Exterior Domain Research Articles

A note on local energy decay results for wave equations with a potential

In this paper, we derive uniform local energy decay results for wave equations with a short-range potential in an exterior domain. In this study, we considered this problem within the framework of non-compactly supported initial data, unlike previously reported studies. The essential parts of analysis are both L 2 -estimates of the solution itself and the weighted energy estimates. Only a multiplier method is used, and we do not rely on any resolvent estimates.

Asymptotic expansion of solutions to the wave equation with space-dependent damping

We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy methods with suitable supersolutions of the corresponding parabolic problem.

A Practical Walk-on-Boundary Method for Boundary Value Problems

We introduce the walk-on-boundary (WoB) method for solving boundary value problems to computer graphics. WoB is a grid-free Monte Carlo solver for certain classes of second order partial differential equations. A similar Monte Carlo solver, the walk-on-spheres (WoS) method, has been recently popularized in computer graphics due to its advantages over traditional spatial discretization-based alternatives. We show that WoB's intrinsic properties yield further advantages beyond those of WoS. Unlike WoS, WoB naturally supports various boundary conditions (Dirichlet, Neumann, Robin, and mixed) for both interior and exterior domains. WoB builds upon boundary integral formulations, and it is mathematically more similar to light transport simulation in rendering than the random walk formulation of WoS. This similarity between WoB and rendering allows us to implement WoB on top of Monte Carlo ray tracing, and to incorporate advanced rendering techniques (e.g., bidirectional estimators with multiple importance sampling, the virtual point lights method, and Markov chain Monte Carlo) into WoB. WoB does not suffer from the intrinsic bias of WoS near the boundary and can estimate solutions precisely on the boundary. Our numerical results highlight the advantages of WoB over WoS as an attractive alternative to solve boundary value problems based on Monte Carlo.

An efficient Fourier-Laguerre spectral-Galerkin method for exterior problems of two-dimensional complex obstacles

In this paper, we propose a Fourier-Laguerre spectral method for exterior problems of two-dimensional complex obstacles based on the mapping method. We first use a polar coordinate transformation to convert the exterior domains of complex obstacles into the exterior domain of the unit disk. Then we apply the polar coordinate transformation to the exterior problem of an elliptic equation, derive its weak formulation and prove the existence and uniqueness of the weak solution. On this basis, we construct the Fourier-Laguerre spectral Galerkin scheme, describe its numerical implementation and analyze the convergence of the numerical solution under the H1-norm. Numerical results indicate that our spectral Galerkin method is easy to implement and possesses high-order accuracy.

Existence and nonexistence criteria for a system of biharmonic wave inequalities in an exterior domain of ℝN

We consider a system of nonlinear biharmonic wave inequalities posed in an exterior domain of [Formula: see text]. Three types of boundary conditions are investigated. For each case, the existence and nonexistence of weak solutions are studied. Our study yields naturally existence and nonexistence results for the corresponding stationary system and equation. Many new results are provided and some open questions are proposed.

Existence and orbital stability/instability of standing waves with prescribed mass for the $$L^{2}$$-supercritical NLS in bounded domains and exterior domains

Existence and orbital stability/instability of standing waves with prescribed mass for the $$L^{2}$$-supercritical NLS in bounded domains and exterior domains

Global wellposedness of the 3D compressible Navier–Stokes equations with free surface in the maximal regularity class

This paper concerns the global well posedness issue of the compressible Navier–Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal L p -L q estimate and the L p -L q decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal L p -L q regularity class for some p > 2 and q > 3. Namely, the solution is bounded as L p in time and L q in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal L p -L q regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.

No-Slip Boundary Condition for Vorticity Equation in 2D Exterior Domain

In this article we derive the no-slip boundary condition for a non-stationary vorticity equation. This condition generates the affine invariant manifold and no-slip integral relations on vorticity can be transferred to a Robin-type boundary condition.

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

AbstractConstruction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains.

Solvability of Iterative Classes of Nonlinear Elliptic Equations on an Exterior Domain

This work explores the possibility that iterative classes of elliptic equations have both single and coupled positive radial solutions. Our approach is based on using the well-known Guo–Krasnoselskii and Avery–Henderson fixed-point theorems in a Banach space. Furthermore, we utilize Rus’ theorem in a metric space, to prove the uniqueness of solutions for the problem. Examples are constructed for the sake of verification.

