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Global Stabilization for the BBM-KP equations on R2

F. A. Gallego, V. H. Gonzalez Martinez, J. C. Muñoz Grajales

TL;DR

The paper addresses exponential stabilization for the BBM-KP equation on $\mathbb{R}^2$ with localized damping. It develops global well-posedness and a unique continuation property, leverages a compactness-uniqueness framework to derive a local and then global observability inequality, and proves global uniform exponential decay of the energy in $H^1_x(\mathbb{R}^2)$ for $p=1$. The authors complement the analysis with a spectral-finite-difference numerical scheme that validates the theory and reveals exponential decay in simulations, including nonlinear regimes. The work advances understanding of long-time behavior for dispersive two-dimensional models on unbounded domains under localized damping and provides a practical numerical approach to study these dynamics, while outlining several open problems for future work.

Abstract

In this paper, we present results on the energy decay of the BBM-KP equations (I and II) posed on $\R^2$ with localized damping. This model offers an alternative to the KP equations, analogous to how the regularized long-wave equation relates to the classical Korteweg-de Vries (KdV) equation. We show that the energy associated with the Cauchy problem decays exponentially when a localized dissipative mechanism is present in a subdomain. Finally, we validate the theoretical results on the exponential stabilization of solutions to the BBM-KP equations with damping through numerical experiments using a spectral-finite difference scheme.

Global Stabilization for the BBM-KP equations on R2

TL;DR

The paper addresses exponential stabilization for the BBM-KP equation on with localized damping. It develops global well-posedness and a unique continuation property, leverages a compactness-uniqueness framework to derive a local and then global observability inequality, and proves global uniform exponential decay of the energy in for . The authors complement the analysis with a spectral-finite-difference numerical scheme that validates the theory and reveals exponential decay in simulations, including nonlinear regimes. The work advances understanding of long-time behavior for dispersive two-dimensional models on unbounded domains under localized damping and provides a practical numerical approach to study these dynamics, while outlining several open problems for future work.

Abstract

In this paper, we present results on the energy decay of the BBM-KP equations (I and II) posed on with localized damping. This model offers an alternative to the KP equations, analogous to how the regularized long-wave equation relates to the classical Korteweg-de Vries (KdV) equation. We show that the energy associated with the Cauchy problem decays exponentially when a localized dissipative mechanism is present in a subdomain. Finally, we validate the theoretical results on the exponential stabilization of solutions to the BBM-KP equations with damping through numerical experiments using a spectral-finite difference scheme.
Paper Structure (15 sections, 12 theorems, 110 equations, 8 figures)

This paper contains 15 sections, 12 theorems, 110 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Setting of the Problem
  4. Preliminar and notations
  5. Main Result
  6. Paper’s outline
  7. Global Well-posedness and Unique Continuation Property
  8. Global Well-posedness
  9. Unique Continuation Property
  10. Global Uniform Stabilization
  11. Numerical approximation of the BBM-KP equation
  12. The numerical scheme
  13. Some exact solutions of the BBM-KP equation
  14. Numerical experiments
  15. Further Comments

Key Result

Theorem 1.1

Let $s>3/2$, $a\in L^\infty(\Omega)$ satisfy Assumption hyp a and $\gamma=\pm 1$ and $p=1$. Then, e1 is globally uniformly exponentially stable in $H^{1}_x(\mathbb{R}^2)$, i.e. there exists a positive constant $\nu$ that does not depend on initial data such that, for any $\varepsilon>0$ and $u_0 \in where $E$ is the energy functional given by energy and $\alpha_{\varepsilon}: \mathbb{R} \rightarro

Figures (8)

  • Figure 1: The cross-sectional profile $u(x,0,t=2)$ of the exact solution \ref{['exact_sol']}, in the linear regime ($\alpha =0$), compared with the numerical solution obtained using the scheme given in \ref{['two_step_scheme']}. The modeling parameters are $\gamma = 1$, $a = 1$. The numerical solution is shown as the pointed line, while the exact solution is displayed by the solid line.
  • Figure 2: The cross-sectional profile $u(x,0,t=2)$ of the exact solution \ref{['exact_sol']}, in the linear regime ($\alpha = 0$), compared with the numerical solution. The modeling parameters are $\gamma = -1$, $a=1$. The numerical solution is shown as the pointed line, while the exact solution is displayed by the solid line.
  • Figure 3: Periodic travelling wave solution \ref{['periodic_wave_BBMKP']} for $\gamma = 1$, at $t = 2$.
  • Figure 4: Periodic travelling wave solution \ref{['periodic_wave_BBMKP']} for $\gamma=-1$, at $t = 2$.
  • Figure 5: Logarithm of the $L^2$-norm $\|u\|_{L^2}$ of a solution $u$ of equation \ref{['BBMKP2']}, as a function of time $t$ for $\gamma=1, p=1$.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.1: Global Uniform Stabilization
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 18 more