Skip to content
  • Publications
  • Jobs
  • enzuazua
  • Seminars
  • Events Calendar
cmc.deusto.eus
  • Home
  • About us
    • About the Chair
    • Head of the Chair
    • Team
    • Past Members
  • Research
    • Projects
    • ERC – DyCon
    • DyCon Blog
    • DyCon Toolbox
    • Industrial & Social TransferenceContents related to the industrial and social transference aspects of the work in the Chair of Computational Mathematics.
  • Publications
    • Publications (All)
    • Publications Relased
      • Publications 2022
      • Publications 2021
      • Publications 2020
      • Publications 2019
      • Publications 2018
      • Publications 2017
      • Publications 2016
    • AcceptedAccepted to be released
    • SubmittedSubmitted publications
  • Activities
    • Events calendar
    • Past Events
    • News
    • Seminars
    • Courses
    • enzuazua
    • Gallery
  • Jobs
  • Contact

Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation

R. Bianchini, Crin-Barat T., M. Paicu. Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation (2022)

Abstract. We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in \dot{H}^{1-\tau}(\mathbb{R}^2) \cap \dot{H}^s(\mathbb{R}^2) with s > 3 and for any 0 \lt \tau \lt 1. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to H^{20} (\mathbb{R}^2) . 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 H^{1 - \tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2) with s > 3 and 0 \lt \tau \lt 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 \|u_2(t)\|_{L^\infty (\mathbb{R}^2)} for initial data only in \dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2) with s >3.

Read Full Paper

arxiv: 2210.02118

Last updated on October 19, 2022

Post navigation

Previous Post
FAU DCN-AvH Seminar: Natural gradient in evolutionary games FAU DCN-AvH Seminar: Natural gradient in evolutionary games
Next Post
Control of certain parabolic models from biology and social sciences

Last Publications

Control of neural transport for normalizing flows

A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation

Gaussian Beam ansatz for finite difference wave equations

Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave

Optimal design of sensors via geometric criteria

  • Control of neural transport for normalizing flows
  • A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation
  • Gaussian Beam ansatz for finite difference wave equations
  • Optimal design of sensors via geometric criteria
  • Eigenvalue bounds for the Gramian operator of the heat equation
  • Control of neural transport for normalizing flows
  • A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation
  • Gaussian Beam ansatz for finite difference wave equations
  • Optimal design of sensors via geometric criteria
  • Eigenvalue bounds for the Gramian operator of the heat equation
Copyright 2016 - 2023 — cmc.deusto.eus. All rights reserved. Chair of Computational Mathematics, Deusto Foundation - University of Deusto
Scroll to Top
  • Aviso Legal
  • Política de Privacidad
  • Política de Cookies
  • Configuración de Cookies
WE USE COOKIES ON THIS SITE TO ENHANCE USER EXPERIENCE. We also use analytics. By navigating any page you are giving your consent for us to set cookies.    more information
Privacidad