Exterior Domain Research Articles

Semiclassical resolvent estimates for Hölder potentials

We first prove semiclassical resolvent estimates for the Schrodinger operator in R d , d ≥ 3, with real-valued potentials which are Holder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d ≥ 2, and real-valued potentials which are Holder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be Holder, Lipschitz or just L ∞ .

A remark on the non-uniqueness in L^infty of the solutions to the two-dimensional Stokes problem in exterior domains

The paper is concerned with the IBVP in exterior domains of the two-dimensional Stokes equations. The goal was to investigate the well-posedness in the set of solutions assuming an initial data u_0in L^infty (Omega ), divergence free, and enjoying the property for all t>0 and c independent of u. For all u_0in L^infty , divergence-free one shows examples of non-uniqueness in the above set of solutions.

Existence and nonexistence of radial solutions for semilinear equations with bounded nonlinearities on exterior domains

In this article we study radial solutions of \(\Delta u + K(r)f(u)= 0\) on the exterior of the ball of radius R>0 centered at the origin in \({\mathbb R}^N\) where f is odd with f<0 on \(0, \beta\), f>0 on \(\beta\, \delta\) \(f\equiv 0 for (u> \delta\)), and where the function K(r) is assumed to be positive and \(K(r)\to 0) as (r \to \infty\)). The primitive \(F(u) = \int_0^u f(t)\), dt has a "hilltop'' at \(u=\delta\). With mild assumptions on f we prove that if \(K(r)\sim r^{-\alpha}\) with \(2< \alpha< 2(N-1\)) then there are n solutions of \(Delta\ u +(K(r)f(u)= 0\) on the exterior of the ball of radius R such that \(u\ to\ 0\) as \(r \to \infty\) if R>0 is sufficiently small. We also show there are no solutions if R>0 is sufficiently large. For more information see https://ejde.math.txstate.edu/Volumes/2020/117/abstr.html

On the asymptotic properties for stationary solutions to the Navier-Stokes equations

In this paper we study solutions of the stationary Navier-Stokes system, and investigate the minimal decay rate for a nontrivial velocity field at infinity in outside of an obstacle. We prove that in an exterior domain if a solution \begin{document}$ v $\end{document} and its derivatives decay like \begin{document}$ O(|x|^{-k}) $\end{document} for sufficiently large \begin{document}$ k $\end{document} , depending on the velocity field, as \begin{document}$ |x|\to \infty $\end{document} , then \begin{document}$ v $\end{document} is zero on that exterior domain. Constructive estimate for \begin{document}$ k $\end{document} is given. In the case where velocity field is only bounded at infinity, we show that the infimum of \begin{document}$ L^2 $\end{document} norm of a velocity field on a unit ball located at distance \begin{document}$ t $\end{document} from an origin is bounded from below as \begin{document}$ Ce^{-\beta t^\frac 43\ln(t)}. $\end{document} The proof of these results are based on the Carleman type estimates, and also the Kelvin transform.

Leray's plane steady state solutions are nontrivial

We study solutions to the obstacle problem for the stationary Navier–Stokes system in a two dimensional exterior domain (flow past a prescribed body). We prove that the classical Leray solution to this problem is always nontrivial. No additional condition (on symmetry or smallness, etc.) is assumed. This is a complete extension of a classical result of C.J. Amick (1988) [1] where nontriviality was proved under symmetry assumption.

Artificial boundary conditions for linearized stationary incompressible viscous flow around rotating and translating body

AbstractWe consider the linearized and nonlinear stationary incompressible flow around rotating and translating body in the exterior domain , where is open and bounded, with Lipschitz boundary. We derive the pointwise estimates for the pressure in both cases. Moreover, we consider the linearized problem in a truncation domain of the exterior domain under certain artificial boundary conditions on the truncating boundary , and then compare this solution with the solution in the exterior domain to get the truncation error estimate.

Finite Time Blow-Up for Wave Equations with Strong Damping in an Exterior Domain

We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power-type nonlinearity $$|u|^p$$ . We will establish blow-up results under some conditions on the initial data and the exponent p, using the method of test function with an appropriate harmonic functions. We also study the existence of mild solution and its relation with the weak formulation.

