[Paper Review] Hölder Continuous Solutions Of Boussinesq Equation with compact support

Tao Tao, Liqun Zhang|arXiv (Cornell University)|Dec 21, 2015

Advanced Mathematical Physics Problems22 references3 citations

TL;DR

This paper establishes the existence of nontrivial, Hölder continuous weak solutions to the 2D Boussinesq equations with compact support in both space and time, using a convex integration framework adapted to include temperature dynamics. The key result is a solution with spatial and temporal Hölder regularity exponents of $\frac{1}{28}-\varepsilon$ for velocity and $\frac{1}{25}-\varepsilon$ for temperature, demonstrating dissipative behavior in the presence of thermal effects.

ABSTRACT

We show the existence of Holder continuous solution of Boussinesq equations in whole space which has compact support both in space and time.

Motivation & Objective

• To establish the existence of Hölder continuous weak solutions to the 2D Boussinesq equations that are compactly supported in both space and time, extending the Onsager-type conjecture to systems with temperature coupling.
• To address the challenge of controlling interactions between velocity and temperature fields in convex integration, which are absent in the Euler case.
• To extend the framework of convex integration for the Euler equations to the Boussinesq system by constructing compatible oscillatory perturbations and a geometric lemma for the temperature-velocity coupling.
• To demonstrate that dissipative, Hölder continuous solutions exist even when thermal effects are present, confirming a nontrivial extension of the Onsager conjecture to the Boussinesq setting.

Proposed method

• Adapts the convex integration method of De Lellis and Székelyhidi to the Boussinesq system, incorporating temperature dynamics into the iterative construction of weak solutions.
• Constructs a family of oscillatory perturbations compatible with the Boussinesq equations, ensuring that the Reynolds stress and temperature flux terms are controlled at each stage of the iteration.
• Establishes a modified geometric lemma that accounts for the coupling between velocity and temperature fields, enabling the construction of solutions with compact spatial and temporal support.
• Uses a recursive iterative scheme with parameters $a$, $b$, and $\varepsilon$ to control the growth of derivatives and ensure convergence in Hölder norms.
• Applies Nash-Moser-type mollification techniques to handle the regularity loss in the iterative steps, particularly for the temperature and pressure components.
• Employs interpolation estimates and decay bounds on the sequence of approximations to prove convergence in Hölder spaces and establish the final regularity exponents.

Experimental results

Research questions

• RQ1 Can Hölder continuous weak solutions with compact support in both space and time be constructed for the 2D Boussinesq equations, despite the coupling between velocity and temperature?
• RQ2 What is the optimal Hölder regularity exponent achievable for velocity and temperature fields in such solutions, and how does it compare to the Euler case?
• RQ3 Can the convex integration framework be extended to include thermal effects in a way that preserves energy dissipation and compact support?
• RQ4 How do the interactions between velocity and temperature fields affect the regularity and convergence of the iterative construction process?
• RQ5 Is it possible to construct nontrivial solutions with $\theta \neq 0$ that are compactly supported and Hölder continuous, even when the temperature field is not identically zero?

Key findings

• The paper constructs a nontrivial, compactly supported weak solution $(v, p, \theta)$ to the 2D Boussinesq equations in $C_c(Q_{2r}; \mathbb{R}^2 \times \mathbb{R} \times \mathbb{R})$, with $v \neq 0$.
• The velocity field $v$ belongs to the Hölder space $C^{\frac{1}{28} - \varepsilon}_{t,x}$, and the temperature field $\theta$ belongs to $C^{\frac{1}{25} - \varepsilon}_{t,x}$, with pressure $p \in C^{1 - \varepsilon}_{t,x}$.
• The solution is constructed via an iterative convex integration scheme that ensures convergence in Hölder norms, with the regularity exponents derived from careful parameter selection in the iteration.
• When $\varepsilon \to 0$, the Hölder exponent for velocity approaches $\frac{1}{28}$, and for temperature, it approaches $\frac{1}{25}$, indicating the sharpness of the constructed regularity.
• The construction confirms that dissipative, Hölder continuous solutions exist for the Boussinesq system, extending the Onsager conjecture to systems with thermal effects.
• The method generalizes the Euler case (where $\theta = 0$) to the Boussinesq system, with the result reducing to Isett and Oh's construction when $\theta = 0$.