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Nontrivial weak solutions of the stationary KdV equation in sharp $L^p$ spaces

Mandon Pathak

Abstract

In this paper we utilize a convex integration scheme to construct non-trivial solutions to the stationary KdV equation which lie in $L^p(\mathbb{T})$, $p < 2$. In addition, we demonstrate this result is sharp in the sense that if $u \in L^2(\mathbb{T})$ is a weak solution then $u \in C^\infty(\mathbb{T})$.

Nontrivial weak solutions of the stationary KdV equation in sharp $L^p$ spaces

Abstract

In this paper we utilize a convex integration scheme to construct non-trivial solutions to the stationary KdV equation which lie in , . In addition, we demonstrate this result is sharp in the sense that if is a weak solution then .
Paper Structure (17 sections, 11 theorems, 109 equations)

This paper contains 17 sections, 11 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Main Result
  4. Outline of Paper
  5. Acknowledgments
  6. Background Theory and Technical Lemmas
  7. Convex Integration Scheme
  8. Inductive Proposition
  9. Proof of Proposition \ref{['prop:ind']}
  10. Parameters
  11. Construction of Increment
  12. Proof of Item \ref{['i:1']}
  13. Proof of Item \ref{['i:2']}
  14. Proof of Item \ref{['i:3']}
  15. Proof of Item \ref{['i:4']}
  16. ...and 2 more sections

Key Result

Theorem 1.3

If $u \in L^2(\mathbb T)$ solves eq:stat_eqn then $u \in C^\infty(\mathbb T)$.

Theorems & Definitions (30)

  • Definition 1.1: Paraproducts in $\dot H^s(\mathbb T)$
  • Definition 1.2: Weak paraproduct solutions to \ref{['eq:stat_eqn']}
  • Theorem 1.3: Rigidity Result
  • proof
  • Theorem 1.4: Flexibility Result
  • Remark 1.5
  • Definition 2.1: Littlewood-Paley projectors
  • Lemma 2.2: $L^p$ boundedness of projection operators
  • Definition 2.3: $\dot H^s$ Sobolev spaces
  • Remark 2.4
  • ...and 20 more