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A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$

Mihaela Ifrim, Thierry Laurens

TL;DR

This work establishes local-in-time a-priori bounds for generalized Korteweg-de Vries equations in $H^{-1}(\\mathbb{R})$, including non-integrable perturbations and models with uneven bottom. The authors develop a bootstrap framework based on the renormalized perturbation determinant $\\alpha(\\kappa,u)$ together with a local smoothing norm $\\mathrm{LS}_{\\kappa}$, leveraging a density-flux structure that is nearly conserved under perturbations. The main results are a local smoothing estimate and a coupled a-priori energy bound with an explicit lifespan $T_0\gtrsim(1+A)^{-4}$, which together yield robust control of the solution in the critical $H^{-1}$ regime. These results extend sharp well-posedness insights from the integrable KdV to a broader class of gKdV equations and are applicable to long-wave models in channels with variable bottoms, including corollaries for the KdV with bottom variation.

Abstract

We prove local-in-time a-priori estimates in $H^{-1}(\mathbb{R})$ for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of the sharp well-posedness theory for KdV. In particular, we show that our analysis applies to models for long waves in a shallow channel of water with an uneven bottom. The proof of our main result is based upon a bootstrap argument for the renormalized perturbation determinant coupled with a local smoothing norm.

A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$

TL;DR

This work establishes local-in-time a-priori bounds for generalized Korteweg-de Vries equations in , including non-integrable perturbations and models with uneven bottom. The authors develop a bootstrap framework based on the renormalized perturbation determinant together with a local smoothing norm , leveraging a density-flux structure that is nearly conserved under perturbations. The main results are a local smoothing estimate and a coupled a-priori energy bound with an explicit lifespan , which together yield robust control of the solution in the critical regime. These results extend sharp well-posedness insights from the integrable KdV to a broader class of gKdV equations and are applicable to long-wave models in channels with variable bottoms, including corollaries for the KdV with bottom variation.

Abstract

We prove local-in-time a-priori estimates in for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of the sharp well-posedness theory for KdV. In particular, we show that our analysis applies to models for long waves in a shallow channel of water with an uneven bottom. The proof of our main result is based upon a bootstrap argument for the renormalized perturbation determinant coupled with a local smoothing norm.

Paper Structure

This paper contains 6 sections, 25 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. The local smoothing norm
  5. The local smoothing estimate
  6. The a-priori estimates

Key Result

Theorem 1.1

There exists $\epsilon >0$ so that, if the coefficients are given smooth functions that satisfy the decay bounds uniformly for $|t|\leq T$ and $x\in\mathbb{R}$: then for any $A>0$ there exist constants $C,T > 0$ so that for any smooth solution $u$ to gKdV in $[-T,T]$ whose initial data satisfies satisfies the uniform bound

Theorems & Definitions (45)

  • Theorem 1.1: A-priori estimate
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5: Local smoothing estimate
  • Lemma 2.1: Killip2018
  • Proposition 2.2: Diagonal Green's function Killip2019
  • Proposition 2.3: Introducing $\alpha$ Killip2019
  • Lemma 2.4
  • Lemma 2.5
  • ...and 35 more