Finite time singularity for the modified SQG patch equation. (English) Zbl 1360.35159

Kiselev, Alexander; Ryzhik, Lenya; Yao, Yao; Zlatoš, Andrej

Ann. Math. (2) 184, No. 3, 909-948 (2016).

The authors of the present paper consider the question of finite time singularity formation for the modified SQG patch equation in the half-plane.
It is well-known that the 2D incompressible Euler equations have globally regular solutions. For a related system, the inviscid surface quasi-geostrophic (SQG) equation, the question of global regularity for its solutions is still open. The modified SQG equation can be seen as an interpolation between the two previous models, hence posing the same question is natural and important.
The modified SQG equation can be described via the equation \[ \partial_t\omega+(u\cdot\nabla)\omega=0 \] with the initial condition \(\omega(\cdot,0)=\omega_0\) and the Biot-Savart law for the velocity \[ u:=\nabla^\perp(-\Delta)^{-1+\alpha}\omega, \] where \(\alpha\in[0,1/2]\) is a parameter. The values \(\alpha=0\) and \(\alpha=1/2\) correspond to the 2D Euler (in vorticity formulation) and to the SQG equations, respectively.
The authors work with a special class of solutions, called vortex patches. These can be written as \[ \omega(x,t)=\sum_k\theta_k\chi_{\Omega_k(t)}(x), \] where \(\theta_j\) are constants and \(\Omega_j(t)\) are time-evolving open sets with nonzero mutual distances and smooth boundaries.
The main results of the authors in the half-plane read as follows:

1.

For \(\alpha=0\) (i.e., Euler case) and any \(\gamma\in(0,1]\), for each \(C^{1,\gamma}\) patch initial data \(\omega_0,\) there exists a unique global in time \(C^{1,\gamma}\) patch solution of the previous system with \(\omega(\cdot,0)=\omega_0\).

2.

For any \(\alpha\in(0,1/24)\) there are \(H^3\) patch-like initial data \(\omega_0\) for which the unique global \(H^3\) patch solution of the above system with \(\omega(\cdot,0)=\omega_0\) becomes singular in finite time.

This well-written paper adds an important and beautiful contribution to the theory of fluid dynamics and PDEs in particular.

Reviewer: Alpár R. Mészáros (Los Angeles)

MSC:

35Q31	Euler equations
35Q86	PDEs in connection with geophysics
35B65	Smoothness and regularity of solutions to PDEs
76B03	Existence, uniqueness, and regularity theory for incompressible inviscid fluids
86A05	Hydrology, hydrography, oceanography

Keywords:

Euler equations; finite time singularity; global regularity; modified SQG equations; patches in domains with boundary

References:

