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Nonlinear maximum principles for dissipative linear nonlocal operators and applications. (English) Zbl 1256.35078
Summary: We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical \(d\)-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2D incompressible Euler equations and generalized fractional dissipative 2D Boussinesq equations.

35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q31 Euler equations
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[1] Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Communications in Mathematical Physics 94(1), 61–66 (1984) · Zbl 0573.76029 · doi:10.1007/BF01212349
[2] Bardos C., Titi E.S.: Euler equations of incompressible ideal fluids. Russian Mathematical Surveys 62(3), 409–451 (2007) · Zbl 1139.76010 · doi:10.1070/RM2007v062n03ABEH004410
[3] Caffarelli L., Vasseur A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Annals of Mathematics 71(3), 1903–1930 (2010) · Zbl 1204.35063 · doi:10.4007/annals.2010.171.1903
[4] C. Cao and J. Wu. Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation. Preprint arXiv:1108.2678v1 [math.AP]. · Zbl 1284.35140
[5] Chae D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Advances in Mathematics 203(2), 497–513 (2006) · Zbl 1100.35084 · doi:10.1016/j.aim.2005.05.001
[6] Chae D., Constantin P., Wu J.: Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations. Archive for Rational Mechanics and Analysis 202(1), 35–62 (2011) · Zbl 1266.76010 · doi:10.1007/s00205-011-0411-5
[7] J.Y. Chemin. Fluides parfaits incompressibles. In: Astérisque, Vol. 230. Soc. Math. France, Paris (1995). · Zbl 0829.76003
[8] P. Constantin. Euler equations, Navier–Stokes equations and turbulence. In: Mathematical Foundation of Turbulent Viscous Flows, 143, Lecture Notes in Math., Vol. 1871. Springer, Berlin (2006).
[9] Constantin P.: On the Euler equations of incompressible fluids. Bulletin of the American Mathematical Society (N.S.) 44(4), 603–621 (2007) · Zbl 1132.76009 · doi:10.1090/S0273-0979-07-01184-6
[10] Constantin P.: Singular, weak and absent: solutions of the Euler equations. Physics D 237(14–17), 1926–1931 (2008) · Zbl 1143.76388 · doi:10.1016/j.physd.2008.01.006
[11] Constantin P., Iyer G., Wu J.: Global regularity for a modified critical dissipative quasi-geostrophic equation. Indiana University Mathematics Journal 57(6), 2681–2692 (2008) · Zbl 1159.35059 · doi:10.1512/iumj.2008.57.3629
[12] Constantin P., Majda A.J., Tabak E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994) · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[13] Constantin P., Wu J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(6), 1103–1110 (2008) · Zbl 1149.76052 · doi:10.1016/j.anihpc.2007.10.001
[14] Constantin P., Wu J.: Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 159–180 (2009) · Zbl 1163.76010 · doi:10.1016/j.anihpc.2007.10.002
[15] Constantin P., Wu J.: Behavior of solutions to 2d quasigeostrophic equations. SIAM Journal of Mathmatical Analysis 30, 937–948 (1999) · Zbl 0957.76093 · doi:10.1137/S0036141098337333
[16] Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Communications in Mathematical Physics 249(3), 511–528 (2004) · Zbl 1309.76026 · doi:10.1007/s00220-004-1055-1
[17] Dabkowski M.: Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation. Geometric and Functional Analysis 21(1), 1–13 (2011) · Zbl 1210.35185 · doi:10.1007/s00039-011-0108-9
[18] Danchin R., Paicu M.: Global existence results for the anisotropic Boussinesq system in dimension two. Mathematical Models and Methods in Applied Sciences 21(3), 421–457 (2011) · Zbl 1223.35249 · doi:10.1142/S0218202511005106
[19] Hou T.Y., Li C.: Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems 12(1), 1–12 (2005) · Zbl 1274.76185
[20] Hmidi T., Keraani S., Rousset F.: Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation. Journal of Differential Equations 249(9), 2147–2174 (2010) · Zbl 1200.35228 · doi:10.1016/j.jde.2010.07.008
[21] Hmidi T., Keraani S., Rousset F.: Global well-posedness for Euler-Boussinesq system with critical dissipation. Communications in Partial Differential Equations 36(3), 420–445 (2011) · Zbl 1284.76089 · doi:10.1080/03605302.2010.518657
[22] Kiselev A., Nazarov F.: Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation. Nonlinearity 23(3), 549–554 (2010) · Zbl 1185.35190 · doi:10.1088/0951-7715/23/3/006
[23] Kiselev A., Nazarov F.: A variation on a theme of Caffarelli and Vasseur. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370, 58–72 (2009) · Zbl 1288.35393
[24] Kiselev A., Nazarov F., Volberg A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Inventiones Mathematicae, 67(3), 445–453 (2007) · Zbl 1121.35115 · doi:10.1007/s00222-006-0020-3
[25] Kiselev A., Nazarov F., Shterenberg R.: Blow up and regularity for fractal Burgers equations. Dynamics of PDE 5, 211–240 (2008) · Zbl 1186.35020
[26] Kozono H., Taniuchi Y.: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Communications in Mathematical Physics 214, 191–200 (2000) · Zbl 0985.46015 · doi:10.1007/s002200000267
[27] A. Larios, E. Lunasin and E.S. Titi. Global well-posedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid voigt-\(\alpha\) regularization. Preprint arXiv:1010.5024v1 [math.AP]. · Zbl 1284.35343
[28] A.J. Majda and A.L. Bertozzi. Vorticity and incompressible flow. In: Cambridge Texts in Applied Mathematics, Vol. 27. Cambridge University Press, Cambridge (2002). · Zbl 0983.76001
[29] S. Resnick. Dynamical problems in nonlinear advective partial differential equations. Ph.D. Thesis, University of Chicago (1995).
[30] Silvestre L.: Eventual regularization for the slightly supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 693–704 (2010) · Zbl 1187.35186 · doi:10.1016/j.anihpc.2009.11.006
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