament dynamics and vortex ring motion revisited”
A Liouville theorem produces Evans-Krylov regularity by blowup
For the complex Monge-Ampère equation, the method works even where classical C^3 estimates fail, on cylinders and cones.
Aggregation-diffusion flows smooth exactly like porous media
One sharp L-infinity estimate buys uniqueness for every finite-mass initial datum, even measures.
· “Smoothing effect and uniqueness for aggregation diffusion models”
On intervals, sign-changing normalized waves are unique
Graph topology decides whether the same mass supports two least-energy sign-changing states.
Stochastic Allen-Cahn equation admits a global stochastic flow
One null set works for all initial conditions, enabling derivatives and random-dynamical-system analysis.
· “A Stochastic Flow for the Stochastic Allen-Cahn Equation with Multiplicative Noise”
One shifted Jacobi method beats Bernstein on all three test problems
The same algorithm yields Legendre and four Chebyshev methods, each more accurate than the published baseline.
Isabelle machine-checks conductivity uniqueness theorem
A week-long LLM-assisted experiment formalizes a piecewise-polynomial inverse problem, with literature results assumed as interfaces.
· “Relative formalization in Isabelle/HOL of a result in inverse problems”
Burkholder function is quasiconvex on symmetric matrices
Sharp weak-type and Lp bounds for the Beurling–Ahlfors transform follow from one integral inequality.
· “Quasiconvexity of the Burkholder function on symmetric matrices”
Charged collapse to an extremal black hole needs |e|r+ > 1/3
A new bound pins down when charged scalar collapse can reach extremality; subextremal collapse carries no such limit.
Instantons in a double well follow a Poisson distribution
On an exponentially long scale, passages between wells are Poisson(mean N), giving the splitting.
Fluid–beam response is real-analytic in the applied force
Near zero force the coupled free-boundary solution exists, is unique, and varies analytically with the data.
A growth indicator predicts every string of resonances
For Schrödinger operators with jump singularities, derivative jumps of the new indicator locate all resonance strings.
Big regular graph patches characterize parabolic uniform rectifiability
Every uniformly rectifiable space-time set contains large smooth graph patches at all scales, and conversely.
Dini-beta finiteness governs caloric measure absolute continuity
For parabolic Lipschitz graphs, a finite square Dini-beta integral is exactly absolute continuity between surface and caloric measure.
· “The parabolic Dini-β condition and absolute continuity of surface and caloric measure”
A symplectic-Hessian bound forces entire solutions to be quadratic
Pointwise curvature estimate extends Bernstein rigidity and sharpens the planar constant to 3.
No convexity needed: Hessian regularity in all dimensions
The proof removes convexity and dimension restrictions for the positive branch, where the Laplacian is positive.
· “Interior Hessian estimates for the quadratic Hessian equation”
Four-wave inflow pattern proven stable even with a strong rarefaction
Rarefaction strength need not be small, resolving the open shock-stability problem for the inflow case.
Triangular symbols pick out holomorphic toroidal operators
For holomorphic Besov and Triebel–Lizorkin spaces, the critical loss is d(1−ρ)|1/p−1/2| and is sharp.
· “Holomorphic Toroidal Pseudodifferential Operators on the Polydisk”
Proven: energy scaling laws for thin-sheet defects
First finite-thickness bounds for disclinations and dislocations, tight in thickness and matching physics conjectures.
· “Energy scaling laws for thin elastic sheets with topological defects”
Nonlocal energy rules out stable solitary waves
The recursion-generated second flow of the modified Burgers' equation cannot balance kink or pulse profiles.
Each boundary T-curvature selects one conformal ball metric
Normalize the Q-curvature and the boundary T-curvature fixes the conformal metric up to conformal motion.
Explicit counterexamples break n-Laplace regularity hopes
Bounded discontinuous solutions with antisymmetric L^n potentials show the Lorentz-space assumption is essential.
As K→0, a Robin transmission gap closes at rate √K
The limit is a Cahn-Hilliard system with an Allen-Cahn dynamic boundary condition.
Noise-free boundary data locate hidden nonlinear anomalies
Monotonicity plus the p-Laplace limit reconstructs outer support or convex hull, even without uniform material contrast.
Stretch-based elastic energy models often lack the mathematical structure needed to prove…
A Cosserat lifting with rotational curvature yields existence of minimizers for stretch-based nonlinear elastic energies, with polar…
· “Polyconvexity for Cosserat nonlinear elasticity and nonlinear couple-stress theory”
Weak solutions exist for all generalized SQG equations
Extends the known existence theory for standard SQG to every β in (0,2), with rough data in a negative Sobolev space.
· “Existence of weak solutions to the generalized SQG equations in Sobolev spaces”
Binary gas shock waves exist for soft potentials
The proof covers unequal masses and all cutoff exponents from γ=-3 to γ=1.
· “Shock profiles for the cutoff Boltzmann equation of a binary gas mixture”
This paper computes the sharp constant and explicitly identifies the optimizers for a…
Under explicit integrability conditions on homogeneous weights, weighted Carlson-Levin inequalities on cones have sharp constants and…
· “From weighted Carlson-Levin inequalities to weighted Gagliardo-Nirenberg inequalities”
p-form scattering collapses to two boundary phases
Co-closed and closed forms each scatter by a phase fixed by the boundary Laplacian, plus a smoothing term.
· “The scattering matrix for the p-form Laplacian on asymptotically conic manifolds”
This paper derives exact conditions for when tiny density ripples in a cell-adhesion…
A weakly nonlinear analysis yields an explicit Landau coefficient depending on mean density, nonlinear exponent, and the kernel's…
This paper proves a critical case of orthonormal Strichartz estimates
For orthonormal systems in Sobolev spaces, the free Schrödinger propagator satisfies the critical orthonormal Strichartz estimate for q=4…
This paper proves a priori interior curvature estimates and smoothness for solutions of…
A priori interior curvature estimates and C^1-to-smooth regularity hold for the scalar curvature equation in dimension 4.
· “Interior estimates and regularity for the scalar curvature equation in dimension 4”
This paper analyzes the one-dimensional quasilinear MMT equation with random initial data…
For the quasilinear MMT equation with random data, the paper proves long-time smooth solutions and derives the first-order wave kinetic…
This paper proves quantitative weak flocking rates for the kinetic Cucker-Smale model…
The kinetic Cucker-Smale equation with nonconvex confinement exhibits weak flocking with optimal algebraic decay for polynomial-tail data…
This mathematics paper proves a sharp unique continuation result at infinity for the…
Any weak solution to ∂̄u=Vu on an exterior domain with bounded V that decays like e^{-C|z|} for C>2||V||∞ must vanish, and this threshold…
Nutrient ring width alone explains constant colony expansion
A moving-boundary model derives the two observed kinetic laws: constant radial speed and vertical slowdown.
· “Kinetics of an Expanding Bacterial Colony: Continuum Modeling and Analysis”
Forests make graph Monge–Ampère label spreading unique and computable
On tree-like graphs it has one solution and a provably convergent fixed-point solver.
Critical Sobolev space yields unique Boltzmann solutions
Local well-posedness, instant smoothing, and global existence follow for very soft potentials.
· “Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces”