Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
This article deals with the asymptotic behavior of the two-dimensional inviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time-decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.
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Keywords: Stratified fluids, Boussinesq approximation, Green function analysis, dispersion relation homogeneous of degree 0, anisotropic decay rates
@article{COCV_2021__27_1_A45_0,
author = {Bianchini, Roberta and Natalini, Roberto},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Asymptotic behavior of {2D} stably stratified fluids with a damping term in the velocity equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021045},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021045/}
}
TY - JOUR AU - Bianchini, Roberta AU - Natalini, Roberto ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021045/ DO - 10.1051/cocv/2021045 LA - en ID - COCV_2021__27_1_A45_0 ER -
%0 Journal Article %A Bianchini, Roberta %A Natalini, Roberto %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021045/ %R 10.1051/cocv/2021045 %G en %F COCV_2021__27_1_A45_0
Bianchini, Roberta; Natalini, Roberto. Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 43. doi: 10.1051/cocv/2021045
[1] , and , Fourier analysis and nonlinear partial differential equations. Vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011).
[2] and , Large time asymptotics for partially dissipative hyperbolic systems. Arch. Ration. Mech. Anal. 199 (2011) 177–227.
[3] , Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling. SIAM J. Math. Anal. 50 (2018) 1877–1899.
[4] , and , Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime. Preprint (2020). | arXiv
[5] , and , Near-critical reflection of internal waves. Anal. PDE 14 (2021) 205–249.
[6] , and , Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60 (2007) 1559–1622.
[7] , and , On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term. Math. Models Methods Appl. Sci. 29 (2019) 1227–1277.
[8] , Global well-posedness and asymptotics for a geophysical fluid system. Commun. Partial Differ. Equ. 29 (2004) 1919–1940.
[9] , , and , Mathematical geophysics. An introduction to rotating fluids and the Navier-Stokes equations. Vol. 32 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, Oxford (2006).
[10] and , Global existence results for the anisotropic Boussinesq system in dimension two. Math. Models Methods Appl. Sci. 21 (2011) 421–457.
[11] , and , Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids. arXiv e-prints (2019).
[12] , , and , Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion. Phys. D 376/377 (2018) 144–159.
[13] , On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation. Arch. Ration. Mech. Anal. 225 (2017) 573–599.
[14] and , Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems. SIAM J. Math. Anal. 47 (2015) 4672–4684.
[15] and , Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Ration. Mech. Anal. 169 (2003) 89–117.
[16] , Perturbation theory for linear operator. Vol. 132 of 2nd ed. Grundlehren der Mathematischen Wissenschaften. Springer, New York (1976).
[17] , Fluid Dynamics: An Introduction. Graduate Texts in Physics. Springer International Publishing (2015).
[18] and , Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14 (1985) 249–275.
[19] , , and , Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion. Arch. Ration. Mech. Anal. 237 (2020) 585–630.
[20] , Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete Contin. Dyn. Syst. 39 (2019) 2709–2730.
[21] , Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space H$$. Commun. Contemp. Math. 22 (2020) 1850063.
[22] and , Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Z. Angew. Math. Phys. 67 (2016) 104.
[23] , Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172 (2004) 247–266.
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