Limit behavior of attractive Bose-Einstein condensates passing an obstacle

In this paper, we mainly investigate ground states of trapped attractive Bose-Einstein condensates (BEC) passing an obstacle in the plane, which can be described by an L2-critical constraint minimization problem in an exterior domain Ω=R2∖ω, where the bounded convex domain ω⊂R2 with smooth boundary denotes the region of the obstacle. It is shown that minimizers (i.e. ground states) exist, if and only if the interaction strength a satisfies a<a⁎=‖Q‖22, where Q>0 is the unique positive radial solution of Δu−u+u3=0 in R2. If the trapping potential V(x) attains its global minima only along the whole boundary ∂Ω, the limit behavior of minimizers is analyzed as a↗a⁎ by employing the Pohozaev identity and the delicate energy analysis, where the mass concentration occurs at the flattest critical point of V(x) on ∂Ω.

Semipositone nonlocal Neumann elliptic system depending on the gradient in exterior domains

We study existence and multiplicity of positive radial solutions for a semipositone coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the Neumann boundary conditions are nonlocal. We use a nonstandard cone to establish existence of multiple solutions by means of fixed point index theory.

Semilinear mixed problems in exterior domains for σ-evolution equations with friction and coefficients depending on spatial variables

The main purpose of this paper is to investigate decay estimates for solutions to the Cauchy problemutt+a1(x)(−Δ)σu+aut=0,u(0,x)=u0(x),ut(0,x)=u1(x)forx∈Rn, as well as the estimates for solutions to the corresponding Cauchy-Dirichlet problem in an exterior domain Ω⊂Rn. Here a is a positive constant. The coefficient a1=a1(x) is supposed to be continuous and positive on the closure Ω‾. The parameter σ∈(0,1) brings to the model the so-called Levi-stable behavior for the corresponding diffusion stochastic process. Finally, we show the global (in time) existence of energy solutions from evolution spaces to the semilinear modelsutt+a1(x)(−Δ)σu+aut=|ut|p,u(0,x)=u0(x),ut(0,x)=u1(x), in domain (0,∞)×Ω with arbitrarily small initial data.

Stability of the radially symmetric stationary wave of the Burgers equation with multi‐dimensional initial perturbations in exterior domain

AbstractThe present paper is concerned with stability of the stationary solution of the Burgers equation in exterior domains in . In the previous papers [5, 6, 7] the asymptotic behavior of radially symmetric solutions for the multi‐dimensional Burgers equation in exterior domains in , has been considered. The results [5, 6, 7] are restricted to stability of radially solutions within the class of spherically one dimensional flow. However, from a viewpoint of fluid dynamics, it is the rare case that such a radially symmetric stationary wave remains to be a radial flow under the initial disturbance. Hence it seems to be natural to handle the non‐radially symmetric perturbed fluid motion even from the radially symmetric one. On the other hand, Kozono and Ogawa [8] showed the asymptotic stability of stationary solutions for the incompressible Navier–Stokes equation on multi‐dimensional spaces. In this paper we apply their method [8] to the multidimensional Burgers equation, and show the asymptotic stability for stationary wave on .

A proof of the instability of AdS for the Einstein-null dust system with an inner mirror

In 2006, Dafermos and Holzegel formulated the so-called AdS instability conjecture, stating that there exist arbitrarily small perturbations to AdS initial data which, under evolution by the Einstein vacuum equations for $\Lambda<0$ with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically symmetric Einstein--scalar field system was initiated by Bizon and Rostworowski, followed by a vast number of numerical and heuristic works by several authors. In this paper, we provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein--massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein--null dust system, allowing for both ingoing and outgoing dust. In order to overcome the break down of this system occuring once the null dust reaches the centre $r=0$, we place an inner mirror at $r=r_{0}>0$ and study the evolution of this system on the exterior domain $\{r\ge r_{0}\}$. The structure of the maximal development and the Cauchy stability properties of general initial data in this setting are studied in our companion paper. The statement of the main theorem is as follows: We construct a family of mirror radii $r_{0\epsilon}>0$ and initial data $\mathcal{S}_{\epsilon}$, $\epsilon\in(0,1]$, converging to the AdS initial data in a suitable norm, such that, for any $\epsilon>0$, the maximal development $(\mathcal{M}_{\epsilon},g_{\epsilon})$ of $\mathcal{S}_{\epsilon}$ contains a black hole region. Our proof is based on purely physical space arguments.

Asymptotic flatness of Morrey extremals

We study the limiting behavior as $$|x|\rightarrow \infty $$ of extremal functions u for Morrey’s inequality on $$\mathbb {R}^n$$ . In particular, we compute the limit of u(x) as $$|x|\rightarrow \infty $$ and show |x||Du(x)| tends to 0. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $$-\Delta _pu=c(\delta _{x_0}-\delta _{y_0})$$ for some $$c\in \mathbb {R}$$ and distinct $$x_0,y_0\in \mathbb {R}^n$$ . More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded p-harmonic functions on exterior domains of $$\mathbb {R}^n$$ for $$p>n$$ .

