Global solutions to the lake equations with isolated vortex regions

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 $\partial {\Omega _t} \in {C^\alpha }$.