Finite time singularity for the modified SQG patch equation

Annals of Mathematics

Volumn 184, Issue 3, 2016, Pages 909-948

  Kiselev, Alexander ^a   Ryzhik, Lenya ^b   Yao, Yao ^c   Zlatoš, Andrej ^d,e  

^a RICE UNIVERSITY   (United States)

^b STANFORD UNIVERSITY   (United States)

^c GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^d UNIVERSITY OF WISCONSIN MADISON   (United States)

^e UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85008690042     PISSN: 0003486X     EISSN: 19398980     Source Type: Journal    
DOI: 10.4007/annals.2016.184.3.7     Document Type: Article

References (56)

1. A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152 (1993), 19-28. MR 1207667. Zbl 0771.76014. Available at http://projecteuclid.org/euclid.cmp/1104252307.
2. T. F. Buttke, The observation of singularities in the boundary of patches of constant vorticity, Physics of Fluids A: Fluid Dynamics 1 (1989), 1283-1285. http://dx.doi.org/10.1063/1.857353.
3. L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. 171 (2010), 1903-1930. MR 2680400. Zbl 1204.35063. http://dx.doi.org/10.4007/annals.2010.171.1903.
4. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, and J. Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. 178 (2013), 1061-1134. MR 3092476. Zbl 1291.35199. http://dx.doi.org/10.4007/annals.2013.178.3.6.
5. Á. Castro, D. Córdoba, C. Fefferman, F. Gancedo, and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. 175 (2012), 909-948. MR 2993754. Zbl 1267.76033. http://dx.doi.org/10.4007/annals.2012.175.2.9.
6. D. Chae, P. Constantin, D. Córdoba, F. Gancedo, and J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math. 65 (2012), 1037-1066. MR 2928091. Zbl 1244.35108. http://dx.doi.org/10.1002/cpa.21390.
7. J.-Y. Chemin, Persistance de structures géométriques dans les uides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup. 26 (1993), 517-542. MR 1235440. Zbl 0779.76011. Available at http://www.numdam.org/item?id=ASENS 1993 4 26 4 517 0.
8. K. Choi, T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao, On the finite-time blowup of a 1D model for the 3D axisymmetric Euler equations, 2014. arXiv 1407.4776.
9. P. Constantin, G. Iyer, and J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 57 (2008), 2681-2692. MR 2482996. Zbl 1159.35059. http://dx.doi.org/10.1512/iumj.2008.57.3629.
10. P. Constantin, A. J. Majda, and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), 1495-1533. MR 1304437. Zbl 0809.35057. Available at http://stacks.iop.org/0951-7715/7/1495.
11. P. Constantin, A. Tarfulea, and V. Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys. 335 (2015), 93-141. MR 3314501. Zbl 1316. 35238. http://dx.doi.org/10.1007/s00220-014-2129-3.
12. P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal. 22 (2012), 1289-1321. MR 2989434. Zbl 1256.35078. http://dx.doi.org/10.1007/s00039-012-0172-9.
13. A. Córdoba, D. Córdoba, and F. Gancedo, Interface evolution: the Hele-Shaw and Muskat problems, Ann. of Math. 173 (2011), 477-542. MR 2753607.Zbl 1229.35204. http://dx.doi.org/10.4007/annals.2011.173.1.10.
14. D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasigeostrophic equation, Ann. of Math. 148 (1998), 1135-1152. MR 1670077. Zbl 0920.35109. http://dx.doi.org/10.2307/121037.
15. D. Cordoba and C. Fefferman, Growth of solutions for QG and 2D Euler equations, J. Amer. Math. Soc. 15 (2002), 665-670. MR 1896236. Zbl 1013. 76011. http://dx.doi.org/10.1090/S0894-0347-02-00394-6.
16. D. Córdoba, M. A. Fontelos, A. M. Mancho, and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005), 5949-5952. MR 2141918. Zbl 1135.76315. http://dx.doi.org/10.1073/pnas.0501977102.