Diffusion phenomenon for indirectly damped hyperbolic systems coupled by velocities in exterior domains

We consider a system of two coupled wave equations in an exterior domain, where only one equation is directly damped. We prove that the solutions are [Formula: see text]-approximated by special functions, classified into three patterns depending on the values of the damping and the coupling terms, as well as on the speeds of the waves. In particular, when the damping term is sufficiently large, the waves are asymptotically equal to solutions of parabolic-type equations as [Formula: see text]. This result generalizes the standard diffusion phenomenon for directly damped hyperbolic systems.

Decay for turbulent solutions of the magneto-hydrodynamic equations in an exterior domain

By using the spectral analysis method and establishing some new inequalities, we address the decay properties of the energy of turbulent solutions to the non-stationary magneto-hydrodynamic equations in exterior domains. Our results reveal that the L2-decay rate of each turbulent solution is generally determined by its linear part.

Decay of local energy for solutions of the free Schrödinger equation in exterior domains

On s’intéresse dans cet article à la décroissance temporelle de la solution du problème extérieur pour l’équation de Schrödinger, dans le cas d’obstacles. Les situations auquelles on s’intéresse sont le cas général (pas d’hypothèse géométrique) ou alors des situations faiblement captantes.

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

SummaryThree‐dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the ‐transform and the convolution quadrature method, we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency‐domain BEMs. Then, taking advantage of a well‐designed high‐frequency approximation, we drastically reduce the number of frequency‐domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regard to the time discretization and for the spacial discretization, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid‐structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity and the scattering of an abrupt wave by simple and realistic geometries.

PML absorbing boundary condition for structure-preserving seismic wave modeling

Some efficient structure preserving algorithms have been proposed for high precision seismic exploration and detection in recent years. However, feasible artificial boundary conditions to truncate unbounded medium are never discussed in structure preserving seismic modeling. We pay our emphasis on application of perfectly matched layer absorbing boundary condition (PML ABC) to fit the structure preserving algorithms. In the domain of interest, a typical structure preserving algorithm is presented to discretize the conservative dynamic system, while in the PML region, traditional central difference is applied to solve the dissipative system. In the adjacent region of the interior and exterior domains, the temporary values of wave field between two time steps are approximated by interpolation using the present and the updated value of the wave fields in the PML region. A numerical experiment is adopted to demonstrate compatibility of the algorithms and no visible reflections of outgoing waves occur on the edge of interior domain.

The radially symmetric Green’s function for Dirichlet problem of Pennes equation outside of a circular domain

We find a general method to obtain the radially symmetric solutions of Dirichlet problem for Pennes bioheat equation in the exterior domain of a circle through the computation of Green’s function of a naturally related operator. We apply this technique to solve a problem in radio-frequency ablation.

The Helmholtz–Weyl decomposition of $$L^r$$ vector fields for two dimensional exterior domains

Let $$\Omega $$ be a two-dimensional exterior domain with smooth boundary $$\partial \Omega $$ and $$1< r < \infty $$ . Then $$L^r(\Omega )^2$$ allows a Helmholtz–Weyl decomposition, i.e., for every $$\mathbf{u}\in L^r(\Omega )^2$$ there exist $$\mathbf{h} \in X^r_{\tiny {\text{ har }}}(\Omega )$$ , $$w \in {\dot{H}}^{1,r}(\Omega )$$ and $$p \in {\dot{H}}^{1,r}(\Omega )$$ such that $$\begin{aligned} \mathbf{u} = \mathbf{h} + \mathrm{rot}\, w + \nabla p. \end{aligned}$$ The function $$\mathbf{h}$$ can be chosen alternatively also from $$V^r_{\tiny {\text{ har }}}(\Omega )$$ , another space of harmonic vector fields subject to different boundary conditions. These spaces $$X^r_{\tiny {\text{ har }}}(\Omega )$$ and $$V^r_{\tiny {\text{ har }}}(\Omega )$$ of harmonic vector fields are known to be finite dimensional. The above decomposition is unique if and only if $$1< r \leqq 2$$ , while in the case $$2< r < \infty $$ , uniqueness holds only modulo a one dimensional subspace of $$L^r(\Omega )^2$$ . The corresponding result for the three dimensional setting was proved in our previous paper, where in contrast to the two dimensional case, there are two threshold exponents, namely $$r=3/2$$ and $$r =3$$ . In our two dimensional situation, $$r=2$$ is the only critical exponent, which determines the validity of a unique Helmholtz–Weyl decomposition.