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Structural stability for the splash singularities of the water waves problem
Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches
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 |
References:
[1] |
A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28.
doi: 10.1007/BF02097055. |
[2] |
M. Berz and K. Makino, New methods for high-dimensional verified quadrature, Reliable Computing, 5 (1999), 13-22.
doi: 10.1023/A:1026437523641. |
[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. |
[4] |
J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup. (4), 26 (1993), 517-542. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
H. Lamb, Hydrodynamics, Cambridge Mathematical Library. Cambridge University Press, Cambridge, sixth edition, 1993. |
[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. |
[19] |
R. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, volume 2, Society for Industrial & Applied Mathematics, 1979. |
[20] |
J. Pedlosky, Geophysical fluid dynamics, Journal of Applied Mechanics, 48 (1981), 684, 1pp.
doi: 10.1115/1.3157711. |
[21] |
S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, PhD thesis, University of Chicago, Department of Mathematics, 1995. |
[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. |
[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. |
[24] |
R. K. Scott, A scenario for finite-time singularity in the quasigeostrophic model, Journal of Fluid Mechanics, 687 (2011), 492-502. |
[25] |
W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011, A short introduction to rigorous computations. |
[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. |
[27] |
V. I. Yudovich., Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066. |
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. |
[2] |
M. Berz and K. Makino, New methods for high-dimensional verified quadrature, Reliable Computing, 5 (1999), 13-22.
doi: 10.1023/A:1026437523641. |
[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. |
[4] |
J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup. (4), 26 (1993), 517-542. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
H. Lamb, Hydrodynamics, Cambridge Mathematical Library. Cambridge University Press, Cambridge, sixth edition, 1993. |
[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. |
[19] |
R. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, volume 2, Society for Industrial & Applied Mathematics, 1979. |
[20] |
J. Pedlosky, Geophysical fluid dynamics, Journal of Applied Mechanics, 48 (1981), 684, 1pp.
doi: 10.1115/1.3157711. |
[21] |
S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, PhD thesis, University of Chicago, Department of Mathematics, 1995. |
[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. |
[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. |
[24] |
R. K. Scott, A scenario for finite-time singularity in the quasigeostrophic model, Journal of Fluid Mechanics, 687 (2011), 492-502. |
[25] |
W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011, A short introduction to rigorous computations. |
[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. |
[27] |
V. I. Yudovich., Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066. |
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