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Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. (English) Zbl 1204.35063
Motivated by the quasi-geostrophic model, the authors study the equation:
\[ \partial_{t}\theta+v\cdot \nabla \theta= -\Lambda \theta, \quad x\in \mathbb R^N,\qquad \operatorname{div} v=0, \] where \(\Lambda \theta = (-\Delta)^{1/2}\theta\) and they prove the following theorems:
(1) Let \(\theta(t,x)\) be a function in \(L^\infty(0,T;L^{2}(\mathbb R^N))\cap L^2(0,T;H^{1/2}(\mathbb R^N))\). For \(\lambda >0\), set \(\theta_\lambda:=(\theta-\lambda)_+\). If \(\theta\) (and \(-\theta\)) satisfies for every \(\lambda>0\) the level set energy inequalities
\[ \int_{\mathbb R^N} \theta_\lambda^2(t_2,x)\,dx+2 \int_{t_1}^{t_2}\int_{\mathbb R^N} |\Lambda^{1/2}\theta_\lambda|^2\,dx\,dt\leq \int_{\mathbb R^N} \theta_\lambda^2(t_1,x)\,dx, \quad 0<t_1<t_2, \] then
\[ \sup_{x\in\mathbb R^N}|\theta(T,x)|\leq C^* \frac{\|\theta_0\|_{L^2}}{T^{N/2}}, \] where \(C^*>0\) is a constant.
(2) Let \(Q_r=[-r,0]\times[-r,r]^N\), for \(r>0\). Assume that \(\theta(t,x)\) is bounded in \([-1,0]\times \mathbb R^N\) and \(v|_{Q_1}\in L^\infty(-1,0;\text{BMO})\); then \(\theta\) is \(C^\alpha\) in \(Q_{1/2}\).
From these two theorems, the regularity of solutions to the quasi-geostrophic equation follows.
(3) Let \(\theta\) be a solution to an equation
\[ \partial_{t}\theta+u\cdot\nabla \theta= -\Lambda \theta, \quad x\in\mathbb R^N,\qquad \operatorname{div} u=0, \] with \(u_j=\overline{R}_j[\theta]\), \(\overline{R}_j\) a singular integral operator. Assume also that \(\theta\) verifies the level set energy inequalities. Then, for every \(t_0>0\), there exists \(\alpha\) such that \(\theta\) is bounded in \(C^\alpha([t_0,\infty[\times\mathbb R^N)\).

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
42B37 Harmonic analysis and PDEs
86A99 Geophysics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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References:
[1] A. Córdoba, D. Córdoba, and M. A. Fontelos, ”Formation of singularities for a transport equation with nonlocal velocity,” Ann. of Math., vol. 162, iss. 3, pp. 1377-1389, 2005. · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377
[2] G. Duvaut and J. -L. Lions, Les inéquations en mécanique et en physique, Paris: Dunod, 1972. · Zbl 0298.73001
[3] A. Córdoba and D. Córdoba, ”A maximum principle applied to quasi-geostrophic equations,” Comm. Math. Phys., vol. 249, iss. 3, pp. 511-528, 2004. · Zbl 1309.76026 · doi:10.1007/s00220-004-1055-1
[4] P. Constantin, ”Euler equations, Navier-Stokes equations and turbulence,” in Mathematical foundation of turbulent viscous flows, New York: Springer-Verlag, 2006, pp. 1-43. · Zbl 1190.76146 · doi:10.1007/11545989_1
[5] P. Constantin, D. Cordoba, and J. Wu, ”On the critical dissipative quasi-geostrophic equation,” Indiana Univ. Math. J., vol. 50, iss. Special Issue, pp. 97-107, 2001. · Zbl 0989.86004 · doi:10.1512/iumj.2001.50.2153
[6] P. Constantin and J. Wu, ”Behavior of solutions to 2D quasi-geostrophic equations,” SIAM J. Math. Anal., vol. 30, pp. 937-948, 1999. · Zbl 0957.76093 · doi:10.1137/S0036141098337333
[7] D. Chae, ”On the regularity conditions for the dissipative quasi-geostrophic equations,” SIAM J. Math. Anal., vol. 37, iss. 5, pp. 1649-1656, 2006. · Zbl 1141.76010 · doi:10.1137/040616954
[8] D. Chae and J. Lee, ”Global well-posedness in the super-critical dissipative quasi-geostrophic equations,” Comm. Math. Phys., vol. 233, iss. 2, pp. 297-311, 2003. · Zbl 1019.86002 · doi:10.1007/s00220-002-0750-z
[9] D. Cordoba, ”Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation,” Ann. of Math., vol. 148, iss. 3, pp. 1135-1152, 1998. · Zbl 0920.35109 · doi:10.2307/121037 · www.math.princeton.edu · eudml:119907 · arxiv:math/9811184
[10] J. Wu, ”Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces,” SIAM J. Math. Anal., vol. 36, iss. 3, pp. 1014-1030, 2004/05. · Zbl 1083.76064 · doi:10.1137/S0036141003435576
[11] J. Wu, ”The quasi-geostrophic equation and its two regularizations,” Comm. Partial Differential Equations, vol. 27, iss. 5-6, pp. 1161-1181, 2002. · Zbl 1012.35067 · doi:10.1081/PDE-120004898
[12] J. Wu, ”Solutions of the 2D quasi-geostrophic equation in Hölder spaces,” Nonlinear Anal., vol. 62, iss. 4, pp. 579-594, 2005. · Zbl 1116.35348 · doi:10.1016/j.na.2005.03.053
[13] M. E. Schonbek and T. P. Schonbek, ”Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows,” Discrete Contin. Dyn. Syst., vol. 13, iss. 5, pp. 1277-1304, 2005. · Zbl 1091.35070 · doi:10.3934/dcds.2005.13.1277
[14] M. E. Schonbek and T. P. Schonbek, ”Asymptotic behavior to dissipative quasi-geostrophic flows,” SIAM J. Math. Anal., vol. 35, iss. 2, pp. 357-375, 2003. · Zbl 1126.76386 · doi:10.1137/S0036141002409362
[15] S. Resnick, ”Dynamical problems in nonlinear advective partial differential equations,” PhD Thesis , University of Chicago, 1995.
[16] A. Kiselev, F. Nazarov, and A. Volberg, ”Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,” Invent. Math., vol. 167, pp. 445-453, 2007. · Zbl 1121.35115 · doi:10.1007/s00222-006-0020-3 · arxiv:math/0604185
[17] A. Vasseur, ”A new proof of partial regularity of solutions to Navier-Stokes equations.,” Nonlinear Differential Equations Appl., vol. 14, pp. 753-785, 2007. · Zbl 1142.35066 · doi:10.1007/s00030-007-6001-4
[18] A. Mellet and A. Vasseur, ”\(L^p\) estimates for quantities advected by a compressible flow,” , preprint. · Zbl 1172.35056 · doi:10.1016/j.jmaa.2009.01.073
[19] E. De Giorgi, ”Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari,” Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., vol. 3, pp. 25-43, 1957. · Zbl 0084.31901
[20] L. Caffarelli and L. Silvestre, ”An extension problem related to the fractional laplacian,” Comm. Partial Differential Equations, vol. 32, pp. 1245-1260, 2007. · Zbl 1143.26002 · doi:10.1080/03605300600987306 · arxiv:math/0608640
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