Nonlinear Analysis-Theory Methods & Applications最新文献

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A regularity result for the Fokker–Planck equation with non-smooth drift and diffusion

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-22 DOI: 10.1016/j.na.2025.113748

Paolo Bonicatto , Gennaro Ciampa , Gianluca Crippa

The goal of this paper is to study weak solutions of the Fokker–Planck equation. We first discuss existence and uniqueness of weak solutions in an irregular context, providing a unified treatment of the available literature along with some extensions. Then, we prove a regularity result for distributional solutions under suitable integrability assumptions, relying on a new, simple commutator estimate in the spirit of DiPerna-Lions’ theory of renormalized solutions for the transport equation. Our result is somehow transverse to Theorem 4.3 of Figally (2008): on the diffusion matrix we relax the assumption of Lipschitz regularity in time at the price of assuming Sobolev regularity in space, and we prove the regularity (and hence the uniqueness) of distributional solutions to the Fokker–Planck equation.

引用次数: 0

Measure-valued solutions of scalar hyperbolic conservation laws, Part 2: Uniqueness

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-16 DOI: 10.1016/j.na.2024.113740

Michiel Bertsch , Flavia Smarrazzo , Andrea Terracina , Alberto Tesei

We prove well-posedness of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, where the singular part of the initial data is a finite superposition of Dirac masses and the flux is a continuous function with possible linear growth at infinity. The uniqueness class consists of signed Radon measure-valued entropy solutions, called admissible, whose regular and singular parts satisfy so-called compatibility conditions and suitable continuity requirements with respect to time.

引用次数: 0

Global solutions of the 3D incompressible inhomogeneous viscoelastic system

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-16 DOI: 10.1016/j.na.2025.113747

Chengfei Ai , Yong Wang

In this paper, we prove the global existence of strong solutions for the 3D incompressible inhomogeneous viscoelastic system. We avoid to use the “initial state” assumption and the “div–curl” structure in the proof of global solutions inspired by the works (Zhu,2018; Zhu,2022). It is a key to transform the original system into a suitable dissipative system by introducing a new effective tensor, which is useful to establish a series of energy estimates with appropriate time weights.

引用次数: 0

Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-16 DOI: 10.1016/j.na.2025.113746

Juan Carlos Sampedro

In this paper we use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation

L u = λ u + {| u |}^{p}

, where

L

is a nonlocal pseudodifferential operator defined as a Fourier multiplier and

λ

is the bifurcation parameter. Our general setting includes the fractional Laplacian

L \equiv {(- Δ)}^{s}

and sharpens the results obtained for this operator to date. As a direct application, we establish the existence of traveling waves for general nonlocal dispersive equations for some velocity ranges.

引用次数: 0

Boundary value problems for Choquard equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-13 DOI: 10.1016/j.na.2024.113745

Chiara Bernardini , Annalisa Cesaroni

We consider the following nonlinear Choquard equation

- Δ u + V u = (I_{α} * {| u |}^{p}) {| u |}^{p - 2} u in Ω \subset R^{N},

where

N \geq 2

p \in (1, + \infty)

V (x)

is a continuous radial function such that

{inf}_{x \in Ω} V > 0

and

I_{α} (x)

is the Riesz potential of order

α \in (0, N)

. Assuming Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when

Ω

is an annulus, or an exterior domain of the form

R^{N} ∖ \bar{B_{r} (0)}

. We also provide a nonexistence result: if

p \geq \frac{N + α}{N - 2}

the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting

α \to 0^{+}

we recover existence results for the corresponding local problem with power-type nonlinearity.

引用次数: 0

Mild regularity of weak solutions to the Navier – Stokes equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-09 DOI: 10.1016/j.na.2024.113744

Ira Herbst

We use a well known integral equation to derive some regularity properties of Leray–Hopf weak solutions of the Navier–Stokes equations in

L^{p} (R^{3}), p \in [1, 2)

引用次数: 0

Convergence of the area functional on spaces with lower Ricci bounds and applications

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-06 DOI: 10.1016/j.na.2024.113738

Alessandro Cucinotta

We show that the heat flow provides good approximation properties for the area functional on proper

RCD (K, \infty)

spaces, implying that in this setting the area formula for functions of bounded variation holds and that the area functional coincides with its relaxation. We then obtain partial regularity and uniqueness results for functions whose hypographs are perimeter minimizing. Finally, we consider sequences of

RCD (K, N)

spaces and we show that, thanks to the previously obtained properties, Sobolev minimizers of the area functional in a limit space can be approximated with minimizers along the converging sequence of spaces. Using this last result, we obtain applications on Ricci-limit spaces.

引用次数: 0

Composition of rough singular integral operators on rearrangement invariant Banach type spaces

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-06 DOI: 10.1016/j.na.2024.113742

Jiawei Tan, Qingying Xue

Let

Ω

be a homogeneous function of degree zero that enjoys the vanishing condition on the unit sphere

S^{n - 1} (n \geq 2)

. Let

T_{Ω}

be the convolution singular integral operator with kernel

Ω (x) {| x |}^{- n}

. In this paper, when

Ω \in L^{\infty} (S^{n - 1})

, we consider quantitative weighted bounds of composite operators of

T_{Ω}

on rearrangement invariant Banach function spaces. These spaces contain classical Lorentz spaces and Orlicz spaces as special examples. Weighted boundedness of the composite operators on rearrangement invariant quasi-Banach spaces are also given.

引用次数: 0

A partially overdetermined problem for the p-Laplace equation in a convex cone

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-01 DOI: 10.1016/j.na.2024.113743

Hui Ma , Mingxuan Yang , Jiabin Yin

We consider a partially overdetermined problem for the

p

-Laplace equation in a convex cone

C

intersected with the exterior of a smooth bounded domain

\bar{Ω}

R^{n}

(

n \geq 2

). First, we establish the existence, regularity, and asymptotic behavior of a capacitary potential. Then, based on these properties of the potential, we obtain a rigidity result under the assumption of orthogonal intersection, by using

P

-function, the isoperimetric inequality, and a Heintze–Karcher type inequality in a convex cone.

引用次数: 0

Local second order regularity of solutions to elliptic Orlicz–Laplace equation

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-01 DOI: 10.1016/j.na.2024.113737

Arttu Karppinen , Saara Sarsa

We consider Orlicz–Laplace equation

- div (\frac{φ^{'} (| \nabla u |)}{| \nabla u |} \nabla u) = f

where

φ

is an Orlicz function and either

f = 0

f \in L^{\infty}

. We prove local second order regularity results for the weak solutions

u

of the Orlicz–Laplace equation. More precisely, we show that if

ψ

is another Orlicz function that is close to

φ

in a suitable sense, then

\frac{ψ^{'} (| \nabla u |)}{| \nabla u |} \nabla u \in W_{loc}^{1, 2}

. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.

引用次数: 0

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