October  2014, 7(5): 1111-1132. doi: 10.3934/dcdss.2014.7.1111

On one multidimensional compressible nonlocal model of the dissipative QG equations

1. 

College of Applied Sciences, Beijing University of Technology, PingLeYuan100, Chaoyang District, Beijing 100124, China, China, China

2. 

Basic Courses Department, Institute of Disaster Prevention, Yanjiao, Sanhe City, Hebei Province, 065201, China

Received  January 2013 Revised  June 2013 Published  May 2014

In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations and discuss the effect of the sign of initial data on the wellposedness of this model. First, we prove the existence and uniqueness of local smooth solutions for the Cauchy problem for the model with the nonnegative initial data, which seems to imply that whether the well-posedness of this model holds or not depends heavily upon the sign of the initial data even for the subcritical case. Secondly, for the sub-critical case $1<\alpha\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Next, we prove the global existence of the weak solution for $0<\alpha\le 2$ and $\nu>0$. Finally, for the sub-critical case $1<\alpha\leq 2$, we establish $H^\beta(\beta\geq 0)$ and $L^p(p\geq 2)$ decay rates of the smooth solution as $t\to\infty$. A inequality for the Riesz transformation is also established.
Citation: Shu Wang, Zhonglin Wu, Linrui Li, Shengtao Chen. On one multidimensional compressible nonlocal model of the dissipative QG equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1111-1132. doi: 10.3934/dcdss.2014.7.1111
References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation,, SIAM J. Math. Anal., 40 (2008), 167. doi: 10.1137/070682319.

[2]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with nonlocal fluxes,, Physics D, 91 (1996), 349. doi: 10.1016/0167-2789(95)00271-5.

[3]

P. Balodis and A. Córdoba, An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations,, Advances in Mathematics, 214 (2007), 1. doi: 10.1016/j.aim.2006.07.021.

[4]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1998), 845. doi: 10.1137/S0036139996313447.

[5]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math. (2), 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903.

[6]

L. Caffarelli and J. Vazquez, Nonlinear porous medium flow with fractional potential pressure,, Arch. Rational Mech. Anal., 202 (2011), 537. doi: 10.1007/s00205-011-0420-4.

[7]

A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux,, Advance in Mathematics, 219 (2008), 1916. doi: 10.1016/j.aim.2008.07.015.

[8]

A. Castro, D. Córdoba, F. Gancedo and R. Orive, Incompressible flow in porous media with fractional diffusion,, Nonlinearity, 22 (2009), 1791. doi: 10.1088/0951-7715/22/8/002.

[9]

D. Chae, A. Córdoba, D. Córdoba and M. A. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophis equations,, Advance in Mathematics, 194 (2005), 203. doi: 10.1016/j.aim.2004.06.004.

[10]

D. Chae and J. Lee, Global well-posedness in the super-critical dissipative quasi-geostrophic equations,, Comm. Math. Phys., 233 (2003), 297.

[11]

P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar,, Nonlinearity, 7 (1994), 1498. doi: 10.1088/0951-7715/7/6/001.

[12]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,, SIAM J. Math. Anal., 30 (1999), 937. doi: 10.1137/S0036141098337333.

[13]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Phys., 249 (2004), 511. doi: 10.1007/s00220-004-1055-1.

[14]

A. P. Calderon and A Zygmund, On singular integrals,, American J of Math., 78 (1956), 289. doi: 10.2307/2372517.

[15]

H. Dong and D. Du, Global well-posedness and a dacay estimate for the critical dissipative quasi-geostrophic equation in the whole space,, Discrete Contin. Dyn. Syst., 21 (2008), 1095. doi: 10.3934/dcds.2008.21.1095.

[16]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,, Invent. Math., 167 (2007), 445. doi: 10.1007/s00222-006-0020-3.

[17]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations,, Commun. Math. Phys., 255 (2005), 161. doi: 10.1007/s00220-004-1256-7.

[18]

T. Laurent, Local and global existence for an aggregation equation,, Comm. in Parti. Diff. Equa., 32 (2007), 1941. doi: 10.1080/03605300701318955.

[19]

D. Li and J. Rodrigo, Wellposedness and regularity of solutions of an aggregation equation,, Rev. Mat. Iberoam., 26 (2010), 261. doi: 10.4171/RMI/601.

[20]

D. Li, J. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem,, Rev. Mat. Iberoam., 26 (2010), 295. doi: 10.4171/RMI/602.

[21]

M. Schonbek, Decay of solutions to parabolic conservation laws,, Commun. Partial Diff Eqns., 5 (1980), 449. doi: 10.1080/0360530800882145.

[22]

M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 88 (1985), 209. doi: 10.1007/BF00752111.

[23]

M. Schonbek and T. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows,, SIAM J. Math. Anal., 35 (2003), 357. doi: 10.1137/S0036141002409362.

[24]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).

[25]

M. Taylor, Pseudodifferential Operators and Nonlinear P.D.E',, Birkhäuser, (1993).

[26]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data,, Electron J. Differ. Eqns., (2001), 1.

