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Formation of singularities in solutions to nonlinear hyperbolic systems with general sources

Johannes Bärlin

Abstract

We consider nonlinear hyperbolic systems with a general source and prove that for appropriately chosen smooth initial data the lifespan of the associated $C^1$-solution $u$ cannot be infinite. We employ ideas of F. John (1974) and L. Hörmander (1987) to show that the derivative $u_x$ of $u$ becomes unbounded in finite time.

Formation of singularities in solutions to nonlinear hyperbolic systems with general sources

Abstract

We consider nonlinear hyperbolic systems with a general source and prove that for appropriately chosen smooth initial data the lifespan of the associated -solution cannot be infinite. We employ ideas of F. John (1974) and L. Hörmander (1987) to show that the derivative of becomes unbounded in finite time.
Paper Structure (4 sections, 8 theorems, 99 equations)

This paper contains 4 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Initial data and smallness properties
  4. The gradient blow-up

Key Result

Theorem 1.1

If A1, A2, A3 hold then there exist smooth compactly supported functions $u^0:\mathbb{R} \mapsto \Omega$ such that while remaining bounded, the unique $C^1$-solution $u$ of conservationlawwithsource with data $u(0,\cdot) = u^0$ exists only for finite time.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.3
  • ...and 8 more