Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


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