Analyticity of solutions for a generalized Euler equation.

*(English)*Zbl 0876.35090The authors investigate analyticity of a class of shallow water equations that include the so-called lake and great lake equations. The great lake equation derived from a non-dimensional expansion in a “small” parameter is given by:
\[
\partial_tv- u^\perp\nabla\wedge v+\nabla(h- 1/2 u\cdot u+ u\cdot v)=0,\;v={\mathcal L}u,\;\nabla\cdot(bu)=0,\;u^{\text{in}}= u(0).
\]
The \(\mathcal L\) operator is defined by
\[
{\mathcal L}u= u+\delta^2((u\cdot\nabla b)\nabla b+ 1/2 b(\nabla\cdot u)\nabla b- 1/2 b^{-1}\nabla(b^2u\cdot\nabla b)- 1/3 b^{-1}\nabla(b^3\nabla\cdot u)).
\]
The equation obtained by retaining only the leading terms in \(\delta\) is called the lake equation. For a flat bottom case the lake and great lake equations are reduced to a two-dimensional Euler equation. These equations can be rewritten in terms of the potential vorticity
\[
\omega= b^{-1}\nabla\wedge v:\qquad \partial_t\omega+ u\cdot\nabla\omega= 0,\;\omega^{\text{in}}= \omega(0),\;u= K\omega.
\]
Analyzing this system, the authors have shown in their previous papers that this system is globally well-posed, and has a classical solution. Using Gevrey classes of functions (that is functions whose derivatives are bounded by \(s\)-powers of the corresponding factorial or the Gamma function), the authors show here that the solutions are real analytic, if the initial data are analytic. The proof given by the authors is new.

Comment: The relation between membership in a Gevrey class and analyticity has been demonstrated in the classical text of R. A. Adamss [Sobolev Spaces, Academic Press, New York (1975; Zbl 0314.46030)]. The converse of this statement is indicated without naming Gevrey classes in the classical series of L. HĂ¶rmander on partial differential equations. The advantage of using Gevrey classes comes from invariance, or retention of membership, under ordinary multiplication.

Comment: The relation between membership in a Gevrey class and analyticity has been demonstrated in the classical text of R. A. Adamss [Sobolev Spaces, Academic Press, New York (1975; Zbl 0314.46030)]. The converse of this statement is indicated without naming Gevrey classes in the classical series of L. HĂ¶rmander on partial differential equations. The advantage of using Gevrey classes comes from invariance, or retention of membership, under ordinary multiplication.

Reviewer: V.Komkov (Roswell)