# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] D. Chae and J. Lee, Global well-posedness in the super-critical dissipative quasi-geostrophic equations,, Comm. Math. Phys., 233 (2003), 297.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. P. Calderon and A Zygmund, On singular integrals,, American J of Math., 78 (1956), 289.  doi: 10.2307/2372517.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Laurent, Local and global existence for an aggregation equation,, Comm. in Parti. Diff. Equa., 32 (2007), 1941.  doi: 10.1080/03605300701318955.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] M. Schonbek, Decay of solutions to parabolic conservation laws,, Commun. Partial Diff Eqns., 5 (1980), 449.  doi: 10.1080/0360530800882145.  Google Scholar [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.  Google Scholar [23] M. Schonbek and T. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows,, SIAM J. Math. Anal., 35 (2003), 357.  doi: 10.1137/S0036141002409362.  Google Scholar [24] E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar [25] M. Taylor, Pseudodifferential Operators and Nonlinear P.D.E',, Birkhäuser, (1993).   Google Scholar [26] J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data,, Electron J. Differ. Eqns., (2001), 1.   Google Scholar [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.  Google Scholar [28] J. Wu, The Quasi-geostrophic equations and its two regularizations,, Comm. Partial Differ. Eqns., 27 (2002), 1161.  doi: 10.1081/PDE-120004898.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] D. Chae and J. Lee, Global well-posedness in the super-critical dissipative quasi-geostrophic equations,, Comm. Math. Phys., 233 (2003), 297.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. P. Calderon and A Zygmund, On singular integrals,, American J of Math., 78 (1956), 289.  doi: 10.2307/2372517.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Laurent, Local and global existence for an aggregation equation,, Comm. in Parti. Diff. Equa., 32 (2007), 1941.  doi: 10.1080/03605300701318955.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] M. Schonbek, Decay of solutions to parabolic conservation laws,, Commun. Partial Diff Eqns., 5 (1980), 449.  doi: 10.1080/0360530800882145.  Google Scholar [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.  Google Scholar [23] M. Schonbek and T. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows,, SIAM J. Math. Anal., 35 (2003), 357.  doi: 10.1137/S0036141002409362.  Google Scholar [24] E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar [25] M. Taylor, Pseudodifferential Operators and Nonlinear P.D.E',, Birkhäuser, (1993).   Google Scholar [26] J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data,, Electron J. Differ. Eqns., (2001), 1.   Google Scholar [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.  Google Scholar [28] J. Wu, The Quasi-geostrophic equations and its two regularizations,, Comm. Partial Differ. Eqns., 27 (2002), 1161.  doi: 10.1081/PDE-120004898.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197 [2] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [3] T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171 [4] May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179 [5] Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525 [6] T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119 [7] Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133 [8] Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361 [9] Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557 [10] Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277 [11] Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293 [12] Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152 [13] M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215 [14] Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure & Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987 [15] Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687 [16] Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047 [17] Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745 [18] Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541 [19] Jin Liang, James H. Liu, Ti-Jun Xiao. Nonlocal Cauchy problems for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 529-535. doi: 10.3934/cpaa.2006.5.529 [20] Young-Sam Kwon, Antonin Novotny. Derivation of geostrophic equations as a rigorous limit of compressible rotating and heat conducting fluids with the general initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 395-421. doi: 10.3934/dcds.2020015

2018 Impact Factor: 0.545