[Paper Review] The incompressible limit in $L^p$ type critical spaces

Raphaël Danchin, Ling-Bing He|arXiv (Cornell University)|Sep 25, 2014

Navier-Stokes equation solutions17 references4 citations

TL;DR

This paper establishes the low Mach number limit for compressible Navier-Stokes equations in $L^p$-type critical Besov spaces, proving convergence to the incompressible Navier-Stokes equations under minimal regularity assumptions. It extends prior $L^2$-based energy methods to critical regularity frameworks, showing that divergence-free initial velocity in $\dot{B}^{d/p-1}_{p,r} \cap \dot{B}^{-1}_{\infty,1}$ suffices for convergence when $p \in [2,4]$ and $r \in [1,\infty]$, with $L^2$-type bounds on low-frequency components being unavoidable due to acoustic waves.

ABSTRACT

This paper aims at justifying the low Mach number convergence to the incompressible Navier-Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to $L^2$ spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the $L^p$ type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space $\dot B^{d/p-1}_{p,r}\cap\dot B^{-1}_{\infty,1}$ for some suitable $(p,r)\in[2,4] imes[1,+\infty].$ We still require $L^2$ type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier-Stokes system for heat conducting fluids.

Motivation & Objective

• To justify the low Mach number limit for viscous compressible flows in the ill-prepared data regime using $L^p$-type critical regularity spaces.
• To extend prior $L^2$-based energy methods to critical Besov spaces, particularly $\dot{B}^{d/p-1}_{p,r}$ and $\dot{B}^{-1}_{\infty,1}$, for the divergence-free part of the initial velocity.
• To identify minimal regularity assumptions that ensure convergence to the incompressible Navier-Stokes equations, while accounting for acoustic wave effects through $L^2$-type bounds on low frequencies of density and potential velocity components.
• To generalize the results to the full Navier-Stokes system for heat-conducting fluids in the same critical framework.

Proposed method

• Uses a homogeneous Littlewood-Paley decomposition $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$ to define critical Besov spaces $\dot{B}^{s}_{p,r}$ and analyze frequency-localized components of the solution.
• Applies Bony's decomposition $uv = T_u v + R(u,v) + T_v u$ to handle nonlinear terms in the compressible Navier-Stokes system.
• Employs paraproduct and commutator estimates, particularly $\| [\dot{S}_{j_0}A(D), T_a]b \|_{\dot{B}^{\sigma+s}_{p,1}} \lesssim \| \nabla a \|_{\dot{B}^{s-1}_{p_1,1}} \| b \|_{\dot{B}^{\sigma}_{p_2,\infty}}$, to control nonlinear interactions.
• Introduces a modified energy functional involving $\| \delta \Theta^\varepsilon \|_{L^\infty(\dot{B}^{(d+1)/p-3/2}_{p,1} + \dot{B}^{d/p-2}_{p,1})}$ and $L^2$-type norms to track convergence rates.
• Uses the scaling invariance of the critical framework: $(a,u) \mapsto (a, \ell u)(\ell^2 t, \ell x)$, to guide the choice of function spaces.
• Applies composition estimates for smooth functions $G$ with $G(0) = 0$, such as $\| G(a) \|_{\dot{B}^s_{p,1}} \lesssim \| a \|_{\dot{B}^s_{p,1}}$, to control nonlinear pressure and viscosity terms.

Experimental results

Research questions

• RQ1 Can the low Mach number limit be justified in $L^p$-type critical spaces rather than $L^2$-based energy spaces?
• RQ2 What is the minimal regularity required for the divergence-free part of the initial velocity to ensure convergence in the ill-prepared data case?
• RQ3 How do acoustic waves influence the required regularity assumptions, particularly in terms of low-frequency components?
• RQ4 Can the critical framework be extended to the full Navier-Stokes system with heat conduction?
• RQ5 What role do commutator and paraproduct estimates play in controlling nonlinearities in the critical regularity setting?

Key findings

• The divergence-free part of the initial velocity only needs to be bounded in $\dot{B}^{d/p-1}_{p,r} \cap \dot{B}^{-1}_{\infty,1}$ for $p \in [2,4]$ and $r \in [1,\infty]$, which is a sharp critical regularity condition.
• The $L^2$-type bounds on low frequencies of the potential velocity and density remain necessary due to persistent acoustic wave effects in the ill-prepared data regime.
• Convergence to the incompressible Navier-Stokes equations is established in the $L^p$-type critical framework, with convergence rates controlled via $\delta X^\varepsilon$-type norms involving $\dot{B}^{(d+1)/p-3/2}_{p,1}$ and $\dot{B}^{d/p-2}_{p,1}$ regularity.
• The results are extended to the full Navier-Stokes system for heat-conducting fluids, maintaining the same critical regularity assumptions.
• The proof relies on a refined energy estimate combining $L^\infty$ and $L^2$-type norms in Besov spaces, with careful control of nonlinear terms via paraproduct and commutator estimates.
• The framework is invariant under the scaling $ (a,u) \mapsto (a, \ell u)(\ell^2 t, \ell x) $, ensuring consistency with the scaling of the limit system.