17. M. Dabkowski, A. Kiselev, L. Silvestre, and V. Vicol, Global wellposedness of slightly supercritical active scalar equations, Anal. PDE 7 (2014), 43-72. MR 3219499. Zbl 1294.35092. http://dx.doi.org/10.2140/apde.2014.7.43.
18. S. A. Denisov, Infinite superlinear growth of the gradient for the twodimensional Euler equation, Discrete Contin. Dyn. Syst. 23 (2009), 755-764. MR 2461825. Zbl 1156.76009. http://dx.doi.org/10.3934/dcds.2009.23.755.
19. S. A. Denisov, Double exponential growth of the vorticity gradient for the two-dimensional Euler equation, Proc. Amer. Math. Soc. 143 (2015), 1199-1210. MR 3293735. Zbl 1315.35150. http://dx.doi.org/10.1090/S0002-9939-2014-12286-6.
20. N. Depauw, Poche de tourbillon pour Euler 2D dans un ouvert à bord, J. Math. Pures Appl. 78 (1999), 313-351. MR 1687165. Zbl 0927.76014. http://dx.doi. org/10.1016/S0021-7824(98)00003-8.
21. D. G. Dritschel and M. E. McIntyre, Does contour dynamics go singular?, Phys. Fluids A 2 (1990), 748-753. MR 1050012. http://dx.doi.org/10.1063/1. 857728.
22. D. G. Dritschel and N. J. Zabusky, A new, but flawed, numerical method for vortex patch evolution in two dimensions, J. Comput. Phys. 93 (1991), 481-484. MR 1104362. Zbl 0726.76029. http://dx.doi.org/10.1016/0021-9991(91)90197-S.
23. A. Dutrifoy, On 3-D vortex patches in bounded domains, Comm. Partial Differential Equations 28 (2003), 1237-1263. MR 1998937. Zbl 1030.76011. http://dx.doi.org/10.1081/PDE-120024362.
24. R. Finn and D. Gilbarg, Asymptotic behavior and uniquenes of plane subsonic flows, Comm. Pure Appl. Math. 10 (1957), 23-63. MR 0086556. Zbl 0077.18801. http://dx.doi.org/10.1002/cpa.3160100102.
25. F. Gancedo, Existence for the α-patch model and the QG sharp front in Sobolev spaces, Adv. Math. 217 (2008), 2569-2598. MR 2397460. Zbl 1148.35099. http: //dx.doi.org/10.1016/j.aim.2007.10.010.
26. F. Gancedo and R. M. Strain, Absence of splash singularities for surface quasigeostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014), 635-639. MR 3181769. http://dx.doi.org/10.1073/pnas.1320554111.
27. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, New York, 2001. MR 1814364. Zbl 1042.35002.
28. R. Hardt and L. Simon, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math. 110 (1979), 439-486. MR 0554379. Zbl 0457.49029. http://dx.doi.org/10.2307/1971233.
29. E. Hölder, Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Flüssigkeit, Math. Z. 37 (1933), 727-738. MR 1545431. Zbl 0008.06902. http://dx.doi.org/10.1007/BF01474611.
30. N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys. 251 (2004), 365-376. MR 2100059. Zbl 1106.35061. http://dx.doi.org/10.1007/s00220-004-1062-2.
31. V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 1032-1066. MR 0158189. http://dx.doi.org/10.1016/0041-5553(63)90247-7.
32. V. I. Judovič, The loss of smoothness of the solutions of Euler equations with time, Dinamika Splošn. Sredy (1974), 71-78, 121. MR 0454419.
33. V. I. Judovič, On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid, Chaos 10 (2000), 705-719. MR 1791984. Zbl 0982.76014. http://dx.doi.org/10.1063/1.1287066.
34. A. Kiselev and F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370 (2009), 58-72, 220. MR 2749211. Zbl 1288.35393. http://dx.doi.org/10.1007/s10958-010-9842-z.
35. A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010), 549-554. MR 2586369. Zbl 1185.35190. http://dx.doi.org/10.1088/0951-7715/23/3/006.
36. A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), 445-453. MR 2276260. Zbl 1121.35115. http://dx.doi.org/10.1007/s00222-006-0020-3.