[1]	A. L. Bertozzi and P. Constantin, ”Global regularity for vortex patches,” Comm. Math. Phys., vol. 152, iss. 1, pp. 19-28, 1993. · Zbl 0771.76014
[2]	T. F. Buttke, ”The observation of singularities in the boundary of patches of constant vorticity,” Physics of Fluids A: Fluid Dynamics, vol. 1, pp. 1283-1285, 1989.
[3]	L. A. Caffarelli and A. Vasseur, ”Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,” Ann. of Math., vol. 171, iss. 3, pp. 1903-1930, 2010. · Zbl 1204.35063
[4]	&. 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., vol. 175, iss. 2, pp. 909-948, 2012. · Zbl 1267.76033
[5]	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., vol. 178, iss. 3, pp. 1061-1134, 2013. · Zbl 1291.35199
[6]	D. Chae, P. Constantin, D. Córdoba, F. Gancedo, and J. Wu, ”Generalized surface quasi-geostrophic equations with singular velocities,” Comm. Pure Appl. Math., vol. 65, iss. 8, pp. 1037-1066, 2012. · Zbl 1244.35108
[7]	J. Chemin, ”Persistance de structures géométriques dans les fluides incompressibles bidimensionnels,” Ann. Sci. École Norm. Sup., vol. 26, iss. 4, pp. 517-542, 1993. · Zbl 0779.76011
[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. · Zbl 1377.35218
[9]	P. Constantin, G. Iyer, and J. Wu, ”Global regularity for a modified critical dissipative quasi-geostrophic equation,” Indiana Univ. Math. J., vol. 57, iss. 6, pp. 2681-2692, 2008. · Zbl 1159.35059
[10]	P. Constantin, A. J. Majda, and E. Tabak, ”Formation of strong fronts in the \(2\)-D quasigeostrophic thermal active scalar,” Nonlinearity, vol. 7, iss. 6, pp. 1495-1533, 1994. · Zbl 0809.35057
[11]	P. Constantin and V. Vicol, ”Nonlinear maximum principles for dissipative linear nonlocal operators and applications,” Geom. Funct. Anal., vol. 22, iss. 5, pp. 1289-1321, 2012. · Zbl 1256.35078
[12]	P. Constantin, A. Tarfulea, and V. Vicol, ”Long time dynamics of forced critical SQG,” Comm. Math. Phys., vol. 335, iss. 1, pp. 93-141, 2015. · Zbl 1316.35238
[13]	A. Córdoba, D. Córdoba, and F. Gancedo, ”Interface evolution: the Hele-Shaw and Muskat problems,” Ann. of Math., vol. 173, iss. 1, pp. 477-542, 2011. · Zbl 1229.35204
[14]	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
[15]	D. Cordoba and C. Fefferman, ”Growth of solutions for QG and 2D Euler equations,” J. Amer. Math. Soc., vol. 15, iss. 3, pp. 665-670, 2002. · Zbl 1013.76011
[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, vol. 102, iss. 17, pp. 5949-5952, 2005. · Zbl 1135.76315
[17]	M. Dabkowski, A. Kiselev, L. Silvestre, and V. Vicol, ”Global well-posedness of slightly supercritical active scalar equations,” Anal. PDE, vol. 7, iss. 1, pp. 43-72, 2014. · Zbl 1294.35092
[18]	S. A. Denisov, ”Infinite superlinear growth of the gradient for the two-dimensional Euler equation,” Discrete Contin. Dyn. Syst., vol. 23, iss. 3, pp. 755-764, 2009. · Zbl 1156.76009
[19]	S. A. Denisov, ”Double exponential growth of the vorticity gradient for the two-dimensional Euler equation,” Proc. Amer. Math. Soc., vol. 143, iss. 3, pp. 1199-1210, 2015. · Zbl 1315.35150
[20]	N. Depauw, ”Poche de tourbillon pour Euler 2D dans un ouvert à bord,” J. Math. Pures Appl., vol. 78, iss. 3, pp. 313-351, 1999. · Zbl 0927.76014
[21]	D. G. Dritschel and M. E. McIntyre, ”Does contour dynamics go singular?,” Phys. Fluids A, vol. 2, iss. 5, pp. 748-753, 1990.
[22]	D. G. Dritschel and N. J. Zabusky, ”A new, but flawed, numerical method for vortex patch evolution in two dimensions,” J. Comput. Phys., vol. 93, iss. 2, pp. 481-484, 1991. · Zbl 0726.76029
[23]	A. Dutrifoy, ”On 3-D vortex patches in bounded domains,” Comm. Partial Differential Equations, vol. 28, iss. 7-8, pp. 1237-1263, 2003. · Zbl 1030.76011
[24]	R. Finn and D. Gilbarg, ”Asymptotic behavior and uniquenes of plane subsonic flows,” Comm. Pure Appl. Math., vol. 10, pp. 23-63, 1957. · Zbl 0077.18801
[25]	F. Gancedo, ”Existence for the \(\alpha\)-patch model and the QG sharp front in Sobolev spaces,” Adv. Math., vol. 217, iss. 6, pp. 2569-2598, 2008. · Zbl 1148.35099
[26]	F. Gancedo and R. M. Strain, ”Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem,” Proc. Natl. Acad. Sci. USA, vol. 111, iss. 2, pp. 635-639, 2014. · Zbl 1355.76065
[27]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 2001. · Zbl 1042.35002
[28]	R. Hardt and L. Simon, ”Boundary regularity and embedded solutions for the oriented Plateau problem,” Ann. of Math., vol. 110, iss. 3, pp. 439-486, 1979. · Zbl 0457.49029
