arXiv 1806.044622018-06math.AP

Local Boundedness of Weak Solutions to the Diffusive Wave Approximation of the Shallow Water Equations

Thomas Singer, Matias Vestberg

备注：: Corrected typos

英文摘要：

In this paper we prove that weak solutions to the Diffusive Wave Approximation of the Shallow Water equations $$ \partial_t u - \nabla\cdot ((u-z)^α|\nabla u|^{γ-1}\nabla u) = f $$ are locally bounded. Here, $u$ describes the height of the water, $z$ is a given function that represents the land elevation and $f$ is a source term accounting for evaporation, infiltration or rainfall.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Boundedness of the Stationary Solution to the Boltzmann Equation with Spatial Smearing, Diffusive Boundary Conditions, and Lions' Collision Kernel

We investigate the Boltzmann equation with spatial smearing, diffusive boundary conditions, and Lions' collision kernel. Both, the physical as well as the velocity space, are assumed to be bounded. Existence and uniqueness of a stationary solution, which is a probability density, has been demonstrated in [3] under a certain smallness assumption on the collision term. We prove that whenever there is a stationary solution then it is a.e. positively bounded from below and above.

2018-07-29math.AP

Existence of Solutions to a Diffusive Shallow Medium Equation

In this article we establish the existence of weak solutions to the shallow medium equation. We proceed by an approximation argument. First we truncate the coefficients of the equation from above and below. Then we prove convergence of the solutions of the truncated problem to a solution to the original equation.

2020-01-22math.AP

On a cross-diffusive SIS epidemic model with power-like nonlinear incidence

In this work, we study global existence, boundedness and convergence of nonnegative classical solutions of a Neumann initial-boundary value problem for a cross diffusive SIS (susceptible-infected-susceptible) epidemic model with power-like infection mechanism generalizing the standard mass action mechanism. Global existence and boundedness of classical solutions are established in certain parameter ranges, and threshold/non-threshold long-time behaviors of global bounded solutions are also detected. Our findings significantly improve and extend previous related studies.

2022-08-20math.AP

On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states

This paper is concerned with a parabolic-elliptic Keller-Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it Dictyostelium discoideum} cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of sol…

2020-05-09math.AP

Initial value problem for one-dimensional rotating shallow water equations

In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we establish the local existence and the uniqueness of a solution to the hyperbolic system, as well as the global existence of a solution to the regularized system. In order to prove this, we use suitable variables that symmetrize the system.

2021-03-09math.AP

一类时间-空间分数阶扩散 Volterra 方程解的渐近行为

Asymptotic behavior of solutions of a time-space fractional diffusive Volterra equation

针对带齐次 Neumann 边界条件和正初值的时间-空间分数阶扩散 Volterra 方程，证明其解在 R^+ 上有界且一致连续，并分析正解的渐近行为。

2025-02-20math.AP

更多相关论文收起更多论文

Global existence and stabilization in a diffusive predator-prey model with population flux by attractive transition

The diffusive Lotka-Volterra predator-prey model \begin{eqnarray*} \left\{ \begin{array}{rcll} u_t &=& \nabla\cdot \left[ d_1\nabla u + χv^2 \nabla \Big(\dfrac{u}{v}\Big)\right] +u(m_1-u+av), \qquad & x\inΩ, \ t>0, \\ v_t &=& d_2Δv+v(m_2-bu-v), \qquad & x\inΩ, \ t>0, \end{array} \right. \end{eqnarray*} is considered in a bounded domain $Ω\subset\mathbb{R}^n$, $n \in\{2,3\}$, under Neumann boundary condition, where $d_1, d_2, m_1, χ, a, b$ are positive constants and $m_2$ is a real constant. The purpose of this paper is to establish global existence and boundedness of classical solutions in the…

2022-03-26math.AP

具有p增长的非线性动力学扩散方程

Nonlinear Kinetic Diffusion Equations with $p$-Growth

Université Paris Cité（巴黎西岱大学）；Sorbonne Université, CNRS, IMJ-PRG（索邦大学）；Institut für Analysis und Numerik, Universität Münster（明斯特大学分析与数值研究所）；Institut für Angewandte Analysis, Universität Ulm（乌尔姆大学应用分析研究所）

本文研究了具有p增长的非线性动力学扩散方程(sub-)解的局部有界性，关键在于推导了动能Gagliardo-Nirenberg不等式，通过估计函数的Lebesgue范数来控制其运输和扩散方向。

2026-05-18math.AP

Global boundedness of the fully parabolic Keller-Segel system with signal-dependent motilities

This paper establishes the global uniform-in-time boundedness of solutions to the following Keller-Setel system with signal-dependent diffusion and chemotaxis \begin{equation}\left\{ \begin{array}{ll} u_t=\nabla\cdot(γ(v)\nabla u - uϕ(v)\nabla v),\quad & x\in Ω, t>0,\\ v_t = dΔv- v+u,\quad & x\in Ω, t>0 \end{array}\right.\end{equation} in a bounded domain $Ω\subset\mathbb{R}^N(N\leq4)$ with smooth boundary, where the density-dependent motility functions $γ(v)$ and $ϕ(v)$ denote the diffusive and chemotactic coefficients, respectively. The model was originally proposed by Keller and Segel in \c…

2020-08-25math.AP

加权Kolmogorov方程：传递估计、局部有界性与Harnack不等式

Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities

Uppsala University（乌普萨拉大学）

本文研究加权Kolmogorov方程，在一般A2权重下证明传递估计、局部有界性、Harnack不等式及强最小值原理，并涵盖分数幂扩展中的特定权重。

2026-09-23math.AP

一类椭圆型Hamilton-Jacobi方程的解的分类

Classification of solutions of an elliptic Hamilton-Jacobi equation

本研究证明半空间中1<p≤2的零边界扩散型Hamilton-Jacobi方程经典解必为一维，改进了此前需有界假设的结论，结合p>2的已有结果，完善了该方程Dirichlet问题的分类体系。

2025-09-30math.AP

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time $[0, T]$, with $T>0$. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time $H^s$-estimates for the model, established in a previous work, combined with the $L^2$-relative entropy estimate and the interpolation properties of the Sobolev spaces.

2018-07-11math.AP

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg-de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the cons…

2020-11-03math-phmath.APmath.MP

动能Sobolev空间

Kinetic Sobolev Spaces

Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Australian National University-CNRS（澳大利亚国立大学-法国国家科学研究中心）；Westfälische Wilhelms-Universität Münster（明斯特西法利亚威廉大学）

本文定义并研究了适应Kolmogorov方程的同源动能Sobolev空间，考虑了局部和非局部扩散，通过Lebesgue空间L^p和Kolmogorov方程的尺度建立正则性假设，证明了这些空间满足精确的嵌入性质，包括正则性转移、时间连续性和积分性增益。

2026-03-18math.AP

修正 Serre-Green-Naghdi 系统的振荡与正则化激波

Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system

本文提出含人工色散与耗散项的修正 Serre-Green-Naghdi 系统，证明其行波解在参数趋零时收敛到浅水熵激波，以解释色散激波及扩散对涌波的作用。

2024-01-22math.AP

用于矢量不变非线性浅水方程的强稳定双配对分部求和有限差分格式——I：数值格式及平面验证

Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: Numerical scheme and validation on the plane

本文基于双配对SBP算子构建能量/熵稳定的高阶有限差分格式，引入耗散熵与拟涡能的非线性超黏性，并通过理论估计及二维典型算例验证其稳定性、收敛性与平衡性。

2024-10-27cs.NAmath.APmath.NA