[Paper Review] Large time decay properties of solutions to a viscous Boussinesq system in a half space

Peiran Han, María E. Schonbek|arXiv (Cornell University)|Sep 17, 2013

Navier-Stokes equation solutions18 references3 citations

TL;DR

This paper establishes sharp large-time decay estimates for solutions to the 3D viscous Boussinesq system in a half-space, using $L^q$-$L^r$ estimates and fractional powers of the Stokes operator. It proves that the temperature and its derivatives decay as $t^{-2 - n/2(1 - 1/q)}$ in $L^q$-norms, with improved rates under small data assumptions, and resolves challenges from the unboundedness of the Helmholtz projection in $L^∞$ via nonlinear term decomposition.

ABSTRACT

We consider the long time behavior of weak and strong solutions of the n-dimensional viscous Boussinesq system in the half space, with $n\geq3$ . The $L^r(R^n_+)$-asymptotics of strong solutions and their first three derivatives, with $1\leq r\leq\infty$, are derived combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^\infty-$asymptotics of the second order derivatives the unboundedness of the projection operator $P: L^\infty(R^n_+) ightarrow L^\infty_σ(R^n_+)$ is dealt by an appropriate decomposition of the nonlinear term.

Motivation & Objective

• To analyze the long-time decay behavior of weak and strong solutions to the viscous Boussinesq system in the half-space $\mathbb{R}^n_+$ for $n \geq 3$.
• To extend $L^2$-decay results from the whole space to the half-space setting, particularly for temperature and velocity fields.
• To address the challenge of unboundedness of the Helmholtz projection in $L^\infty$ when analyzing second-order derivatives.
• To derive precise decay rates for the temperature and its derivatives in $L^q$-norms, including $L^\infty$-type estimates.

Proposed method

• Combines $L^q$-$L^r$ estimates with properties of fractional powers of the Stokes operator to analyze decay of strong solutions.
• Employs a nonlinear term decomposition to handle the unboundedness of the Helmholtz projection $P: L^\infty(\mathbb{R}^n_+) \to L^\infty_\sigma(\mathbb{R}^n_+)$ in $L^\infty$-estimates.
• Uses iterative integral representation and heat kernel estimates to bound higher-order derivatives of the temperature and velocity fields.
• Applies the method of majorizing kernels and energy-type estimates to control nonlinear terms involving $u \cdot \nabla \theta$ and $u \cdot \nabla u$.
• Derives decay estimates for $\nabla^3 \theta(t)$ in $L^q(\mathbb{R}^n_+)$ for $1 \leq q \leq \infty$ via bootstrapping and interpolation.
• Implements a weak convergence argument to pass from approximate solutions to the limit solution in $L^q$-norms.

Experimental results

Research questions

• RQ1 How do the decay rates of solutions to the viscous Boussinesq system in a half-space compare to those in the whole space?
• RQ2 What is the precise large-time decay behavior of the temperature field and its derivatives in $L^q$-norms for $n \geq 3$?
• RQ3 How can the unboundedness of the Helmholtz projection in $L^\infty$ be overcome when estimating second-order derivatives?
• RQ4 What conditions on initial data lead to improved decay rates for the velocity and temperature fields?
• RQ5 Can sharp decay estimates for $\nabla^3 \theta(t)$ be derived in both $L^q$ and $L^\infty$ norms?

Key findings

• The temperature $\theta(t)$ decays as $\|\theta(t)\|_{L^2(\mathbb{R}^n_+)} \leq C(1+t)^{-(n+2)/4}$ under small initial data conditions, improving upon the standard $L^2$-decay rate.
• The velocity field $u(t)$ decays as $\|u(t)\|_{L^2(\mathbb{R}^n_+)} \to 0$ as $t \to \infty$ for $n \geq 3$, with rates $O((1+t)^{1/4})$ in 3D, $O(\log(1+t))$ in 4D, and $O(1)$ in $n \geq 5$.
• For $1 \leq q < \infty$, the third-order spatial derivatives of the temperature satisfy $\|\nabla^3 \theta(t)\|_{L^q(\mathbb{R}^n_+)} \leq C t^{-2 - n/2(1 - 1/q)}$ for $t \geq 1$, with explicit dependence on $n$ and $q$.
• In the $L^\infty$-norm, $\|\nabla^3 \theta(t)\|_{L^\infty(\mathbb{R}^n_+)} \leq \widetilde{C}_{\ast\ast\ast} t^{-2 - n/2}$ for $t \geq 1$, achieved via careful decomposition of the nonlinear term.
• The decay rate for $\nabla^3 \theta(t)$ in $L^q$-norms is sharp and consistent across $1 \leq q \leq \infty$, with the exponent depending on the integrability index $q$.
• The results are robust under small initial data and extend previous $L^2$-decay results to higher-order derivatives and broader function spaces.