Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation

Bianchini R, Crin-Barat T, Paicu M (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Archive for Rational Mechanics and Analysis Springer Verlag (Germany)

Book Volume: 248

Article Number: 2

Journal Issue: 1

DOI: 10.1007/s00205-023-01945-x

Abstract

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in H˙ 1-τ(R2) ∩ H˙ s(R2) with s> 3 and for any 0 < τ< 1 . Such a result improves upon the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to H20(R2) . More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in H1-τ(R2) ∩ H˙ s(R2) with s> 3 and 0 < τ< 1 . Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity ‖u2(t)‖L∞(R2) for initial data only in H˙ 1-τ(R2) ∩ H˙ s(R2) with s> 3 .

Authors with CRIS profile

Timothée Crin-Barat Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Consiglio Nazionale delle Ricerche (CNR) / National Research Council of Italy IT Italy (IT) Université de Bordeaux FR France (FR)

How to cite

APA:

Bianchini, R., Crin-Barat, T., & Paicu, M. (2024). Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation. Archive for Rational Mechanics and Analysis, 248(1). https://dx.doi.org/10.1007/s00205-023-01945-x

MLA:

Bianchini, Roberta, Timothée Crin-Barat, and Marius Paicu. "Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation." Archive for Rational Mechanics and Analysis 248.1 (2024).

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