An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches

. We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter α ≤ 14 . This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all α ∈ (0 , 1)). As a corollary, we obtain an improved global regularity criterion for H 3 patch solutions when α ≤ 1 4 , namely that finite time singularities cannot occur while the H 3 norms of patch boundaries remain bounded.

[1]  Alexander Kiselev,et al.  On Nonexistence of Splash Singularities for the $$\alpha $$ α -SQG Patches , 2021, J. Nonlinear Sci..

[2]  Neel Patel,et al.  Well-Posedness for SQG Sharp Fronts with Unbounded Curvature , 2021, Mathematical Models and Methods in Applied Sciences.

[3]  F. Gancedo,et al.  On the local existence and blow-up for generalized SQG patches , 2018, Annals of PDE.

[4]  L. Ryzhik,et al.  Finite time singularity for the modified SQG patch equation , 2016 .

[5]  Antonio C'ordoba,et al.  Uniqueness for SQG patch solutions , 2016, 1605.06663.

[6]  A. Kiselev,et al.  Local Regularity for the Modified SQG Patch Equation , 2015, 1508.07611.

[7]  Thomas Y. Hou,et al.  Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation , 2014, Multiscale Model. Simul..

[8]  A. Kiselev,et al.  Small scale creation for solutions of the incompressible two dimensional Euler equation , 2013, 1310.4799.

[9]  T. Hou,et al.  Potentially singular solutions of the 3D axisymmetric Euler equations , 2013, Proceedings of the National Academy of Sciences.

[10]  S. Shkoller,et al.  On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations , 2012, 1201.4919.

[11]  C. Fefferman,et al.  Finite time singularities for the free boundary incompressible Euler equations , 2011, 1112.2170.

[12]  P. Constantin,et al.  Generalized surface quasi‐geostrophic equations with singular velocities , 2011, 1101.3537.

[13]  L. Debnath.  Geophysical Fluid Dynamics , 2008 .

[14]  Peter Constantin,et al.  Global regularity for a modified critical dissipative quasi-geostrophic equation , 2008, 0803.1318.

[15]  F. Gancedo.  Existence for the α-patch model and the QG sharp front in Sobolev spaces , 2007, math/0701447.

[16]  Jose L. Rodrigo,et al.  On the evolution of sharp fronts for the quasi‐geostrophic equation , 2005 .

[17]  M. Fontelos,et al.  Evidence of singularities for a family of contour dynamics equations , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[18]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[19]  Andrew J. Majda,et al.  Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar , 1994 .

[20]  K. Swanson,et al.  Spectra of local and nonlocal two-dimensional turbulence , 1994 .

[21]  W. Wolibner.  Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long , 1933 .

[22]  E. Hölder.  Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Flüssigkeit , 1933 .

[23]  K. S. S M I T H,et al.  Turbulent diffusion in the geostrophic inverse cascade , 2002 .

[24]  J. Chemin,et al.  Persistance de structures géométriques dans les fluides incompressibles bidimensionnels , 1993 .