All posts under category "Mathematics / Math.AP"
In this paper, we build a procedure that allows to establish regularity and controls in time for probabilistic solutions to PDEs. Probabilistic approaches to global wellposedness problems usually provide ensemble bounds on the solutions. These bounds are the main tools to ensure convergence procedur
We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, Müller, Székelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Székelyhidi on optimal Meyers' regularity theory via this framework.
The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of 'streak-roll-critical layer'. As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form ($U(y,
In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.
The rates at which individuals memorize and forget environmental information strongly influence their movement paths and long-term space use. To understand how these cognitive time scales shape population-level patterns, we propose and analyze a nonlocal population model with a cognitive map. The po
In this paper, we characterize the wave front sets of solutions to fractional Schrödinger equations (ipartial_{t}u =(-Δ)^{θ/2}u + V(x)u) with $0<θ<2$ via the wave packet transform (short-time Fourier transform). We clarify the relationship between the order (θ) of the fractional Laplacian and the
Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, isotropic interaction ranges naturally become anisotropic, leading to the development of a
We study a competitive nonlinear Schrödinger system in $mathbb{R}^N$ whose nonlinear potential is localized in small regions that shrink to isolated points. Within a variational framework based on a fully sign-changing Nehari constraint and Krasnosel'skii genus, we construct, for all $varepsilon>0$,
In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we
In this paper, we study large-time asymptotics for heat and fractional heat equations in two discrete settings: the full lattice (mathbb Z^d) and finite connected subgraphs with Dirichlet boundary condition. These results provide a unified discrete theory of long-time asymptotics for local and nonlo
We study the existence of solutions of the following nonlinear Schrödinger equation $$ -Δu+V(x)u-frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:mathbb{R}^Ntomathbb{R}$ and $f:mathbb{R}^Ntimes mathbb{R}to mathbb{R}$ are periodic with respect to $xinmathbb{R}^N.$ We assume that $V$ has positive essential i
We consider the two-dimensional incompressible Navier-Stokes equations with measure initial vorticity. By means of improved Nash inequalities, we establish quantitative estimates for the enstrophy depending on the absolute vorticity decay on balls. The bounds are optimal in several aspects and yield
The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the Order Completion Method for nonlinear PDEs. In this regard, we consider an appropriate space ML(X) of normal lower semi-continuous functions. The space ML(X)= appears in the ring theory o
A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Non-constant steady-s
The author describes some of his experiences as a mathematician cooperating with diabetes specialists.
We investigate the global Gevrey hypoellipticity of a class of first-order differential operators associated with tube-type involutive structures on $Mtimesmathbb{T}^m$, where $M$ is a non-compact manifold diffeomorphic to the interior of a compact manifold with boundary and $mathbb{T}^m$ is the $m$
We study global minimizers of the Landau-de Gennes energy for the nematic liquid crystals in simply connected, bounded, smooth domains of dimension 3, subject to a weak planar anchoring. The boundary condition is degenerate. In the regime where the elastic constant tends to zero and the temperature
We consider the thin-film equation with linear mobility and a stabilizing second-order porous-medium type term modeling gravity. The model admits self-similar solutions, and our goal is to analyze their stability. We reformulate the problem in mass-Lagrangian coordinates and exploit the underlying g
The spaces D, S and E' over mathbb{R}^(n) are known to be flat modules over A=mathbb{C}[partial_{1},...,partial_{n}], whereas their duals D', S' and E are known to be injective modules over the same ring. Let A be a Noetherian k-algebra (k=mathbb{R} or mathbb{C}). The above observation leads us to s
We outline an extension of paraproduct decompositions for compositions of the form $A(f)$ where $A in C^{d}(mathbb{R}), f in Λ_α([0,1]^d)$ developed in [arXiv:2503.12629] and [arXiv:2508.13322] to settings where $(A in C^1(mathbb{R}),f in Λ_α(X))$ and $ (A in C^2(mathbb{R}),f in Λ_α(X times Y))$. To
In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to mixed-boundary problems in domains of either small volume or n
The proof of cite[Proposition 2.1]{DKS}[arXiv:1104.0232] is partially incorrect. In this short note, we provide a new proof, which requires an additional hypothesis. A modification of this new proof also corrects the proof of cite[Proposition 2.1]{Ch}[arXiv:2310.17456], where the incorrect argument
We study the effective behavior of heterogeneous energies arising in the modeling of material voids in geometrically linear elastic materials. Specifically, we consider functionals featuring bulk terms depending on the symmetrized gradient of the displacement and terms comparable to the surface area
We derived here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. The full (Euler) model for th
We establish uniqueness and structural stability of a class of parallel flows in a 2D straight, infinite channel, under perturbations with either globally or locally bounded Dirichlet integrals. The significant feature of our result is that it does not require any restriction on the size of the flux
The generalized Taylor expansion including a secret auxiliary parameter $h$ which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of $h$ can't be understood in the frame of the homotopy analysis method. This i
In this paper, our primary objective is to establish the time decay properties of solutions $u(t, x)$ with initial data $u_0, u_1 in mathcal{Y}^q$ (pseudo-measure spaces) to the linear damped wave equation in the spaces $L^2left(mathbb{R}^nright)$ and $dot{H}^sleft(mathbb{R}^nright)$ (for $s leq 0$
We consider a semilinear wave equation in the whole space with a deep potential well. We prove that as the depth of the well tends to infinity, the solutions of the equation converge to the solutions of a wave equation defined on the bottom of the well, with Dirichlet condition on the boundary.
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