[Paper Review] On the Cauchy problem for a Boussinesq type system

Felipe Linares, Mahendra Panthee+1 more|arXiv (Cornell University)|Jun 17, 2012

Advanced Mathematical Physics Problems9 references3 citations

TL;DR

This paper establishes local well-posedness for the Cauchy problem of a two-dimensional Boussinesq-type system by transforming it into a coupled system of KdV-type equations via diagonalization. Using sharp smoothing estimates and Strichartz-type analysis, the authors prove unique local solutions in $H^s(\mathbb{R}^2) \times \mathcal{V}^{s+1}(\mathbb{R}^2)$ for $s > 3/2$, improving upon prior results that required $s > 2$. The method relies on handling nonlinearities through derivative enlargement and space-time norm estimates.

ABSTRACT

We consider the initial value problem (IVP) associated to a Boussinesq type system. After rewriting the system in an equivalent form of coupled KdV-type equations, we prove that this is locally well-posed in $(H^s(\R^2))^4$, $s>3/2$, using sharp smoothing estimates. Consequently we obtain the local well-posedness result for the original system in $H^s imes \mathcal{V}^{s+1}$ for $s>3/2$ (see below for the definition of $\mathcal{V}^{s}$).

Motivation & Objective

• To improve the local well-posedness threshold for the Cauchy problem of a 2D Boussinesq-type system beyond the $s > 2$ result previously established in [9].
• To develop a rigorous framework for handling the nonlinear couplings in the system by transforming it into a symmetric system of coupled KdV-type equations.
• To establish the existence and uniqueness of local solutions in a function space that reflects the intrinsic regularity of the original physical variables $\eta$ and $\Phi$.
• To provide a complete and detailed proof of the equivalence between the original Boussinesq system and the transformed KdV-type system, including the recovery of solutions in the original variables.

Proposed method

• The system is rewritten in an equivalent form of coupled KdV-type equations by diagonalizing the linear part using the operator $i\sqrt{-\Delta}(1 - \Delta)$.
• The original unknowns $\eta$ and $\Phi$ are transformed into $u = \partial_{x_1}\Phi$ and $v = \partial_{x_2}\Phi$, leading to a symmetric system in $u$ and $v$.
• Nonlinear terms are handled by differentiating the system in $x_1$ and $x_2$, resulting in a system of four equations for the partial derivatives of $u$ and $v$, which linearizes the nonlinearities.
• The solution is constructed via a contraction mapping argument in a space of space-time norms, using Strichartz, maximal, and smoothing estimates.
• The solution space $\mathcal{X}_T^s$ is defined via norms involving $L^q_t L^\infty_x$ and $L^2_t L^2_x$ components, with control on derivatives and Riesz transforms.
• The original solution is recovered via the inverse transformation $\eta = i\sqrt{-\Delta}(u - v)$, $\Phi = u + v$, ensuring continuity in time with values in $H^s \times \mathcal{V}^{s+1}$.

Experimental results

Research questions

• RQ1 Can the local well-posedness threshold for the 2D Boussinesq system be improved from $s > 2$ to $s > 3/2$?
• RQ2 Is it possible to establish local well-posedness using a KdV-type framework after diagonalizing the system’s linear part?
• RQ3 How can the nonlinear couplings in the Boussinesq system be effectively managed to allow for a contraction mapping argument?
• RQ4 What is the precise function space in which the solution exists and is unique, and how does it relate to the original physical variables?
• RQ5 Can the equivalence between the original system and the transformed KdV-type system be rigorously established, including solution recovery?

Key findings

• The Cauchy problem for the 2D Boussinesq-type system is locally well-posed in $H^s(\mathbb{R}^2) \times \mathcal{V}^{s+1}(\mathbb{R}^2)$ for all $s > 3/2$, improving upon the prior $s > 2$ threshold.
• The solution is constructed in a space $\mathcal{Y}_T^s$ that is continuously embedded in $C([0,T]; H^s \times \mathcal{V}^{s+1})$, ensuring strong continuity in time.
• The method relies on transforming the system into a coupled KdV-type system via diagonalization, which symmetrizes the dispersive structure and simplifies analysis.
• Sharp smoothing estimates and space-time norms (Strichartz-type) are used to control the nonlinear terms, particularly those involving derivatives and Riesz transforms.
• The solution map is shown to be a contraction in a suitable function space, guaranteeing uniqueness and existence for small time $T > 0$ depending on the initial data size.
• The original variables $\eta$ and $\Phi$ are uniquely recovered from the transformed variables $u$ and $v$ via $\eta = i\sqrt{-\Delta}(u - v)$, $\Phi = u + v$, preserving the solution structure.