Long time existence results for bore-type initial data for BBM-Boussinesq systems

Ill-posedness issues on $(abcd)$-Boussinesq system

In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of an inviscid fluid, and derived as a first-order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of the one-dimensional BBM-BBM case by Chen and Liu. Most of results established here, we obtain the optimal result for two-dimensional B…

Long-Time Asymptotic Behavior of Dissipative Boussinesq System

In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems which model two-directional waves.

Existence, Uniqueness, Analyticity, and Borel Summability for Boussinesq Equations

Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…

Modulational instability in nonlinear nonlocal equations of regularized long wave type

We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin- Bona-Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg-de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently…

Existence, Uniqueness, Analyticity, and Borel Summability of Boussinesq and Magnetic Benard Equations

Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…

平面域上的Bona-Smith型Boussinesq系统：理论与数值分析

Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis

研究二维Bona-Smith型Boussinesq系统在有界平面域上的局部适定性，并针对反射边界条件提出一种修正Galerkin方法，证明其在L²范数下达到最优收敛速率，数值实验模拟了复杂平面域中的二维表面波并比较了不同系统解。