Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher-order asymptotic profiles of the solutions to the viscous Fornberg-Whitham equation

Ikki Fukuda, Kenta Itasaka

TL;DR

The paper studies the viscous Fornberg-Whitham equation with nonlocal dispersion and viscosity, proving global existence of small-data solutions and establishing convergence to a nonlinear diffusion wave $\chi(x,t)$, a Burgers-type self-similar profile. By rewriting the nonlocal term as $\alpha(b^{2}-\partial_{x}^{2})^{-1}\partial_{x}^{3}u+\alpha u_x$ with $\alpha=2B/b$, the authors connect the dynamics to Burgers and KdV-Burgers behavior and develop higher-order asymptotics. They construct a second asymptotic profile $V$ to capture dispersion-induced corrections and a third profile $Q$ to reveal finer structure, including the influence of the nonlocal term, and provide precise decay rates in $L^p$ norms for $u-\chi$, $u-\chi- V$, and $u-\chi- V- Q$. The analysis shows how nonlocal dispersion shapes higher-order long-time behavior, and contrasts the results with the KdV-Burgers equation by exposing additional derivative terms in the third-order correction. Overall, the work yields a detailed, multi-profile asymptotic description of solutions, with explicit formulas for the profiles and optimal decay rates.

Abstract

We consider the initial value problem for the viscous Fornberg-Whitham equation which is one of the nonlinear and nonlocal dispersive-dissipative equations. In this paper, we establish the global existence of the solutions and study its asymptotic behavior. We show that the solution to this problem converges to the self-similar solution to the Burgers equation called the nonlinear diffusion wave, due to the dissipation effect by the viscosity term. Moreover, we analyze the optimal asymptotic rate to the nonlinear diffusion wave and the detailed structure of the solution by constructing higher-order asymptotic profiles. Also, we investigate how the nonlocal dispersion term affects the asymptotic behavior of the solutions and compare the results with the ones of the KdV-Burgers equation.

Higher-order asymptotic profiles of the solutions to the viscous Fornberg-Whitham equation

TL;DR

The paper studies the viscous Fornberg-Whitham equation with nonlocal dispersion and viscosity, proving global existence of small-data solutions and establishing convergence to a nonlinear diffusion wave , a Burgers-type self-similar profile. By rewriting the nonlocal term as with , the authors connect the dynamics to Burgers and KdV-Burgers behavior and develop higher-order asymptotics. They construct a second asymptotic profile to capture dispersion-induced corrections and a third profile to reveal finer structure, including the influence of the nonlocal term, and provide precise decay rates in norms for , , and . The analysis shows how nonlocal dispersion shapes higher-order long-time behavior, and contrasts the results with the KdV-Burgers equation by exposing additional derivative terms in the third-order correction. Overall, the work yields a detailed, multi-profile asymptotic description of solutions, with explicit formulas for the profiles and optimal decay rates.

Abstract

We consider the initial value problem for the viscous Fornberg-Whitham equation which is one of the nonlinear and nonlocal dispersive-dissipative equations. In this paper, we establish the global existence of the solutions and study its asymptotic behavior. We show that the solution to this problem converges to the self-similar solution to the Burgers equation called the nonlinear diffusion wave, due to the dissipation effect by the viscosity term. Moreover, we analyze the optimal asymptotic rate to the nonlinear diffusion wave and the detailed structure of the solution by constructing higher-order asymptotic profiles. Also, we investigate how the nonlocal dispersion term affects the asymptotic behavior of the solutions and compare the results with the ones of the KdV-Burgers equation.

Paper Structure

This paper contains 6 sections, 22 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Global Existence and Decay Estimates
  3. Basic Estimates and Auxiliary Problem
  4. Asymptotic Behavior
  5. Second Asymptotic Profile
  6. Third Asymptotic Profile

Key Result

Theorem 1.1

Let $s\ge1$. Assume that $u_{0}\in L^{1}(\R)\cap H^{s}(\R)$ and $E_{s, 0}$ is sufficiently small. Then VFW has a unique global solution $u(x, t) \in C^{0}([0, \infty); H^{s})$. Moreover, if $u_{0}\in L^{1}_{1}(\R)\cap H^{s}(\R)$, for all ${\varepsilon}>0$, the estimate holds for any $p\in [2, \infty]$ and integer $0\le l\le s-1$, where $\chi(x, t)$ is defined by chi1 with $\alpha=\frac{2B}{b}$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 29 more