In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying L∞-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.We use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation.Moreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data.Additionally, we provide results concerning exponential decay of solutions of the nonlinear equation.2010 Mathematics Subject Classification. Primary: 35L75, 35Q35; Secondary: 35B40, 35B65.
Abstract. The Blackstock-Crighton equation models nonlinear acoustic wave propagation in monatomic gases. In the present work, we investigate the associated inhomogeneous Dirichlet and Neumann boundary value problems in a bounded domain and prove long-time well-posedness and exponential stability for sufficiently small data. The solution depends analytically on the data. In the Dirichlet case, the solution decays to zero and the same holds for Neumann conditions if the data have zero mean. We choose an optimal L p -setting, where the regularity of the initial and boundary data is necessary and sufficient for existence, uniqueness and regularity of the solution. The linearized model with homogeneous boundary conditions is represented as an abstract evolution equation for which we show maximal L p -regularity. In order to eliminate inhomogeneous boundary conditions, we establish a general higher regularity result for the heat equation. We conclude that the linearized model induces a topological linear isomorphism and then solves the nonlinear problem by means of the implicit function theorem.
Abstract. We develop singular Weyl-Titchmarsh-Kodaira theory for onedimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Lieberman type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.
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.
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