Cauchy problem for the Kuznetsov equation

Abstract

We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2 -stability estimate, obtained between a regular and a less regular solutions.

Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

“…Proof. The existence of u and u has already been shown in [11] and given in Theorem 4.1. The proof of the approximation estimate follows exactly the proof of Theorem 2.2 and hence it is omitted.…”

“…We subtract the Kuznetsov equation from the approximated Kuznetsov equation (see system (27)), multiply by (u − u) t and integrate over Ω to obtain, as in Ref. [11], the following stability estimate:…”

“…Approximation of the solutions of the Kuznetsov equation with the solutions of the NPE equation. Now let us go back to the NPE equation (11) and consider its ansatz (see [12] for the derivation of the NPE equation from the isentropic Navier-Stokes system or the Euler system). In contrast with Eq.…”

“…To prove the global well-posedness of the periodic in time boundary value problem (21) for the Kuznetsov equation we use its boundary condition as the initial condition of the corresponding Cauchy problem in R n and we combine the maximal regularity result for system (56) with [48,1.5 Cor.,p. 368] (see also [11,Thm. 4.2]) applying the same method as previously done for the Cauchy problem associated with the Kuznetsov equation [11].…”

“…the KZK and NPE equations. For the well posedness of the Cauchy problem for the Kuznetsov equation we cite [11] and for boundary value problems in regular bounded domains see [25,29,40].…”