Asymptotic stability for monotone traveling kinks of the dissipative Boussinesq problem

Technical Summary: Asymptotic Stability for Monotone Traveling Kinks of the Dissipative Boussinesq Problem

Problem Statement
This paper investigates the long-time dynamics of the dissipative Boussinesq equation in one spatial dimension, a model describing the propagation of long waves with small amplitude on shallow water surfaces, incorporating viscosity effects. The governing equation is given by:
$$u_{tt} - c u_{txx} + b u_{xxxx} - u_{xx} = \partial_{xx}(f(u))$$
where $b, c > 0$. While the inviscid case ($c=0$) corresponds to the "good" Boussinesq equation, the inclusion of the damping term $-c u_{txx}$ (analogous to the BBM equation's viscosity) introduces significant changes to the solution structure. The primary focus is on the stability of monotone traveling kink solutions (heteroclinic orbits connecting two distinct constant states $\phi_-$ and $\phi_+$). Specifically, the paper addresses whether these kinks are orbitally stable and, more strongly, asymptotically stable under small perturbations in the energy space $H^1(\mathbb{R})$.

Methodology
The analysis proceeds through several rigorous steps:

1. Existence and Characterization of Kinks:
   The authors first establish the existence of smooth, monotone decreasing traveling wave solutions $\phi(x-\sigma t)$ for the viscous Boussinesq equation. By reducing the PDE to an ODE, they derive the Rankine-Hugoniot relation linking the wave speed $\sigma$ to the asymptotic states $\phi_\pm$. They prove that unique smooth kinks exist provided the parameters satisfy specific constraints, notably a condition on the viscosity parameter $B$ and the wave speed ensuring the monotonicity of the front. The asymptotic behavior of these kinks at infinity is explicitly characterized via exponential decay rates determined by the roots of the linearized operator.

2. Equivalent Formulation:
   To facilitate the stability analysis, the second-order in time equation is recast into a first-order system in time. By introducing an auxiliary variable $v$ such that $u_t - \alpha u_{xx} = v_x$, the problem is transformed into a coupled system of parabolic-hyperbolic type. This formulation allows the authors to treat the perturbation around the traveling wave as a localized Cauchy problem with data in $L^2$ and $H^1$.

3. Green's Function and Local Well-Posedness:
   The linearized operator around the zero solution is analyzed via the spatial Fourier transform. The authors construct the Green's function for the free viscous Boussinesq problem, establishing decay estimates that exhibit parabolic behavior (specifically, $|\hat{G}(t, \xi)| \leq C e^{-c_0 t |\xi|^2}$ for large frequencies). These estimates are used to prove local well-posedness for the nonlinear perturbation equation via a fixed-point argument in appropriate Sobolev spaces.

4. Energy Estimates and Continuity Argument:
   The core of the stability proof relies on a continuity argument. Assuming a solution exists up to a maximal time $t^*$, the authors derive a priori energy estimates for the perturbation variables $(U, V)$.

   • Modulation: To handle the translational invariance of the kink, the authors introduce time-dependent modulation parameters $\mu(t)$ and $\nu(t)$ to track the position of the wave. The parameter $\mu(t)$ is chosen specifically to eliminate the secular terms arising from the linearized operator's kernel (orthogonality conditions).
   • Lyapunov Functional: A modified energy functional is constructed, incorporating terms like $\int (1+\phi)U^2$, $\int U^3$, and $\gamma \|U'\|^2$. By carefully selecting the modulation functions and utilizing the monotonicity of the kink ($\phi' < 0$), the authors show that the time derivative of this energy is non-positive, up to higher-order terms that can be controlled for sufficiently small initial data.
   • Global Existence: These estimates demonstrate that the $H^1$ norm of the perturbation cannot blow up in finite time, thereby extending the local solution to a global one.

5. Asymptotic Decay:
   To prove asymptotic stability, the authors utilize the dissipation inherent in the system. The energy estimates yield an $L^2_t$ bound on the spatial derivatives of the perturbation (i.e., $\int_0^\infty \|U'(t)\|^2 dt < \infty$). By showing that the function $t \mapsto \|U'(t)\|^2$ is uniformly continuous, they invoke Barbalat's Lemma to conclude that $\|U'(t)\|_{L^2} \to 0$ as $t \to \infty$. Finally, Gagliardo-Nirenberg interpolation inequalities are used to extend this decay to $L^p$ norms for $2 < p \leq \infty$.

Key Contributions and Results
The paper establishes the following main results:

• Existence: A rigorous proof of the existence of unique, smooth, monotone decreasing traveling kink solutions for the dissipative Boussinesq equation under specific parameter constraints.
• Orbital Stability: The kinks are shown to be orbitally stable in the $H^1(\mathbb{R})$ norm. This means that if the initial data is sufficiently close to a kink, the solution remains close to the family of translated kinks for all time.
• Asymptotic Stability: The paper proves that these kinks are asymptotically stable. Specifically, the spatial derivative of the solution converges to the derivative of the kink in the $L^2$ norm, and the solution itself converges to the kink in $L^p$ norms for $2 < p \leq \infty$, modulo a time-dependent shift.
• Technical Framework: The work provides a robust framework for analyzing stability in dissipative dispersive systems by combining modulation techniques with energy methods tailored to the specific Green's function decay of the viscous Boussinesq operator.

Significance
The paper claims significance by resolving the stability question for a physically relevant model that bridges the gap between the conservative "good" Boussinesq equation and viscous fluid dynamics. While the "good" Boussinesq equation has been extensively studied, the precise form of viscous corrections and their effect on the stability of heteroclinic connections (kinks) has been a subject of inquiry since Whitham's early work. This work demonstrates that the inclusion of viscosity not only ensures the existence of monotone kinks but also stabilizes them asymptotically, a property that is generally more difficult to establish in conservative dispersive systems. The results confirm that the dissipative mechanism dominates the dispersive effects in the long-time limit, driving the system toward the traveling kink profile.