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On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation
Bifurcation of rotating patches from Kirchhoff vortices
1. | IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France |
2. | Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain |
References:
[1] |
A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge texts in applied Mathematics, Cambridge University Press, Cambridge, 2002. |
[2] |
J. Burbea, Motions of vortex patches, Lett. Math. Phys., 6 (1982), 1-16.
doi: 10.1007/BF02281165. |
[3] |
J. Burbea and M. Landau, The Kelvin waves in vortex dynamics and their stability, Journal of Computational Physics, 45 (1982), 127-156.
doi: 10.1016/0021-9991(82)90106-1. |
[4] |
A. Castro, D. Córdoba and J. Gómez-Serrano, Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations, Duke Math. J., 165 (2016), 935-984.
doi: 10.1215/00127094-3449673. |
[5] |
A. Castro, D. Córdoba and J. Gómez-Serrano, Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE, 2 (2016), Art. 1, 34 pp.
doi: 10.1007/s40818-016-0007-3. |
[6] |
C. Cerretelli and C. H. K. Williamson, A new family of uniform vortices related to vortex configurations before fluid merger, J. Fluid Mech., 493 (2003), 219-229.
doi: 10.1017/S0022112003005536. |
[7] |
J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, 1998. |
[8] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. of Func. Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[9] |
G. S. Deem and N. J. Zabusky, Vortex waves: Stationary "V-states", interactions, recurrence, and breaking, Phys. Rev. Lett., 40 (1978), 859-862. |
[10] |
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. |
[11] |
D. G. Dritschel, The nonlinear evolution of rotating configurations of uniform vorticity, J. Fluid Mech., 172 (1986), 157-182.
doi: 10.1017/S0022112086001696. |
[12] |
G. R. Flierl and L. M. Polvani, Generalized Kirchhoff vortices, Phys. Fluids, 29 (1986), 2376-2379. |
[13] |
Z. Hassainia and T. Hmidi, On the V-States for the generalized quasi-geostrophic equations, Comm. Math. Phys., 337 (2015), 321-377.
doi: 10.1007/s00220-015-2300-5. |
[14] |
Z. Hassainia, T. Hmidi and F. de la Hoz, Doubly connected V-states for the generalized surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 220 (2016), 1209-1281.
doi: 10.1007/s00205-015-0953-z. |
[15] |
T. Hmidi, J. Mateu and J. Verdera, Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal., 209 (2013), 171-208.
doi: 10.1007/s00205-013-0618-8. |
[16] |
T. Hmidi, J. Mateu and J. Verdera, On rotating doubly connected vortices, J. Differential Equations, 258 (2015), 1395-1429.
doi: 10.1016/j.jde.2014.10.021. |
[17] |
T. Hmidi, F. de la Hoz, J. Mateu and J. Verdera, Doubly connected V-states for the planar Euler equations,, Preprint, ().
|
[18] |
Y. Guo, C. Hallstrom and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys., 245 (2004), 297-354.
doi: 10.1007/s00220-003-1017-z. |
[19] |
J. R. Kamm, Shape and Stability of Two-Dimensional Uniform Vorticity Regions, PhD thesis, California Institute of Technology, 1987. |
[20] |
G. Kirchhoff, Vorlesungen Uber Mathematische Physik, (Leipzig, 1874). |
[21] | |
[22] |
A. E. H. Love, On the Stability of certain Vortex Motions, Proc. London Math. Soc., 25 (1893), 18-43.
doi: 10.1112/plms/s1-25.1.18. |
[23] |
P. Luzzatto-Fegiz and C. H. K. Williamson, Stability of elliptical vortices from "Imperfect-Velocity-Impulse" diagrams, Theor. Comput. Fluid Dyn., 24 (2010), 181-188. |
[24] |
J. Mateu, J. Orobitg and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl., 91 (2009), 402-431.
doi: 10.1016/j.matpur.2009.01.010. |
[25] |
T. B. Mitchell and L. F. Rossi, The evolution of Kirchhoff elliptic vortices, Physics of Fluids, 20 (2008), 054103.
doi: 10.1063/1.2912991. |
[26] |
E. A. II Overman, Steady-state solutions of the Euler equations in two dimensions. II. Local analysis of limiting V-states, SIAM J. Appl. Math., 46 (1986), 765-800.
doi: 10.1137/0146049. |
[27] |
P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992. |
[28] |
Y. Tang, Nonlinear stability of vortex patches, Trans. Amer. Math. Soc., 304 (1987), 617-638.
doi: 10.1090/S0002-9947-1987-0911087-X. |
[29] |
Y. H. Wan, The stability of rotating vortex patches, Comm. Math. Phys., 107 (1986), 1-20.
doi: 10.1007/BF01206950. |
[30] |
H. M. Wu, E. A. Overman II and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases I. Algorithms ans results, J. Comput. Phys., 53 (1984), 42-71.
