Large time decay and growth for solutions of a viscous Boussinesq system

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as t→∞ in the sense that the energy and the L-norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from (u0, θ0) with zero-mean for the initial temperature θ0 have a special behavior as |x| or t tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

[1]  Maria E. Schonbek,et al.  Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation , 2007 .

[2]  L. Brandolese.  ASYMPTOTIC BEHAVIOR OF THE ENERGY AND POINTWISE ESTIMATES FOR SOLUTIONS TO THE NAVIER-STOKES EQUATIONS , 2004 .

[3]  L. Ferreira,et al.  Well-posedness and asymptotic behaviour for the convection problem in , 2006 .

[4]  R. Danchin,et al.  Les theoremes de Leray et de Fujita-Kato pour le systeme de Boussinesq partiellement visqueux. The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity , 2008, 0806.4083.

[5]  B. Jin,et al.  Weighted estimate of the asymptotic profiles of the Navier–Stokes flow in Rn , 2008 .

[6]  J. R. Cannon,et al.  The initial value problem for the Boussinesq equations with data in Lp , 1980 .

[7]  Taoufik Hmidi,et al.  On the global well-posedness for Boussinesq system , 2007 .

[8]  T. Miyakawa,et al.  On optimal decay rates for weak solutions to the Navier-Stokes equations in $R^n$ , 2001 .

[9]  G. Karch,et al.  Self-similarity in viscous Boussinesq equations , 2007 .

[11]  Antonio Córdoba,et al.  Communications in Mathematical Physics A Maximum Principle Applied to Quasi-Geostrophic Equations , 2004 .

[12]  Long-time asymptotics of the Navier-Stokes and vorticity equations on R^3 , 2001 .

[13]  T. Ogawa,et al.  Energy Decay for a Weak Solution of the Navier–Stokes Equation with Slowly Varying External Forces , 1997 .

[14]  Pierre Gilles Lemarié-Rieusset,et al.  Recent Developments in the Navier-Stokes Problem , 2002 .

[15]  Thierry Gallay,et al.  Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ3 , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  Dongho Chae,et al.  Global regularity for the 2D Boussinesq equations with partial viscosity terms , 2006 .

[17]  Tetsuro Miyakawa,et al.  OnL2 decay of weak solutions of the Navier-Stokes equations inRn , 1986 .

[18]  Min Chen,et al.  Long-time asymptotic behavior of two-dimensional dissipativeBoussinesq systems , 2009 .

[19]  T. Cazenave,et al.  Chaotic behavior of solutions of the Navier-Stokes system in $\Bbb R^N$ , 2005, Advances in Differential Equations.

[20]  Yong Zhou,et al.  A note on regularity criterion for the 3D Boussinesq system with partial viscosity , 2009, Appl. Math. Lett..

[21]  R. Danchin,et al.  Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces , 2008, 0806.4084.

[22]  J. Serrin.  On the interior regularity of weak solutions of the Navier-Stokes equations , 1962 .

[23]  Rafael de la Llave,et al.  On Squirt Singularities in Hydrodynamics , 2004, SIAM J. Math. Anal..

[24]  R. Danchin,et al.  Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data , 2008, 0806.4081.

[25]  K. Masuda.  Weak solutions of Navier-Stokes equations , 1984 .

[26]  Miguel Escobedo,et al.  Large time behavior for convection-diffusion equations in RN , 1991 .

[27]  H. Abidi,et al.  On the global regularity of axisymmetric Navier-Stokes-Boussinesq system , 2009, 0908.0894.

[28]  F. Rousset,et al.  Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data , 2009, 0912.1353.

[29]  On the suitable weak solutions for the Cauchy problem of the Boussinesq equations , 1996 .

[30]  R. Temam.  Navier-Stokes Equations , 1977 .

[31]  増田 久弥,et al.  Weak solutions of Navier-Stokes equations , 1985 .

[32]  Maria E. Schonbek,et al.  L2 decay for weak solutions of the Navier-Stokes equations , 1985 .

[33]  Y. Taniuchi,et al.  On the Boussinesq Flow with Nondecaying Initial Data , 2004 .

[34]  L. Brandolese,et al.  New asymptotic profiles of nonstationary solutions of the Navier–Stokes system , 2007, 0706.1489.

[35]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[36]  Michael Wiegner,et al.  Decay Results for Weak Solutions of the Navier–Stokes Equations on Rn , 1987 .

[37]  Maria E. Schonbek,et al.  Lower bounds of rates of decay for solutions to the Navier-Stokes equations , 1991 .

[38]  L. Brandolese.  Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations , 2008, 0803.0210.

[39]  Large-time behavior of solutions to a dissipative Boussinesq system , 2006, math/0607627.