37. A. Kiselev and V. Šverák, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. of Math. 180 (2014), 1205-1220. MR 3245016. Zbl 1304.35521. http://dx.doi.org/10.4007/annals.2014.180.3.9.
38. A. Kiselev, Y. Yao, and A. Zlatoš, Local regularity for the modified SQG patch equation, preprint.
39. G. Luo and T. Y. Hou, Potentially singular solutions of the 3D axisymmetric Euler equations, Proc. Nat. Acad. Sci. USA 111 (2014), 12968-12973. http://dx.doi.org/10.1073/pnas.1405238111.
40. G. Luo and T. Y. Hou, Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation, Multiscale Model. Simul. 12 (2014), 1722-1776. MR 3278833. Zbl 1316.35235. http://dx.doi.org/10.1137/140966411.
41. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (1986), S187-S220. MR 0861488. Zbl 0595.76021. http://dx.doi.org/10.1002/cpa.3160390711.
42. A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts Appl. Math. 27, Cambridge Univ. Press, Cambridge, 2001. MR 1867882. Zbl 0983.76001. http://dx.doi.org/10.1017/CBO9780511613203.
43. A. M. Mancho, Numerical studies on the self-similar collapse of the α-patches problem, Commun. Nonlinear Sci. Numer. Simul. 26 (2015), 152-166. MR 3332042. http://dx.doi.org/10.1016/j.cnsns.2015.02.009.
44. C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Appl. Math. Sci. 96, Springer-Verlag, New York, 1994. MR 1245492. Zbl 0789.76002. http://dx.doi.org/10.1007/978-1-4612-4284-0.
45. N. S. Nadirashvili,Wandering solutions of the two-dimensional Euler equation, Funktsional. Anal. i Prilozhen. 25 (1991), 70-71. MR 1139875. Zbl 0769.35048. http://dx.doi.org/10.1007/BF01085491.
46. J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Zbl 0713.76005. http://dx.doi.org/10.1007/978-1-4612-4650-3.
47. R. T. Pierrehumbert, I. M. Held, and K. L. Swanson, Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons Fractals 4 (1994), 1111- 1116. Zbl 0823.76034.
48. D. I. Pullin, Contour dynamics methods, in Annual Review of Fluid Mechanics, Vol. 24, Annual Reviews, Palo Alto, CA, 1992, pp. 89-115. MR 1145007. Zbl 0743.76021.
49. S. G. Resnick, Dynamical Problems in Non-linear Advective Partial Differential Equations, ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.), The University of Chicago. MR 2716577. Available at http://gateway.proquest.com/openurl?url ver=Z39.88-2004&rft val fmt=info:ofi/fmt:kev:mtx:dissertation&res dat=xri:pqdiss&rft dat=xri:pqdiss:9542767.
50. J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math. 58 (2005), 821-866. MR 2142632. Zbl 1073. 35006. http://dx.doi.org/10.1002/cpa.20059.
51. K. S. Smith, G. Boccaletti, C. C. Henning, I. Marinov, C. Y. Tam, I. M. Held, and G. K. Vallis, Turbulent diffusion in the geostrophic inverse cascade, J. Fluid Mech. 469 (2002), 13-48. MR 1932826. Zbl 1152.76402. http://dx.doi.org/10.1017/S0022112002001763.
52. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095. Zbl 0207.13501.
53. W. Wolibner, Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z. 37 (1933), 698-726. MR 1545430. Zbl 0008.06901. http://dx.doi.org/10.1007/BF01474610.
54. S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math. 177 (2009), 45-135. MR 2507638. Zbl 1181.35205. http://dx.doi.org/10. 1007/s00222-009-0176-8.
55. S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math. 184 (2011), 125-220. MR 2782254. Zbl 1221.35304. http://dx.doi.org/10.1007/ s00222-010-0288-1.
56. A. Zlatoš, Exponential growth of the vorticity gradient for the Euler equation on the torus, Adv. Math. 268 (2015), 396-403. MR 3276599. Zbl 1308.35194. http://dx.doi.org/10.1016/j.aim.2014.08.012.