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December  2014, 34(12): 5045-5059. doi: 10.3934/dcds.2014.34.5045

Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches

Angel Castro 1, , Diego Córdoba 2, , Javier Gómez-Serrano 3, and  Alberto Martín Zamora 2,

1. 

Departamento de Matemáticas de la UAM, Instituto de Ciencias Matemáticas del CSIC, Campus de Cantoblanco, 28049 Madrid, Spain
2. 

Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolas Cabrera, 13-15, 28049 Madrid, Spain, Spain
3. 

Department of Mathematics, Princeton University, 1102 Fine Hall, Washington Rd, Princeton, NJ 08544, United States

Received  January 2014 Revised  May 2014 Published  June 2014

Guided by numerical simulations, we present the proof of two results concerning the behaviour of SQG sharp fronts and $\alpha$-patches. We establish that ellipses are not rotational solutions and we prove that initially convex interfaces may lose this property in finite time.
Keywords: computer-assisted, patches., contour dynamics, Surface quasigeostrophic, incompressible flow.
Mathematics Subject Classification: Primary: 76B47, 35Q31; Secondary: 65G3.
Citation: Angel Castro, Diego Córdoba, Javier Gómez-Serrano, Alberto Martín Zamora. Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5045-5059. doi: 10.3934/dcds.2014.34.5045
References:
[1]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28. doi: 10.1007/BF02097055.  Google Scholar

[2]

M. Berz and K. Makino, New methods for high-dimensional verified quadrature, Reliable Computing, 5 (1999), 13-22. doi: 10.1023/A:1026437523641.  Google Scholar

[3]

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. doi: 10.1002/cpa.21390.  Google Scholar

[4]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup. (4), 26 (1993), 517-542.  Google Scholar

[5]

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. doi: 10.1088/0951-7715/7/6/001.  Google Scholar

[6]

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, 02 (2005), 5949-5952. Google Scholar

[7]

G. S. Deem and N. J. Zabusky, Vortex waves: Stationary "V-states", interactions, recurrence, and breaking, Physical Review Letters, 40 (1978), 859-862. doi: 10.1103/PhysRevLett.40.859.  Google Scholar

[8]

S. A. Denisov, The Sharp Corner Formation in 2d Euler Dynamics of Patches: Infinite Double Exponential rate of Merging, ArXiv e-prints, Jan, 2012. Google Scholar

[9]

F. Gancedo, Existence for the $\alpha$-patch model and the QC sharp front in Sobolev spaces, Adv. Math., 217 (2008), 2569-2598. doi: 10.1016/j.aim.2007.10.010.  Google Scholar

[10]

F. Gancedo and R. M. Strain, Absence of splash singularities for SQG sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA, 111 (2014), 635-639. doi: 10.1073/pnas.1320554111.  Google Scholar

[11]

J. Gómez-Serrano and R. Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: A computer-assisted proof, Nonlinearity, 27 (2014), 1471-1498. doi: 10.1088/0951-7715/27/6/1471.  Google Scholar

[12]

I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., 282 (1995), 1-20. doi: 10.1017/S0022112095000012.  Google Scholar

[13]

T. Hmidi, J. Mateu and J. Verdera, Boundary regularity of rotating vortex patches, Archive for Rational Mechanics and Analysis, 209 (2013), 171-208. doi: 10.1007/s00205-013-0618-8.  Google Scholar

[14]

W. Hofschuster and W. Krämer, C-XSC 2.0-A C++ library for extended scientific computing, In Numerical software with result verification, Springer, 2991 (2004), 15-35. doi: 10.1007/978-3-540-24738-8_2.  Google Scholar

[15]

O. Holzmann, B. Lang and H. Schütt, Newton's constant of gravitation and verified numerical quadrature, Reliable Computing, 2 (1996), 229-239. doi: 10.1007/BF02391697.  Google Scholar

[16]

W. Krämer and S. Wedner, Two adaptive Gauss-Legendre type algorithms for the verified computation of definite integrals, Reliable Computing, 2 (1996), 241-253. doi: 10.1007/BF02391698.  Google Scholar

[17]

H. Lamb, Hydrodynamics, Cambridge Mathematical Library. Cambridge University Press, Cambridge, sixth edition, 1993.  Google Scholar

[18]

B. Lang, Derivative-based subdivision in multi-dimensional verified gaussian quadrature, In G. Alefeld, J. Rohn, S. Rump, and T. Yamamoto, editors, Symbolic Algebraic Methods and Verification Methods, Springer Vienna (2001), 145-152.  Google Scholar

[19]

R. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, volume 2, Society for Industrial & Applied Mathematics, 1979.  Google Scholar

[20]

J. Pedlosky, Geophysical fluid dynamics, Journal of Applied Mechanics, 48 (1981), 684, 1pp. doi: 10.1115/1.3157711.  Google Scholar

[21]

S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, PhD thesis, University of Chicago, Department of Mathematics, 1995. Google Scholar

[22]

J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059.  Google Scholar

[23]

R. Scott and D. Dritschel, A self-similar cascade of instabilities in the surface quasigeostrophic system, Phys. Rev. Lett., 112 (2014), 144505 (5 pages). doi: 10.1103/PhysRevLett.112.144505.  Google Scholar

