[Paper Review] Relaxation Limit in Besov Spaces for Compressible Euler Equations

Jiang Xu, Zejun Wang|arXiv (Cornell University)|Sep 18, 2011

Navier-Stokes equation solutions20 references3 citations

TL;DR

This paper establishes the global existence of classical solutions and the relaxation limit to the porous medium equation for multidimensional compressible Euler equations with relaxation in critical Besov spaces. Using Chemin-Lerner's time-space Besov spaces and a refined commutator estimate, it proves that as the relaxation time τ→0, the density converges to the solution of the porous medium equation, with optimal convergence rates in the critical regularity regime σ = 1 + d/2.

ABSTRACT

The relaxation limit in critical Besov spaces for the multidimensional compressible Euler equations is considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem with initial data close to an equilibrium state are constructed in the Chemin-Lerner's spaces with critical regularity. Furthermore, it is shown that the density converges towards the solution to the porous medium equation, as the relaxation time tends to zero. Several important estimates are achieved, including a crucial estimate of commutator.

Motivation & Objective

• To establish the existence of global classical solutions for the compressible Euler equations with relaxation in critical Besov spaces with regularity σ = 1 + d/2.
• To justify the relaxation limit as τ → 0, showing convergence of the density to the solution of the porous medium equation.
• To improve prior results in Sobolev spaces by working in the larger, critical Besov space $ B^{ ho}_{2,1}( r^d) $, which is a subalgebra of $ W^{1,rown} $.
• To develop and apply a new commutator estimate in Besov spaces to control nonlinear terms in the energy estimates.

Proposed method

• Constructing uniform global classical solutions in Chemin-Lerner's time-space Besov spaces $ \widetilde{L}^\theta_T(B^{\sigma}_{2,1}) $ with critical regularity σ = 1 + d/2.
• Employing Littlewood-Paley decomposition and Bony's para-product formula to analyze the structure of nonlinear terms.
• Deriving a crucial commutator estimate for $ [\varrho, \Delta_q]\mathrm{div}\mathbf{v} $ and $ [\mathbf{v}, \Delta_q]\cdot\nabla\varrho $ in $ L^p $-based Besov norms.
• Applying Aubin-Lions compactness lemma to pass to the limit as τ → 0 in the relaxed system.
• Using the embedding $ B^{\sigma}_{2,1} \hookrightarrow \mathcal{W}^{1,\infty} $ to ensure sufficient regularity for classical solutions.
• Establishing uniform bounds in $ \widetilde{L}^\theta_T(B^{\sigma}_{2,1}) $-norms to justify the relaxation limit.

Experimental results

Research questions

• RQ1 Does the compressible Euler system with relaxation admit global classical solutions in critical Besov spaces with regularity σ = 1 + d/2?
• RQ2 What is the asymptotic behavior of the solution as the relaxation time τ → 0 in this critical regularity setting?
• RQ3 Can the relaxation limit be justified in Besov spaces, and does the density converge to the solution of the porous medium equation?
• RQ4 What is the role of the commutator estimate in controlling nonlinearities in the energy estimates for the relaxation limit?
• RQ5 How does the choice of $ B^{\sigma}_{2,1} $ space improve the regularity and convergence results compared to Sobolev spaces?

Key findings

• The Cauchy problem for the compressible Euler equations with relaxation admits a unique global classical solution in $ \mathcal{C}^1(\mathbb{R}^+ \times \mathbb{R}^d) $ for initial data in $ B^{\sigma}_{2,1} $ with $ \sigma = 1 + d/2 $, provided the initial perturbation is sufficiently small.
• The density $ \rho $ converges to the solution of the porous medium equation $ \partial_s \mathcal{N} - \Delta P(\mathcal{N}) = 0 $ as $ \tau \to 0 $, in the sense of weak-* convergence in $ L^\infty $ and strong convergence in $ L^p $ for $ p \in [1, \infty) $.
• A new commutator estimate is derived: $ 2^{q\sigma}\|[ abla \varrho, \Delta_q]\mathrm{div}\mathbf{v}\|_{L^\theta_T(L^{2d/(d+2)})} \leq C c_q \|\nabla \varrho\|_{\widetilde{L}^{\theta_1}_T(B^{\sigma-1}_{2,1})} \|\mathbf{v}\|_{\widetilde{L}^{\theta_2}_T(B^{\sigma}_{2,1})} $ with $ \|(c_q)\|_{\ell^1} \leq 1 $.
• The convergence rate of the solution to the equilibrium state is $ (1+t)^{-d/2(1-1/p)} $ in $ L^p $-norms for $ 1 < p \leq \infty $, matching the optimal rate in previous Sobolev space results.
• The framework of Chemin-Lerner's spaces $ \widetilde{L}^\theta_T(B^{\sigma}_{2,1}) $ allows for sharper control of the time and space regularity, enabling the relaxation limit proof in critical regularity.
• The result extends prior work in Sobolev spaces by working in the larger, critical Besov space $ B^{\sigma}_{2,1} $, which ensures the necessary algebraic and embedding properties for the analysis.