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A two spaces extension of Cauchy-Lipschitz Theorem

Charles Bertucci, Pierre Louis Lions

Abstract

We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider $\dot{x} = f(x,x)$ for a function $f: V\times E \to E$ where $E$ is a Banach space and $V \hookrightarrow E$ a normed vector space. This structure allows us to distinguish between the two dependencies of $f$ in $x$ and allows to generalize classical results. We also prove a similar results for partial differential equations.

A two spaces extension of Cauchy-Lipschitz Theorem

Abstract

We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider for a function where is a Banach space and a normed vector space. This structure allows us to distinguish between the two dependencies of in and allows to generalize classical results. We also prove a similar results for partial differential equations.
Paper Structure (8 sections, 5 theorems, 19 equations)

This paper contains 8 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Assumptions and notation
  3. Preliminaries
  4. Main result
  5. The case of partial differential equations
  6. Comments
  7. On the assumptions
  8. Natural extension

Key Result

Lemma 1

Under Hypothesis hyp1, for all $K > R_0$, there exists $t_1 > 0$ depending only on $R_0$ and $K$ such that for ${\left\vert\left\vert\left\vert y \right\vert\right\vert\right\vert}_{\infty,t_1} \leq K$ implies that the unique solution of $x$ of eqxy satisfies ${\left\vert\left\vert\left\vert x \righ

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2