Global solutions to the lake equations with isolated vortex regions
Author:
Chaocheng Huang
Journal:
Quart. Appl. Math. 61 (2003), 613-638
MSC:
Primary 76B03; Secondary 35Q35, 86A05
DOI:
https://doi.org/10.1090/qam/2019615
MathSciNet review:
MR2019615
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Abstract: The vorticity formulation for the lake equations in ${R^2}$ is studied. We assume that the initial vorticity has the form $\omega \left ( x, 0 \right ) = {\omega _0}\left ( x \right ){\chi _{{\bar \Omega }_0}}$, where the initial vortex region ${\Omega _0}$ is a ${C^{1 + \alpha }}$ domain and ${\omega _0} \in {C^\alpha }\left ( {\bar \Omega _0} \right )$ . It is shown that the Cauchy problem can be formulated as an integral system. Global existence and uniqueness of the ${C^{1 + \alpha }}$ solution to the integral system are established. Consequently, the lake equation admits a unique weak solution, global in time, in the form of $\omega \left ( x, t \right ) = {\omega _t}\left ( x \right ){\chi _{{{\bar \Omega }_t}}}$, where ${\omega _t}\left ( x \right ) \in C_x^\alpha \left ( {\bar \Omega _t} \right )$ and $\partial {\Omega _t} \in {C^\alpha }$.
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A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152 (1993), 19–28.
R. Camassa, D. D. Holm, and C. D. Levermore, Long-time shallow water equations with a varying bottom, J. Fluid Mech. 349 (1997), 173–189
J.-Y. Chemin, Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel, Invention Math. 103 (1991), 599–629.
E. A. Coddington and J. L. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1995.
A. Friedman and C. Huang, Averaged motion of charged particles under their self-induced electric field, Indiana Univ. Math. J. 43 (1994), 1167–1225.
A. Friedman and J. L. Velázquez, A time-dependent free boundary problem modeling the visual image in electrophotography, Archive Rat. Meth. Anal. 123 (1993), 259–303.
C. Huang, On boundary regularity of non-constant vortex patches, Comm. Appl. Math. 3 (1999), 449–459.
C. Huang and T. Svobodny, Evolution of mixed-state region in type-II superconductors, SIAM J. Math. Anal. 29 (1998), 1002–1021.
S. Itô, Diffusion Equations, Translations of Mathematical Monographs 114, Providence, RI: Amer. Math. Soc., 1992.
C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations 133 (1997), 321–339.
C. D. Levermore, M. Oliver, and E. S. Titi, Global well-posedness for the lake equations, Phys. D 98 (1996), 492–502.
C. D. Levermore, M. Oliver and E. S. Titi, Global well-posedness for models of shallow water equations, Indiana Univ. Math. J., Vol. 45 (1996), 479–510.
A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (1986), S187–220.
M. Oliver, Classical solutions for a generalized Euler equation in two dimensions, J. Math. Anal. Appl. 215 (1997), 471–483.
V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, Zh. Vych. Mat. 3 (1963), 1032–1066 (In Russian).
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© Copyright 2003
American Mathematical Society