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On decay of solutions to some perturbations of the Korteweg-de Vries equation

Alexander Munoz Garcia

TL;DR

The paper studies dissipative perturbations of the KdV equation on weighted and asymmetrically weighted Sobolev spaces, connecting decay properties to Sobolev regularity within a Kato-type well-posedness framework. It develops a comprehensive analytic toolbox—including Kato–Ponce type product estimates, linear smoothing for the dissipative group, and energy-type integration formulas—to obtain local well-posedness in $H^s(R)$ and in weighted spaces $Z_{s,r}$, along with a persistence result. A key contribution is the identification of an optimal decay–regularity relation $r\le s/(2k)$ for a subfamily with even leading power $p=2k$, showing stronger decay yields greater regularity. The results extend to asymmetrically weighted spaces $H^s(R) cap L^2(e^{2rx}dx)$ under suitable forms of $\Phi_1$, establishing local well-posedness with exponential decay control and highlighting the broader applicability to dissipative dispersive models.

Abstract

This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces.

On decay of solutions to some perturbations of the Korteweg-de Vries equation

TL;DR

The paper studies dissipative perturbations of the KdV equation on weighted and asymmetrically weighted Sobolev spaces, connecting decay properties to Sobolev regularity within a Kato-type well-posedness framework. It develops a comprehensive analytic toolbox—including Kato–Ponce type product estimates, linear smoothing for the dissipative group, and energy-type integration formulas—to obtain local well-posedness in and in weighted spaces , along with a persistence result. A key contribution is the identification of an optimal decay–regularity relation for a subfamily with even leading power , showing stronger decay yields greater regularity. The results extend to asymmetrically weighted spaces under suitable forms of , establishing local well-posedness with exponential decay control and highlighting the broader applicability to dissipative dispersive models.

Abstract

This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces.
Paper Structure (10 sections, 20 theorems, 138 equations)

This paper contains 10 sections, 20 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Notation
  5. Preliminary estimates
  6. Linear estimates
  7. Integration formulae
  8. Local theory results
  9. Regularity vs. decay
  10. Asymmetrically weighted Sobolev spaces

Key Result

Theorem A

Let $\eta>0$ and $p>3$. For any data $u_0\in H^s(\mathbb{R})$ with $s>-p/2$ there exists a time $T(\|u_0\|_{H^s})$ and a unique solution $v$ to the IVP P in $C([0,T];H^s(\mathbb{R}))$. Moreover, the map $v_0\mapsto v$ is smooth from $H^s(\mathbb{R})$ to $C([0,T];H^s(\mathbb{R}))\cap \cdots$.

Theorems & Definitions (36)

  • Theorem A
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 26 more