Regularity and wellposedness of a problem to one parameter and its behavior at the limit. (English) Zbl 1393.35199

Ayala, Yolanda Santiago; Romero, Santiago Rojas

Bull. Allahabad Math. Soc. 32, No. 2, 207-230 (2017).

Summary: In this article we prove that the Cauchy problem associated to a model of waves in a viscous fluid, proposed by R. J. Iorio jun. and V. de Magalhães Iorio [Fourier analysis and partial differential equations. Cambridge: Cambridge University Press (2001; Zbl 0970.35001)], is globally well posed. We do this in an intuitive way using Fourier theory and in a fine version using semigroups theory, getting \(H^\infty\) regularity.
Also, we analyze the behavior of solutions of Cauchy problems to one parameter and prove that their limit is solution of a Cauchy problem whose associated semigroup is the restriction of a group.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35G10	Initial value problems for linear higher-order PDEs
47D06	One-parameter semigroups and linear evolution equations
35B40	Asymptotic behavior of solutions to PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65	Smoothness and regularity of solutions to PDEs
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35Q35	PDEs in connection with fluid mechanics
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Keywords:

groups and semigroups theory; waves equation in viscous fluid; existence of solution; KdV-Kuramoto-Sivashinski equation; periodic Sobolev spaces; Fourier theory

Citations:

Zbl 0970.35001