# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  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  G. Duvaut and J. -L. Lions, Les inéquations en mécanique et en physique, Paris: Dunod, 1972. · Zbl 0298.73001  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  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  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  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  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  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  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  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  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  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  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  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  S. Resnick, ”Dynamical problems in nonlinear advective partial differential equations,” PhD Thesis , University of Chicago, 1995.  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  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  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  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  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.