Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling

Abstract

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.

Sensitivity Analysis for Shape Optimization of a Focusing Acoustic Lens in Lithotripsy

“…with δ ∈ (0, 1), q ≥ 1, q > d − 1, d ∈ {1, 2, 3}, which allowed to show existence of weak solutions with W 1,q+1 regularity in space, and in turn well-posedness of the acoustic-acoustic coupling problem. The case of the acoustic-acoustic coupling, which we are interested in, is modeled by the presence of spatially varying coefficients in the weak form of the equation (1.2) (see [2] for the linear and [4] and [27] for the nonlinear case) as follows: λ(x) (u) 2 φ} dx ds = 0 holds for all test functions φ ∈X, with (u,u)| t=0 = (u 0 , u 1 ), and appropriately chosen test spaceX. In this model b denotes the quotient between the diffusivity and the bulk modulus, while the other coefficients retain their meaning.…”

On higher regularity for the Westervelt equation with strong nonlinear damping

“…We will show that u ∈ H 1 (0, T ; H 2 loc (Ω)) and |∇u| q−1 2 ∇u ∈ L 2 (0, T ; H 1 loc (Ω)). Although q-Laplace and parabolic q-Laplace equation have been extensively studied in the past (see [15], [19], [6], [7] and references given therein), regularity results in literature on hyperbolic equations with damping of the q-Laplace type are sparse and have so far been concerned with local and global well-posedness (see [21], [9], [4], [17]).…”

“…In [4], the issue of possible degeneracy of the Westervelt equation due to the factor 1 − 2ku is resolved by means of the embedding W 1,q+1 0 (Ω) ֒→ L ∞ (Ω), valid for q > d − 1, and the following estimate…”

“…This model was studied (in an equivalent one domain formulation) in [4]. We will utilize the following well-posedness result (cf.…”

Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions

“…in the coupling of acoustic with acoustic or elastic regions with different material parameters. In [2], Brunnhuber, Kaltenbacher and Radu treated this issue by introducing nonlinear damping terms to the Westervelt equation and considering the following equations…”