Nonlocal nonlinear advection-diffusion equations

Peter Constantin

Chinese Annals of Mathematics, Series B ›› 2017, Vol. 38 ›› Issue (1) : 281 -292.

PDF
Chinese Annals of Mathematics, Series B ›› 2017, Vol. 38 ›› Issue (1) : 281 -292. DOI: 10.1007/s11401-016-1071-4
Article

Nonlocal nonlinear advection-diffusion equations

Author information +
History +
PDF

Abstract

The author reviews some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian.

Keywords

Nonlocal / Advection / Diffusion / Fractional Laplacian

Cite this article

Download citation ▾
Peter Constantin. Nonlocal nonlinear advection-diffusion equations. Chinese Annals of Mathematics, Series B, 2017, 38(1): 281-292 DOI:10.1007/s11401-016-1071-4

登录浏览全文

4963

注册一个新账户 忘记密码

References

[1]

Alibaud N., Droniou J., Vovelle J.. Occurence and non-appearance of shocks in fractal Burgers equations. J. Hyp. Diff. Eqns., 2007, 4: 479-499

[2]

Beale J. T., Kato T., Majda A.. Remarks on the breakdown of smooth solutions for the 3D Euler equations. Comm. Math. Phys., 1984, 94: 61-66

[3]

Cabre X., Tan J.. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math., 2010, 224(5): 2052-2093

[4]

Caffarelli L., Vasseur A.. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Annals of Mathematics, 2010, 171: 1903-1930

[5]

Chae D., Constantin P., Wu J.. Inviscid models generalizing the 2D Euler and the surface quasigeostrophic equations. ARMA, 2011, 202: 35-62

[6]

Chae D., Constantin P., Córdoba D. Generalized surface quasi-geostrophic equations with singular velocities. CPAM, 2012, 65(8): 1037-1066

[7]

Constantin, P. and Ignatova, M., Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications, to appear. arXiv: 1511.00147[math.AP]

[8]

Constantin P., Iyer G., Wu J.. Global regularity for a modified quasi-geostrophic equation. Indiana University Mathematics Journal, 2008, 57: 2681-2692

[9]

Constantin P., Majda A., Tabak E.. Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity, 1994, 7: 1495-1533

[10]

Constantin P., Tarfulea A., Vicol V.. Long time dynamics of forced critical SQG. Commun. Math. Phys., 2015, 335: 93-141

[11]

Constantin P., Vicol V.. Nonlinear maximum principles for dissipative linear nonlocal operators and applications. GAFA, 2012, 22: 1289-1321

[12]

Constantin P., Wu J.. Regularity of Hölder continuous solutions of the supercritical quasigeostrophic equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire, 2008, 25: 1103-1110

[13]

Constantin P., Wu J.. Hölder continuity of solutions of supercritical dissipative hydrodynamice transport equations, Ann. Inst. Henri Poincaré. Analyse Non Linéaire, 2009, 26: 159-180

[14]

Córdoba A., Córdoba D.. A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys., 2004, 249: 511-528

[15]

Coti-Zelati, M. and Vicol, V., On the global regularity for the supercritical SQG equation, Indiana University Math. J., to appear. arXiv: 1410.3186[math.AP]

[16]

Dabkowski M., Kiselev A., Vicol V.. Global well-posedness for a slightly supercritical surface quasigeostrophic equation. Nonlinearity, 2012, 25: 1525-1535

[17]

Dabkowski M., Kiselev A., Silvestre L., Vicol V.. Global well-posedness of slightly supercritical active scalar equations. Analysis and PDE, 2014, 7: 43-72

[18]

Dabkowski M.. Eventual regularity of solutions to the supercritical dissipative quasi-geostrophic equation. GAFA, 2011, 21: 1-13

[19]

Dong H., Pavlovic N.. A regularity criterion for the dissipative quasi-geostrophic equations. Annales de l’Institut Henri Poincaré (C) Non Linear Analysis, 2007, 26: 1607-1619

[20]

Held I., Pierrehumbert R., Garner S., Swanson K.. Surface quasi-geostrophic dynamics. J. Fluid. Mech., 1995, 282: 1-20

[21]

Kiselev A., Nazarov F., Volberg A.. Global well-posedness for the critical 2D dissipative quasigeostrophic equation. Invent. Math., 2007, 167: 445-453

[22]

Kiselev A., Nazarov F.. A variation on a theme of Caffarelli and Vasseur. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 2009, 370: 58-72

[23]

Kiselev A., Nazarov F., Shterenberg R.. Blow up and regularity for fractal Burgers equation. Dyn. PDE, 2008, 5: 211-240

[24]

Kiselev A.. Nonlocal maximum principles for active scalars. Adv. Math., 2011, 227: 1806-1826

[25]

Resnick S. G.. Dynamical problems in non-linear advective partial differential equations, 1995, MI: The University of Chicago, ProQuest LLC, Ann. Arbor.

[26]

Silvestre, L., Hölder estimates for advection fractional diffusion equations, Ann. Sc. Norm. Pisa, to appear. arXiv: 1009.5723(AP)

[27]

Silvestre L.. Eventual regularization for the slightly supercritical quasi-geostrophic equation. Annales de l’Institut Henri Poincaré (C) Anal. Non Linéaire, 2010, 27: 693-704

[28]

Silvestre L., Vicol V., Zlatos A.. On the loss of continuity for super-critical drift-diffusion equations. ARMA, 2013, 207: 845-877

[29]

Xue L., Zheng X.. Note on the well-posedness of a slightly supercritical surface quasi-geostrophic equation. J. Diff. Eqns., 2012, 253: 795-813

AI Summary AI Mindmap
PDF

172

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/