Well-posedness and exponential decay of solutions for the Blackstock–Crighton–Kuznetsov equation

Abstract

Abstract. The present work provides well-posedness and exponential decay results for the Blackstock-Crighton-Kuznetsov equation arising in the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.First, we treat the associated linear equation by means of operator semigroups. Moreover, we derive energy estimates which we will use in a fixed-point argument in order to obtain well-posedness of the Blackstock-Crighton-Kuznetsov equation. Using a classical barrier argument we prove exponential decay of solutions.

Optimal regularity and exponential stability for the Blackstock–Crighton equation in L p -spaces with Dirichlet and Neumann boundary conditions

“…The parameter s ∈ {0, 1} allows us to switch between (1.1) and (1.2). We point out that the present work extends the results from [Bru15] in several ways. First, while in [Bru15] the Blackstock-Crighton equation was considered with homogeneous Dirichlet boundary conditions, we also allow for inhomogeneous Dirichlet as well as Neumann boundary conditions.…”

“…Moreover, in (1.4) we consider potential diffusivity as in (1.3). Therewith, we arrive at equation (1.1) for which in [Bru15] the name Blackstock-Crighton-Kuznetsov equation has been introduced. For a more rigorous derivation of (1.1) we refer to Section 2 in [Bru15].…”

“…On the contrary, mathematical research on higher order partial differential equations arising in nonlinear acoustics is still in an early stage. Well-posedness and exponential decay results for the homogeneous Dirichlet boundary value problems associated with (1.1) and (1.2) in an L 2 (Ω)-setting where Ω ⊂ R n , n ∈ {1, 2, 3}, have been shown in [Bru15] and [BK14], respectively. In the present work we consider (1.1) and (1.2) with inhomogeneous Dirichlet and Neumann boundary conditions in L p (Ω)-spaces where the spatial domain Ω is of dimension n ∈ N. We show global well-posedness and long-time behavior of solutions in an optimal functional analytic setting in the sense that the regularity of the solution is necessary and sufficient for the regularity of the initial and boundary data.…”

“…In the present work we consider (1.1) and (1.2) with inhomogeneous Dirichlet and Neumann boundary conditions in L p (Ω)-spaces where the spatial domain Ω is of dimension n ∈ N. We show global well-posedness and long-time behavior of solutions in an optimal functional analytic setting in the sense that the regularity of the solution is necessary and sufficient for the regularity of the initial and boundary data. While in [Bru15] and [BK14] the results were proved by means of appropriate energy estimates and the Banach fixed-point theorem, the techniques used in the present paper are based on maximal L p -regularity for parabolic problems and the implicit function theorem in Banach spaces.…”

Nonlinear Acoustics: Blackstock–Crighton Equations with a Periodic Forcing Term

“…where we have used the same notation k := 1 c 2 (1 − s) + B 2A and s ∈ {0, 1} as in [2]. The corresponding Neumann problem reads…”