[27]

J. Wu, Global solutions of the 2D dissipative quasi-geostrophic in Besov spaces,, SIAM J. Math. Anal., 36 (2005), 1014. doi: 10.1137/S0036141003435576.

[28]

J. Wu, The Quasi-geostrophic equations and its two regularizations,, Comm. Partial Differ. Eqns., 27 (2002), 1161. doi: 10.1081/PDE-120004898.

[29]

X. Yu, Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic,, J. Math. Anal. Appl., 339 (2008), 359. doi: 10.1016/j.jmaa.2007.06.064.

[30]

Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows,, Nonlinearity, 21 (2008), 2061. doi: 10.1088/0951-7715/21/9/008.

show all references

References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation,, SIAM J. Math. Anal., 40 (2008), 167. doi: 10.1137/070682319.

[2]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with nonlocal fluxes,, Physics D, 91 (1996), 349. doi: 10.1016/0167-2789(95)00271-5.

[3]

P. Balodis and A. Córdoba, An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations,, Advances in Mathematics, 214 (2007), 1. doi: 10.1016/j.aim.2006.07.021.

[4]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1998), 845. doi: 10.1137/S0036139996313447.

[5]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math. (2), 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903.

[6]

L. Caffarelli and J. Vazquez, Nonlinear porous medium flow with fractional potential pressure,, Arch. Rational Mech. Anal., 202 (2011), 537. doi: 10.1007/s00205-011-0420-4.

[7]

A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux,, Advance in Mathematics, 219 (2008), 1916. doi: 10.1016/j.aim.2008.07.015.

[8]

A. Castro, D. Córdoba, F. Gancedo and R. Orive, Incompressible flow in porous media with fractional diffusion,, Nonlinearity, 22 (2009), 1791. doi: 10.1088/0951-7715/22/8/002.

[9]

D. Chae, A. Córdoba, D. Córdoba and M. A. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophis equations,, Advance in Mathematics, 194 (2005), 203. doi: 10.1016/j.aim.2004.06.004.

[10]

D. Chae and J. Lee, Global well-posedness in the super-critical dissipative quasi-geostrophic equations,, Comm. Math. Phys., 233 (2003), 297.

[11]

P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar,, Nonlinearity, 7 (1994), 1498. doi: 10.1088/0951-7715/7/6/001.

[12]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,, SIAM J. Math. Anal., 30 (1999), 937. doi: 10.1137/S0036141098337333.

[13]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Phys., 249 (2004), 511. doi: 10.1007/s00220-004-1055-1.

[14]

A. P. Calderon and A Zygmund, On singular integrals,, American J of Math., 78 (1956), 289. doi: 10.2307/2372517.

[15]

H. Dong and D. Du, Global well-posedness and a dacay estimate for the critical dissipative quasi-geostrophic equation in the whole space,, Discrete Contin. Dyn. Syst., 21 (2008), 1095. doi: 10.3934/dcds.2008.21.1095.

[16]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,, Invent. Math., 167 (2007), 445. doi: 10.1007/s00222-006-0020-3.

[17]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations,, Commun. Math. Phys., 255 (2005), 161. doi: 10.1007/s00220-004-1256-7.

[18]

T. Laurent, Local and global existence for an aggregation equation,, Comm. in Parti. Diff. Equa., 32 (2007), 1941. doi: 10.1080/03605300701318955.

[19]

D. Li and J. Rodrigo, Wellposedness and regularity of solutions of an aggregation equation,, Rev. Mat. Iberoam., 26 (2010), 261. doi: 10.4171/RMI/601.

[20]

D. Li, J. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem,, Rev. Mat. Iberoam., 26 (2010), 295. doi: 10.4171/RMI/602.

[21]

M. Schonbek, Decay of solutions to parabolic conservation laws,, Commun. Partial Diff Eqns., 5 (1980), 449. doi: 10.1080/0360530800882145.

[22]

M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 88 (1985), 209. doi: 10.1007/BF00752111.

[23]

M. Schonbek and T. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows,, SIAM J. Math. Anal., 35 (2003), 357. doi: 10.1137/S0036141002409362.

[24]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).

[25]

M. Taylor, Pseudodifferential Operators and Nonlinear P.D.E',, Birkhäuser, (1993).

[26]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data,, Electron J. Differ. Eqns., (2001), 1.

[27]

J. Wu, Global solutions of the 2D dissipative quasi-geostrophic in Besov spaces,, SIAM J. Math. Anal., 36 (2005), 1014. doi: 10.1137/S0036141003435576.

[28]

J. Wu, The Quasi-geostrophic equations and its two regularizations,, Comm. Partial Differ. Eqns., 27 (2002), 1161. doi: 10.1081/PDE-120004898.

[29]

X. Yu, Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic,, J. Math. Anal. Appl., 339 (2008), 359. doi: 10.1016/j.jmaa.2007.06.064.

[30]

Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows,, Nonlinearity, 21 (2008), 2061. doi: 10.1088/0951-7715/21/9/008.

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