The wave equation with acoustic boundary conditions on non-locally reacting surfaces
Abstract: Chapter 1. Introduction and main results vii 1.1. Presentation of the problem and literature overview vii 1.2. Main results I: well-posedness ix 1.3. Main results II: solutions' behavior xii 1.4. Physical discussion of the model xvi 1.5. The case of Γ 1 disconnected and paper organization xviii Chapter 2. Background and preliminaries xix 2.1. Notation xix 2.2. Assumptions xix 2.3. Geometric preliminaries xxii 2.4. Lebesgue spaces on Γ xxvii Chapter 3. Sobolev spaces on Γ and operators xxix 3.1. Spaces of integ… Show more
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Cited by 3 publications
References 22 publications
The aim of the paper is to study the problem $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}+du_t-c^2\Delta u=0 \qquad &{}\text {in}\, {\mathbb {R}}\times \Omega ,\\ \mu v_{tt}- \textrm{div}_\Gamma (\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+\rho u_t =0\qquad &{}\text {on}\, {\mathbb {R}}\times \Gamma _1,\\ v_t =\partial _\nu u\qquad &{}\text {on}\, {\mathbb {R}}\times \Gamma _1,\\ \partial _\nu u=0 &{}\text {on}\, {\mathbb {R}}\times \Gamma _0,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) &{} \text {in}\, \Omega ,\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) &{} \text {on}\, \Gamma _1, \end{array}\right. } \end{aligned}$$ u tt + d u t - c 2 Δ u = 0 in R × Ω , μ v tt - div Γ ( σ ∇ Γ v ) + δ v t + κ v + ρ u t = 0 on R × Γ 1 , v t = ∂ ν u on R × Γ 1 , ∂ ν u = 0 on R × Γ 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) in Ω , v ( 0 , x ) = v 0 ( x ) , v t ( 0 , x ) = v 1 ( x ) on Γ 1 , where $$\Omega $$ Ω is a open domain of $${\mathbb {R}}^N$$ R N with uniformly $$C^r$$ C r boundary ($$N\ge 2$$ N ≥ 2 , $$r\ge 1$$ r ≥ 1 ), $$\Gamma =\partial \Omega $$ Γ = ∂ Ω , $$(\Gamma _0,\Gamma _1)$$ ( Γ 0 , Γ 1 ) is a relatively open partition of $$\Gamma $$ Γ with $$\Gamma _0$$ Γ 0 (but not $$\Gamma _1$$ Γ 1 ) possibly empty. Here $$\textrm{div}_\Gamma $$ div Γ and $$\nabla _\Gamma $$ ∇ Γ denote the Riemannian divergence and gradient operators on $$\Gamma $$ Γ , $$\nu $$ ν is the outward normal to $$\Omega $$ Ω , the coefficients $$\mu ,\sigma ,\delta , \kappa , \rho $$ μ , σ , δ , κ , ρ are suitably regular functions on $$\Gamma _1$$ Γ 1 with $$\rho ,\sigma $$ ρ , σ and $$\mu $$ μ uniformly positive, d is a suitably regular function in $$\Omega $$ Ω and c is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when $$\Omega $$ Ω is bounded, $$\Gamma _1$$ Γ 1 is connected, $$r=2$$ r = 2 , $$\rho $$ ρ is constant and $$\kappa ,\delta ,d\ge 0$$ κ , δ , d ≥ 0 .
The aim of the paper is to study the problem $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}+du_t-c^2\Delta u=0 \qquad &{}\text {in}\, {\mathbb {R}}\times \Omega ,\\ \mu v_{tt}- \textrm{div}_\Gamma (\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+\rho u_t =0\qquad &{}\text {on}\, {\mathbb {R}}\times \Gamma _1,\\ v_t =\partial _\nu u\qquad &{}\text {on}\, {\mathbb {R}}\times \Gamma _1,\\ \partial _\nu u=0 &{}\text {on}\, {\mathbb {R}}\times \Gamma _0,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) &{} \text {in}\, \Omega ,\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) &{} \text {on}\, \Gamma _1, \end{array}\right. } \end{aligned}$$ u tt + d u t - c 2 Δ u = 0 in R × Ω , μ v tt - div Γ ( σ ∇ Γ v ) + δ v t + κ v + ρ u t = 0 on R × Γ 1 , v t = ∂ ν u on R × Γ 1 , ∂ ν u = 0 on R × Γ 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) in Ω , v ( 0 , x ) = v 0 ( x ) , v t ( 0 , x ) = v 1 ( x ) on Γ 1 , where $$\Omega $$ Ω is a open domain of $${\mathbb {R}}^N$$ R N with uniformly $$C^r$$ C r boundary ($$N\ge 2$$ N ≥ 2 , $$r\ge 1$$ r ≥ 1 ), $$\Gamma =\partial \Omega $$ Γ = ∂ Ω , $$(\Gamma _0,\Gamma _1)$$ ( Γ 0 , Γ 1 ) is a relatively open partition of $$\Gamma $$ Γ with $$\Gamma _0$$ Γ 0 (but not $$\Gamma _1$$ Γ 1 ) possibly empty. Here $$\textrm{div}_\Gamma $$ div Γ and $$\nabla _\Gamma $$ ∇ Γ denote the Riemannian divergence and gradient operators on $$\Gamma $$ Γ , $$\nu $$ ν is the outward normal to $$\Omega $$ Ω , the coefficients $$\mu ,\sigma ,\delta , \kappa , \rho $$ μ , σ , δ , κ , ρ are suitably regular functions on $$\Gamma _1$$ Γ 1 with $$\rho ,\sigma $$ ρ , σ and $$\mu $$ μ uniformly positive, d is a suitably regular function in $$\Omega $$ Ω and c is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when $$\Omega $$ Ω is bounded, $$\Gamma _1$$ Γ 1 is connected, $$r=2$$ r = 2 , $$\rho $$ ρ is constant and $$\kappa ,\delta ,d\ge 0$$ κ , δ , d ≥ 0 .
A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions
In this work we present a unified error analysis for abstract space discretizations of wave-type equations with nonlinear quasi-monotone operators. This yields an error bound in terms of discretization and interpolation errors that can be applied to various equations and space discretizations fitting in the abstract setting. We use the unified error analysis to prove novel convergence rates for a non-conforming finite element space discretization of wave equations with nonlinear acoustic boundary conditions and illustrate the error bound by a numerical experiment.
粘弹声学边界条件和超临界源项的波动方程爆 破解
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