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On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach

Aissa Boukarou, Lamia Seghour

TL;DR

This work establishes sharp temporal Gevrey regularity for the 2+1D fifth-order KP-II equation: if the initial data are Gevrey of order $\sigma$ in space, the solution is Gevrey of order $5\sigma$ in time, and it cannot belong to $G^z$ for any $1 \le z < 5\sigma$. The authors introduce a refined majorant-series method in Gevrey–Bourgain spaces to track the dispersive term $\partial_x^5u$ and nonlinear interactions, and they prove a general principle for dispersive equations of the form $\partial_t u = \pm \partial_x^\alpha u + P(u)$ that temporal Gevrey regularity below $\alpha\sigma$ cannot hold. A key part of the argument is constructing data that forces growth like $((5j)!)^\sigma$ in time derivatives, establishing the optimality of the $5\sigma$ rate. The results extend prior work by Boukarou et al. and provide a robust analytic framework for temporal Gevrey regularity in higher-order dispersive PDEs, with the Kawahara equation discussed as a canonical example.

Abstract

In this paper, we improve and extend the results obtained by Boukarou et al. \cite{boukarou1} on the Gevrey regularity of solutions to a fifth-order Kadomtsev-Petviashvili-II equation. We establish Gevrey regularity in the time variable for solutions in $2+1$ dimensions, providing a sharper result obtained through a new analytical approach. Assuming that the initial data are Gevrey regular of order $σ\geq 1$ in the spatial variables, we prove that the corresponding solution is Gevrey regular of order $5 σ$ in time. Moreover, we show that the function $u(x, y, t)$, viewed as a function of $t$, does not belong to $G^z$ for any $1 \leq z<5 σ$. Our proof introduces a new analytical method that establishes a general principle for dispersive equations of the form $ \partial_t u = \pm\partial_x^αu + P(u),$ where $\partial_x^α$ is the highest spatial derivative and $P(u)$ a polynomial in spatial derivatives of total order at most $α-1$, the solution cannot belong to the Gevrey class $G^z$ in time for any $z$ satisfying $1 \leq z<ασ$.

On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach

TL;DR

This work establishes sharp temporal Gevrey regularity for the 2+1D fifth-order KP-II equation: if the initial data are Gevrey of order in space, the solution is Gevrey of order in time, and it cannot belong to for any . The authors introduce a refined majorant-series method in Gevrey–Bourgain spaces to track the dispersive term and nonlinear interactions, and they prove a general principle for dispersive equations of the form that temporal Gevrey regularity below cannot hold. A key part of the argument is constructing data that forces growth like in time derivatives, establishing the optimality of the rate. The results extend prior work by Boukarou et al. and provide a robust analytic framework for temporal Gevrey regularity in higher-order dispersive PDEs, with the Kawahara equation discussed as a canonical example.

Abstract

In this paper, we improve and extend the results obtained by Boukarou et al. \cite{boukarou1} on the Gevrey regularity of solutions to a fifth-order Kadomtsev-Petviashvili-II equation. We establish Gevrey regularity in the time variable for solutions in dimensions, providing a sharper result obtained through a new analytical approach. Assuming that the initial data are Gevrey regular of order in the spatial variables, we prove that the corresponding solution is Gevrey regular of order in time. Moreover, we show that the function , viewed as a function of , does not belong to for any . Our proof introduces a new analytical method that establishes a general principle for dispersive equations of the form where is the highest spatial derivative and a polynomial in spatial derivatives of total order at most , the solution cannot belong to the Gevrey class in time for any satisfying .

Paper Structure

This paper contains 6 sections, 13 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Functional spaces and main results
  3. Gevrey regularity
  4. Failure of $G^z$-Regularity in Time for $1\leq z < 5\sigma$
  5. $G^{5\sigma}$ regularity in $t$
  6. Conclusion

Key Result

Theorem 2.5

Let $s_{1},s_{2} \geq 0, \delta>0$, $\sigma\geq 1$ and $b = \frac{1}{2}+$. For initial data $\varphi$ in the space $G^{\delta,\sigma, s_{1},s_{2}} (\mathbb{R}^{2})$ and $\vert \xi\vert^{-1}\widehat{\varphi}(\xi,\mu)\in L^{2}$,there exists $T > 0$, which depends on $\varphi$, such that the Cauchy pro Furthermore, the data-to-solution map is continuous.

Theorems & Definitions (29)

  • Definition 2.1: Gevrey space
  • Remark 2.2
  • Definition 2.3: Gevrey Bourgain space
  • Definition 2.4: Time restricted space
  • Theorem 2.5: boukarou1
  • Definition 2.6: Gevrey Class $G^\sigma$
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Proposition 3.1: Spatial Gevrey Regularity
  • ...and 19 more