doi: 10.1016/0021-9991(84)90051-2. |
[31] |
V. I. Yudovich, Non-stationnary flows of an ideal incompressible fluid, Zhurnal Vych Matematika, 3 (1963), 1032-1066. |
show all references
References:
[1] |
A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge texts in applied Mathematics, Cambridge University Press, Cambridge, 2002. |
[2] |
J. Burbea, Motions of vortex patches, Lett. Math. Phys., 6 (1982), 1-16.
doi: 10.1007/BF02281165. |
[3] |
J. Burbea and M. Landau, The Kelvin waves in vortex dynamics and their stability, Journal of Computational Physics, 45 (1982), 127-156.
doi: 10.1016/0021-9991(82)90106-1. |
[4] |
A. Castro, D. Córdoba and J. Gómez-Serrano, Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations, Duke Math. J., 165 (2016), 935-984.
doi: 10.1215/00127094-3449673. |
[5] |
A. Castro, D. Córdoba and J. Gómez-Serrano, Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE, 2 (2016), Art. 1, 34 pp.
doi: 10.1007/s40818-016-0007-3. |
[6] |
C. Cerretelli and C. H. K. Williamson, A new family of uniform vortices related to vortex configurations before fluid merger, J. Fluid Mech., 493 (2003), 219-229.
doi: 10.1017/S0022112003005536. |
[7] |
J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, 1998. |
[8] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. of Func. Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[9] |
G. S. Deem and N. J. Zabusky, Vortex waves: Stationary "V-states", interactions, recurrence, and breaking, Phys. Rev. Lett., 40 (1978), 859-862. |
[10] |
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. |
[11] |
D. G. Dritschel, The nonlinear evolution of rotating configurations of uniform vorticity, J. Fluid Mech., 172 (1986), 157-182.
doi: 10.1017/S0022112086001696. |
[12] |
G. R. Flierl and L. M. Polvani, Generalized Kirchhoff vortices, Phys. Fluids, 29 (1986), 2376-2379. |
[13] |
Z. Hassainia and T. Hmidi, On the V-States for the generalized quasi-geostrophic equations, Comm. Math. Phys., 337 (2015), 321-377.
doi: 10.1007/s00220-015-2300-5. |
[14] |
Z. Hassainia, T. Hmidi and F. de la Hoz, Doubly connected V-states for the generalized surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 220 (2016), 1209-1281.
doi: 10.1007/s00205-015-0953-z. |
[15] |
T. Hmidi, J. Mateu and J. Verdera, Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal., 209 (2013), 171-208.
doi: 10.1007/s00205-013-0618-8. |
[16] |
T. Hmidi, J. Mateu and J. Verdera, On rotating doubly connected vortices, J. Differential Equations, 258 (2015), 1395-1429.
doi: 10.1016/j.jde.2014.10.021. |
[17] |
T. Hmidi, F. de la Hoz, J. Mateu and J. Verdera, Doubly connected V-states for the planar Euler equations,, Preprint, ().
|
[18] |
Y. Guo, C. Hallstrom and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys., 245 (2004), 297-354.
doi: 10.1007/s00220-003-1017-z. |
[19] |
J. R. Kamm, Shape and Stability of Two-Dimensional Uniform Vorticity Regions, PhD thesis, California Institute of Technology, 1987. |
[20] |
G. Kirchhoff, Vorlesungen Uber Mathematische Physik, (Leipzig, 1874). |
[21] | |
[22] |
A. E. H. Love, On the Stability of certain Vortex Motions, Proc. London Math. Soc., 25 (1893), 18-43.
doi: 10.1112/plms/s1-25.1.18. |
[23] |
P. Luzzatto-Fegiz and C. H. K. Williamson, Stability of elliptical vortices from "Imperfect-Velocity-Impulse" diagrams, Theor. Comput. Fluid Dyn., 24 (2010), 181-188. |
[24] |
J. Mateu, J. Orobitg and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl., 91 (2009), 402-431.
doi: 10.1016/j.matpur.2009.01.010. |
[25] |
T. B. Mitchell and L. F. Rossi, The evolution of Kirchhoff elliptic vortices, Physics of Fluids, 20 (2008), 054103.
doi: 10.1063/1.2912991. |
[26] |
E. A. II Overman, Steady-state solutions of the Euler equations in two dimensions. II. Local analysis of limiting V-states, SIAM J. Appl. Math., 46 (1986), 765-800.
doi: 10.1137/0146049. |
[27] |
P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992. |
[28] |
Y. Tang, Nonlinear stability of vortex patches, Trans. Amer. Math. Soc., 304 (1987), 617-638.
doi: 10.1090/S0002-9947-1987-0911087-X. |
[29] |
Y. H. Wan, The stability of rotating vortex patches, Comm. Math. Phys., 107 (1986), 1-20.
doi: 10.1007/BF01206950. |
[30] |
H. M. Wu, E. A. Overman II and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases I. Algorithms ans results, J. Comput. Phys., 53 (1984), 42-71.
doi: 10.1016/0021-9991(84)90051-2. |
[31] |
V. I. Yudovich, Non-stationnary flows of an ideal incompressible fluid, Zhurnal Vych Matematika, 3 (1963), 1032-1066. |
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