[29]	E. Hölder, ”Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Flüssigkeit,” Math. Z., vol. 37, iss. 1, pp. 727-738, 1933. · Zbl 0008.06902
[30]	N. Ju, ”Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space,” Comm. Math. Phys., vol. 251, iss. 2, pp. 365-376, 2004. · Zbl 1106.35061
[31]	V. I. Judovivc, ”The loss of smoothness of the solutions of Euler equations with time,” Dinamika Splo\vsn. Sredy, iss. Vyp. 16 Nestacionarnye Problemy Gidrodinamiki, pp. 71-78, 121, 1974.
[32]	A. Kiselev and F. Nazarov, ”Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation,” Nonlinearity, vol. 23, iss. 3, pp. 549-554, 2010. · Zbl 1185.35190
[33]	A. Kiselev and F. Nazarov, ”A variation on a theme of Caffarelli and Vasseur,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. \((\)POMI\()\), vol. 370, iss. Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 40, pp. 58-72, 220, 2009. · Zbl 1288.35393
[34]	A. Kiselev, F. Nazarov, and A. Volberg, ”Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,” Invent. Math., vol. 167, iss. 3, pp. 445-453, 2007. · Zbl 1121.35115
[35]	A. Kiselev and V. vSverák, ”Small scale creation for solutions of the incompressible two-dimensional Euler equation,” Ann. of Math., vol. 180, iss. 3, pp. 1205-1220, 2014. · Zbl 1304.35521
[36]	A. Kiselev, Y. Yao, and A. Zlatovs, Local regularity for the modified SQG patch equation. · Zbl 1371.35220
[37]	G. Luo and T. Y. Hou, ”Potentially singular solutions of the 3D axisymmetric Euler equations,” Proc. Nat. Acad. Sci. USA, vol. 111, pp. 12968-12973, 2014.
[38]	G. Luo and T. Y. Hou, ”Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation,” Multiscale Model. Simul., vol. 12, iss. 4, pp. 1722-1776, 2014. · Zbl 1316.35235
[39]	A. Majda, ”Vorticity and the mathematical theory of incompressible fluid flow,” Comm. Pure Appl. Math., vol. 39, iss. S, suppl., p. s187-s220, 1986. · Zbl 0595.76021
[40]	A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge: Cambridge Univ. Press, 2001, vol. 27. · Zbl 0983.76001
[41]	A. M. Mancho, ”Numerical studies on the self-similar collapse of the \(\alpha\)-patches problem,” Commun. Nonlinear Sci. Numer. Simul., vol. 26, iss. 1-3, pp. 152-166, 2015.
[42]	C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, New York: Springer-Verlag, 1994, vol. 96. · Zbl 0789.76002
[43]	N. S. Nadirashvili, ”Wandering solutions of the two-dimensional Euler equation,” Funktsional. Anal. i Prilozhen., vol. 25, iss. 3, pp. 70-71, 1991. · Zbl 0769.35048
[44]	J. Pedlosky, Geophysical Fluid Dynamics, New York: Springer-Verlag, 1987. · Zbl 0713.76005
[45]	R. T. Pierrehumbert, I. M. Held, and K. L. Swanson, ”Spectra of local and nonlocal two-dimensional turbulence,” Chaos, Solitons Fractals, vol. 4, pp. 1111-1116, 1994. · Zbl 0823.76034
[46]	D. I. Pullin, ”Contour dynamics methods,” in Annual Review of Fluid Mechanics, Vol. 24, Palo Alto, CA: Annual Reviews, 1992, pp. 89-115. · Zbl 0743.76021
[47]	S. G. Resnick, Dynamical Problems in Non-linear Advective Partial Differential Equations, Ann Arbor, MI: ProQuest LLC, 1995.
[48]	J. L. Rodrigo, ”On the evolution of sharp fronts for the quasi-geostrophic equation,” Comm. Pure Appl. Math., vol. 58, iss. 6, pp. 821-866, 2005. · Zbl 1073.35006
[49]	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., vol. 469, pp. 13-48, 2002. · Zbl 1152.76402
[50]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 30. · Zbl 0207.13501
[51]	W. Wolibner, ”Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long,” Math. Z., vol. 37, iss. 1, pp. 698-726, 1933. · Zbl 0008.06901
[52]	S. Wu, ”Almost global wellposedness of the 2-D full water wave problem,” Invent. Math., vol. 177, iss. 1, pp. 45-135, 2009. · Zbl 1181.35205
[53]	S. Wu, ”Global wellposedness of the 3-D full water wave problem,” Invent. Math., vol. 184, iss. 1, pp. 125-220, 2011. · Zbl 1221.35304
[54]	V. I. Judovivc, ”On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid,” Chaos, vol. 10, iss. 3, pp. 705-719, 2000. · Zbl 0982.76014
[55]	V. I. Judovivc, ”Non-stationary flows of an ideal incompressible fluid,” Ž. Vy\vcisl. Mat. i Mat. Fiz., vol. 3, pp. 1032-1066, 1963. · Zbl 0147.44303
[56]	A. Zlatovs, ”Exponential growth of the vorticity gradient for the Euler equation on the torus,” Adv. Math., vol. 268, pp. 396-403, 2015. · Zbl 1308.35194