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Local existence of classical solution to the chemotaxis-shallow water system with vacuum in $\mathbb{R}^2$

Li Chen, Zhen Luo, Yucheng Wang

Abstract

In this paper, we consider the chemotaxis-shallow water system in $\mathbb{R}^2$. We establish the local existence of classical solution without assuming the initial height is small or has a small perturbation near a constant. The far field behavior of the height is a constant which could be either vacuum or non-vacuum. The initial data is allowed vacuum and the spatial measure of the set of vacuum can be arbitrarily large.

Local existence of classical solution to the chemotaxis-shallow water system with vacuum in $\mathbb{R}^2$

Abstract

In this paper, we consider the chemotaxis-shallow water system in . We establish the local existence of classical solution without assuming the initial height is small or has a small perturbation near a constant. The far field behavior of the height is a constant which could be either vacuum or non-vacuum. The initial data is allowed vacuum and the spatial measure of the set of vacuum can be arbitrarily large.
Paper Structure (6 sections, 31 theorems, 253 equations)

This paper contains 6 sections, 31 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local existence of strong solution
  4. Uniform a priori estimates for the linearized problem
  5. The local existence of strong solution
  6. Regularity analysis

Key Result

Theorem 1.1

For $\widetilde{h}\geq 0$ and $\Omega=\mathbb{R}^2$, assume that the initial data $(n_{0}, c_{0}, h_{0}, \mathbf{u}_{0})$ satisfy and the compatibility condition (A3). Moreover, if $\widetilde{h}=0$, in addition to (A3) and (303), suppose that and where Then there exists a small time $T^*>0$ and a unique strong solution $(n, c, h, \mathbf{u})$ to the Cauchy problem (A)-(A2) on $\mathbb{R}^2\ti

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 48 more