Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the blowup of solutions for a nonlocal multi-dimensional transport equation

Wanwan Zhang

Abstract

In this paper, we revisit the problem of finite-time blowup for a multi-dimensional nonlocal transport equation studied in [Dong, Adv. Math. 264 (2014) 747-761]. Inspired by a one-dimensional analogous model considered in [Li-Rodrigo, Adv. Math. 374 (2020) 1-26], we establish a new weighted nonlinear inequality implying the blow-up by a completely real variable based technique.

On the blowup of solutions for a nonlocal multi-dimensional transport equation

Abstract

In this paper, we revisit the problem of finite-time blowup for a multi-dimensional nonlocal transport equation studied in [Dong, Adv. Math. 264 (2014) 747-761]. Inspired by a one-dimensional analogous model considered in [Li-Rodrigo, Adv. Math. 374 (2020) 1-26], we establish a new weighted nonlinear inequality implying the blow-up by a completely real variable based technique.
Paper Structure (4 sections, 8 theorems, 108 equations)

This paper contains 4 sections, 8 theorems, 108 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Two weighted inequalities
  4. Proof of Theorem \ref{['the-1']}

Key Result

Theorem 1.1

Let the initial data $\theta_0$ be a radial Schwartz function. There exists a constant $A(n,\alpha)>0$ depending only on $n$ and $\alpha$ such that if then the smooth solution $\theta$ to M-CCF-T blows up in finite time.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Proposition 3.1
  • Remark 3.1
  • Proposition 3.2