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A geometric proof of the Quasi-linearity of the water-waves system

Ayman Rimah Said

TL;DR

The paper proves that dispersive perturbations of Burgers-type equations with a nonlocal term $\partial_x|D|^{\alpha-1}u$ (for $0\le\alpha<2$) induce quasi-linearity: the flow map on bounded $H^s$-balls is not uniformly continuous and cannot be $C^1$ into $H^{s-1+(\alpha-1)^++\varepsilon}$. This result holds in any dimension and extends to a broad class of nonlinear transport-dispersive equations, including the Whitham equation and paralinearized water-waves systems, with optimality statements at $\alpha=2$ on the torus via the Benjamin-Ono case. The second part uses a geometric transport/paradifferential framework to deduce the quasi-linearity of both gravity-capillary and gravity water-waves equations, showing that the nonlinear transport dominates the evolution and precludes Picard-type solvability. The approach hinges on crafting high-frequency geometric Ansatz, paracomposition/pseudo-differential changes of variables, and energy estimates for pulled-back equations, yielding robust instability of the flow map and informing the regularity thresholds for well-posedness in water-wave models.

Abstract

In the first part of this paper we prove that the flow associated to the Burgers equation with a non local term of the form $\partial_x |D|^{α-1} u$ fails to be uniformly continuous from bounded sets of $H^s({\mathbb D})$ to $C^0([0,T],H^s({\mathbb D}))$ for $T>0$, $s>\frac{1}{2}+2$, $0\leq α<2$, ${\mathbb D}={\mathbb R} \ \text{or} \ {\mathbb T} $. Furthermore we show that the flow cannot be $C^1$ from bounded sets of $H^s({\mathbb D})$ to $C^0([0,T],H^{s-1+(α-1)^+ +ε}({\mathbb D}))$ for $ε>0$. We generalize this result to a large class of nonlinear transport-dispersive equations in any dimension, that in particular contains the Whitham equation and the paralinearization of the water waves system with and without surface tension. The current result is optimal in the sense that for $α=2$ and ${\mathbb D}={\mathbb T}$ the flow associated to the Benjamin-Ono equation is Lipschitz on function with $0$ mean value $H^s_0$. In the second part of this paper we apply this method to deduce the quasi-linearity of the water waves system, which is the main result of this paper.

A geometric proof of the Quasi-linearity of the water-waves system

TL;DR

The paper proves that dispersive perturbations of Burgers-type equations with a nonlocal term (for ) induce quasi-linearity: the flow map on bounded -balls is not uniformly continuous and cannot be into . This result holds in any dimension and extends to a broad class of nonlinear transport-dispersive equations, including the Whitham equation and paralinearized water-waves systems, with optimality statements at on the torus via the Benjamin-Ono case. The second part uses a geometric transport/paradifferential framework to deduce the quasi-linearity of both gravity-capillary and gravity water-waves equations, showing that the nonlinear transport dominates the evolution and precludes Picard-type solvability. The approach hinges on crafting high-frequency geometric Ansatz, paracomposition/pseudo-differential changes of variables, and energy estimates for pulled-back equations, yielding robust instability of the flow map and informing the regularity thresholds for well-posedness in water-wave models.

Abstract

In the first part of this paper we prove that the flow associated to the Burgers equation with a non local term of the form fails to be uniformly continuous from bounded sets of to for , , , . Furthermore we show that the flow cannot be from bounded sets of to for . We generalize this result to a large class of nonlinear transport-dispersive equations in any dimension, that in particular contains the Whitham equation and the paralinearization of the water waves system with and without surface tension. The current result is optimal in the sense that for and the flow associated to the Benjamin-Ono equation is Lipschitz on function with mean value . In the second part of this paper we apply this method to deduce the quasi-linearity of the water waves system, which is the main result of this paper.

Paper Structure

This paper contains 43 sections, 39 theorems, 312 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Assumptions on the domain
  3. The equations
  4. Gravity water waves: Pressure and Taylor Coefficients
  5. Quasi-linearity of the water Wave system
  6. Strategy of the proof
  7. Quasi-linearity of the Burgers equation
  8. Quasi-linearity of problem \ref{['geometric proof quasi WW_study model pbm_subsec prereq cauchy pbm_eq1']}
  9. Acknowledgement
  10. Study of the model equation
  11. Prerequisites on the Cauchy Problems
  12. Proof of Theorem \ref{['geometric proof quasi WW_introduction_ theorem on dispersive Burgers eq']}
  13. Definition of the Ansatz
  14. Change of variables by transport
  15. Key Lemma and proof of the Theorem
  16. ...and 28 more sections

Key Result

Theorem 1.1

Consider three real numbers $\alpha\in [0,2[$, $s\in]2+\frac{1}{2},+\infty[$, $r>0$ and $u_0 \in H^{s}(\mathbb D)$. Then there exists $T>0$ such that for all $v_0$ in the ball $\hbox{B}(u_0,r)\subset H^{s}(\mathbb D)$ there exists a unique $v\in C([0,T],H^s(\mathbb D))$ solving the Cauchy problem: where, Moreover for all $R>0$, the flow map: is not uniformly continuous. Considering a weaker co

Figures (3)

  • Figure 1: The lines (1) and (2) are the characteristic curves from $Q^{in}_0$ and $P^{in}_0$. T is the time of formation of the shock wave.
  • Figure 2: Graph of the ansatz.
  • Figure 3: Transport of the ansatz.

Theorems & Definitions (67)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • ...and 57 more