Norm inflation for the Boussinesq system

Zongyuan Li; Weinan Wang

doi:10.3934/dcdsb.2020353

Article Contents

2021, Volume 26, Issue 10: 5449-5463. Doi: 10.3934/dcdsb.2020353

This issue Previous Article Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise Next Article On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type

Norm inflation for the Boussinesq system

Zongyuan Li^1, and
Weinan Wang^2,

1.
Department of Mathematics, Rutgers University, Hill Center - Busch Campus 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2.
Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson, AZ 85721, USA

Received: February 2020

Early access: November 2020

Published: October 2021

The second author was supported in part by NSF grant DMS-1907992

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Abstract

We prove the norm inflation phenomena for the Boussinesq system on $ \mathbb T^3 $. For arbitrarily small initial data $ (u_0,\rho_0) $ in the negative-order Besov spaces $ \dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty} $, the solution can become arbitrarily large in a short time. Such largeness can be detected in $ \rho $ in Besov spaces of any negative order: $ \dot{B}^{-s}_{\infty, \infty} $ for any $ s>0 $.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 76D33.

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