Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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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  • [1] A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152 (1993), 19-28. MR 1207667
  • [2] 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 MR 1480071
  • [3] J.-Y. Chemin, Sur le mouvement des particules d'un fluide parfait incompressible bidimensionnel, Invention Math. 103 (1991), 599-629. MR 1091620
  • [4] E. A. Coddington and J. L. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1995.
  • [5] A. Friedman and C. Huang, Averaged motion of charged particles under their self-induced electric field, Indiana Univ. Math. J. 43 (1994), 1167-1225. MR 1322616
  • [6] 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.
  • [7] C. Huang, On boundary regularity of non-constant vortex patches, Comm. Appl. Math. 3 (1999), 449-459. MR 1706742
  • [8] C. Huang and T. Svobodny, Evolution of mixed-state region in type-II superconductors, SIAM J. Math. Anal. 29 (1998), 1002-1021. MR 1617698
  • [9] S. Itô, Diffusion Equations, Translations of Mathematical Monographs 114, Providence, RI: Amer. Math. Soc., 1992. MR 1195786
  • [10] C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations 133 (1997), 321-339. MR 1427856
  • [11] C. D. Levermore, M. Oliver, and E. S. Titi, Global well-posedness for the lake equations, Phys. D 98 (1996), 492-502.
  • [12] 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. MR 1414339
  • [13] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (1986), S187-220. MR 861488
  • [14] M. Oliver, Classical solutions for a generalized Euler equation in two dimensions, J. Math. Anal. Appl. 215 (1997), 471-483. MR 1490763
  • [15] V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, Zh. Vych. Mat. 3 (1963), 1032-1066 (In Russian).

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DOI: https://doi.org/10.1090/qam/2019615
Article copyright: © Copyright 2003 American Mathematical Society

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