[24]

R. K. Scott, A scenario for finite-time singularity in the quasigeostrophic model, Journal of Fluid Mechanics, 687 (2011), 492-502. Google Scholar

[25]

W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011, A short introduction to rigorous computations.  Google Scholar

[26]

H. M. Wu, E. A. Overman and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: Rotating and translating $V$-states with limiting cases. I. Numerical algorithms and results, J. Comput. Phys., 53 (1984), 42-71. doi: 10.1016/0021-9991(84)90051-2.  Google Scholar

[27]

V. I. Yudovich., Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.  Google Scholar

show all references

References:
[1]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28. doi: 10.1007/BF02097055.  Google Scholar

[2]

M. Berz and K. Makino, New methods for high-dimensional verified quadrature, Reliable Computing, 5 (1999), 13-22. doi: 10.1023/A:1026437523641.  Google Scholar

[3]

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. doi: 10.1002/cpa.21390.  Google Scholar

[4]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup. (4), 26 (1993), 517-542.  Google Scholar

[5]

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. doi: 10.1088/0951-7715/7/6/001.  Google Scholar

[6]

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, 02 (2005), 5949-5952. Google Scholar

[7]

G. S. Deem and N. J. Zabusky, Vortex waves: Stationary "V-states", interactions, recurrence, and breaking, Physical Review Letters, 40 (1978), 859-862. doi: 10.1103/PhysRevLett.40.859.  Google Scholar

[8]

S. A. Denisov, The Sharp Corner Formation in 2d Euler Dynamics of Patches: Infinite Double Exponential rate of Merging, ArXiv e-prints, Jan, 2012. Google Scholar

[9]

F. Gancedo, Existence for the $\alpha$-patch model and the QC sharp front in Sobolev spaces, Adv. Math., 217 (2008), 2569-2598. doi: 10.1016/j.aim.2007.10.010.  Google Scholar

[10]

F. Gancedo and R. M. Strain, Absence of splash singularities for SQG sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA, 111 (2014), 635-639. doi: 10.1073/pnas.1320554111.  Google Scholar

[11]

J. Gómez-Serrano and R. Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: A computer-assisted proof, Nonlinearity, 27 (2014), 1471-1498. doi: 10.1088/0951-7715/27/6/1471.  Google Scholar

[12]

I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., 282 (1995), 1-20. doi: 10.1017/S0022112095000012.  Google Scholar

[13]

T. Hmidi, J. Mateu and J. Verdera, Boundary regularity of rotating vortex patches, Archive for Rational Mechanics and Analysis, 209 (2013), 171-208. doi: 10.1007/s00205-013-0618-8.  Google Scholar

[14]

W. Hofschuster and W. Krämer, C-XSC 2.0-A C++ library for extended scientific computing, In Numerical software with result verification, Springer, 2991 (2004), 15-35. doi: 10.1007/978-3-540-24738-8_2.  Google Scholar

[15]

O. Holzmann, B. Lang and H. Schütt, Newton's constant of gravitation and verified numerical quadrature, Reliable Computing, 2 (1996), 229-239. doi: 10.1007/BF02391697.  Google Scholar

[16]

W. Krämer and S. Wedner, Two adaptive Gauss-Legendre type algorithms for the verified computation of definite integrals, Reliable Computing, 2 (1996), 241-253. doi: 10.1007/BF02391698.  Google Scholar

[17]

H. Lamb, Hydrodynamics, Cambridge Mathematical Library. Cambridge University Press, Cambridge, sixth edition, 1993.  Google Scholar

[18]

B. Lang, Derivative-based subdivision in multi-dimensional verified gaussian quadrature, In G. Alefeld, J. Rohn, S. Rump, and T. Yamamoto, editors, Symbolic Algebraic Methods and Verification Methods, Springer Vienna (2001), 145-152.  Google Scholar

[19]

R. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, volume 2, Society for Industrial & Applied Mathematics, 1979.  Google Scholar

[20]

J. Pedlosky, Geophysical fluid dynamics, Journal of Applied Mechanics, 48 (1981), 684, 1pp. doi: 10.1115/1.3157711.  Google Scholar

[21]

S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, PhD thesis, University of Chicago, Department of Mathematics, 1995. Google Scholar

[22]

J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059.  Google Scholar

[23]

R. Scott and D. Dritschel, A self-similar cascade of instabilities in the surface quasigeostrophic system, Phys. Rev. Lett., 112 (2014), 144505 (5 pages). doi: 10.1103/PhysRevLett.112.144505.  Google Scholar

[24]

R. K. Scott, A scenario for finite-time singularity in the quasigeostrophic model, Journal of Fluid Mechanics, 687 (2011), 492-502. Google Scholar

[25]

W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011, A short introduction to rigorous computations.  Google Scholar

[26]

H. M. Wu, E. A. Overman and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: Rotating and translating $V$-states with limiting cases. I. Numerical algorithms and results, J. Comput. Phys., 53 (1984), 42-71. doi: 10.1016/0021-9991(84)90051-2.  Google Scholar

[27]

V. I. Yudovich., Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.  Google Scholar

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