Analyticity of Solutions for a Generalized Euler Equation

Abstract

We consider the so-called lake and great lake equations, which are shallow water equations that describe the long-time motion of an inviscid, incompressible fluid contained in a shallow basin with a slowly spatially varying bottom, a free upper surface, and vertical side walls, under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude. If these equations are posed on a space-periodic domain and the initial data are real analytic, the solution remains real analytic for all times. The proof is based on a characterization of Gevrey classes in terms of decay of Fourier coefficients. In particular, our result recovers known results for the Euler equations in two and three spatial dimensions. We believe the proof is new. Academic Press

Landau Damping: Paraproducts and Gevrey Regularity

“…As in [10], the transport terms in (2.5) are treated via an adaptation of the Gevrey energy methods of [25,50]. The essential content is a commutator estimate to take advantage of the cancellations inherent in the natural transport structure.…”

“…Norms such as G λ(t),σ ;s are common when dealing with analytic or Gevrey regularity, for example, see the works [10,21,22,25,27,45,50,67]. The Sobolev correction σ is included mostly for technical convenience.…”

“…The local existence of analytic solutions can be proved with an abstract Cauchy-Kovalevskaya theorem, see for example [69,70]. The propagation of analyticity by Sobolev regularity can be proved by a variant of the arguments in [50] along with the inequality Bρ 2 α≤M v α Bg 2 for all Fourier multipliers B (with our notation (1.13)) and all integers M > d/2. We omit the proof of Lemma 2.1 for brevity.…”

“…See the classical results [12,41,54,74,78] for the global existence of strong solutions (from which analyticity can be propagated by a variant of e.g. [50]). We remark that in d ≥ 4, finite time blow-up is possible at least for gravitational interactions [48], however, this case is still covered by Theorem 1.…”

“…A paraproduct is again used to decompose the nonlinearity into reaction, transport and remainder terms. As in [10], the transport term is treated using an adaptation of the methods of [25,45,50]. However, perhaps more like [67], the reaction term is treated using (2.11c).…”