On the Jordan–Moore–Gibson–Thompson Wave Equation in Hereditary Fluids with Quadratic Gradient Nonlinearity

Abstract

We prove global solvability of the third-order in time Jordan–More–Gibson–Thompson acoustic wave equation with memory in $${\mathbb {R}}^n$$ R n , where $$n \ge 3$$ n ≥ 3 . This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global existence is based on a sequence of high-order energy bounds that are uniform in time, and derived under the assumption of an exponentially decaying memory kernel and sufficiently small and regular initial data.

On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits

“…The second propose in this paper is to demonstrate global (in time) well-posedness and derive estimates of solutions to the viscous JMGT equation (10) of Kuznetsov-type and Westervelt-type, respectively, in Section 4. These results will give complements to the previous works of [38,39]. Basing on some derived estimates in the linearized problem (9), we construct suitable time-weighted Sobolev spaces of fractional orders and control nonlinear terms by some tools in Harmonic Analysis.…”

“…where δ > 0 stands for the viscous case (from the Navier-Stokes-Cattaneo model) and δ = 0 stands for the inviscid case (from the Euler-Cattaneo model), whose modeling has been shown in [38,Section 2]. Note that δ is relatively small in fluids and gases, which incentivizes this paper taking into consideration some properties of solutions with δ ↓ 0.…”

“…It has been considered in [6,32,38,39,40] and references therein. Precisely, the global (in time) well-posedness for Cauchy problem for the viscous JMGT equation ( 6) of Kuznetsov-type was proved in [38] for n 3 by the history framework of Dafermos transform combined with some energy bounds. But, estimates of solutions (even energies) are still undiscovered.…”

“…In particular, our approach makes use of growth rate or decay rate for the linearized problem, and it provides some refined estimates inheriting from the linear problem to the nonlinear problems, which is different from the previous researches. Under the exponential decay memory kernel (7), we give some simple proofs in terms of [38,39].…”

Global well-posedness of the Cauchy problem for the Jordan--Moore--Gibson--Thompson equation with arbitrarily large higher-order Sobolev norms

“…The MGT and JMGT equations with a memory term have been also investigated recently. For the MGT with memory, the reader is refereed to [2,9,31] and to [19,24,25] for the JMGT with memory. The singular limit problem when τ → 0 has been rigorously justified in [16].…”