A Perturbed Cauchy Viscoelastic Problem in an Exterior Domain

A Cauchy viscoelastic problem perturbed by an inverse-square potential, and posed in an exterior domain of RN, is considered under a Dirichlet boundary condition. Using nonlinear capacity estimates specifically adapted to the non-local nature of the problem, the potential function and the boundary condition, we establish sufficient conditions for the nonexistence of weak solutions.

Positive solutions for Kirchhoff equation in exterior domains with small Sobolev critical perturbation

ABSTRACT The paper concerns with the existence of positive solutions for the following Kirchhoff problem { − ( a + b ∫ Ω | ∇u | 2 d x ) Δu + u = | u | p − 2 u + ε | u | 4 u in Ω , u ∈ H 0 1 ( Ω ) , where a, b>0, 4<p<6, 0 $ ]]> ε > 0 is a parameter and Ω ⊂ R 3 is an exterior domain, that is, Ω is an unbounded domain with R 3 ∖Ω nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when R 3 ∖Ω is contained in a small ball and 0 $ ]]> ε > 0 is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309–330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation.

A system of wave inequalities with inverse-square potentials in an exterior domain

ABSTRACT We study the existence and nonexistence of solutions to a system of wave inequalities with inverse-square potentials in an exterior domain of R N , under inhomogeneous Dirichlet-type boundary conditions. Our study yields naturally existence and nonexistence results for the corresponding stationary wave system.

Blow-up of solutions for the Moore-Gibson-Thompson equations on exterior domain in two dimensions

This paper is devoted to investigating blow-up of solutions to initial boundary value problem of the Moore-Gibson-Thompson (MGT) equations on exterior domain in two dimensions. More precisely, upper bound lifespan estimates of solutions to the problem with power nonlinearity $|u|^{p}$, derivative nonlinearity $|u_{t}|^{p}$ and combined nonlinearities $|u_{t}|^{p} + |u|^{q}$ in the sub-critical and critical cases are established, respectively. The proofs are based on the new test function technique. Our main new contribution is that we improve the results in [36] in two dimensions. It is worth to mention that lifespan estimates of solutions are associated with the well-known Strauss and Glassey conjecture. Moreover, the variation trend of wave for the Cauchy problem of linear MGT equation with different initial values in two dimensions is showed by numerical simulation.

The Singular Limit of Second-Grade Fluid Equations in a 2D Exterior Domain

The Singular Limit of Second-Grade Fluid Equations in a 2D Exterior Domain

Global classical solutions to the viscous two-phase flow model with slip boundary conditions in 3D exterior domains

We consider the two-phase flow model in 3D exterior domains with slip boundary conditions. We establish the global existence of classical solutions of this system, provided that the initial energy is suitably small. Furthermore, we prove that the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions.

Computation of plane potential flow around multi-element airfoils using the Schottky–Klein prime function

Potential flow is computed around multi-element airfoils in two dimensions using numerical conformal mapping combined with the calculus of the Schottky–Klein prime function. The corners at the trailing edges of the airfoils are successively removed by Kármán–Trefftz maps. The map from the domain exterior to disks to the domain exterior to the smooth images of the airfoils is computed using an extension of a Fourier series method for the disk due to Fornberg. The velocity potential for flow in the circle domain, with circulation calculated to satisfy the Kutta condition in the airfoil domain, is computed using the Schottky–Klein prime function. The computations are compared to examples from the aerospace engineering literature.

On global existence for semilinear wave equations with space-dependent critical damping

The global existence for semilinear wave equations with space-dependent critical damping $\partial_{t}^{2} u - \Delta u + \frac{V_0}{|x|} \partial_{t} u = f(u)$ in an exterior domain is dealt with, where $f(u) = |u|^{p-1} u$ and $f(u) = |u|^{p}$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata–Todorova–Yordanov [J. Math. Soc. Japan (2013), 183–236] but the argument in this paper clarifies the precise dependence of the location of the support of initial data. The blowup phenomena are verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.

On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space

We establish blow-up results for a system of semilinear hyperbolic inequalities in an exterior domain of the half-space. The considered system is investigated under an inhomogeneous Dirichlet-type boundary condition depending on both time and space variables. In certain cases, an optimal criterium of Fujita-type is derived. Our results yield naturally sharp nonexistence criteria for the corresponding stationary wave system and equation.

Blow-up of solutions to the semilinear wave equation with scale invariant damping on exterior domain

This paper is concerned with the blow-up of solutions to the initial boundary value problem for the wave equation with scale invariant damping term on the exterior domain, where the nonlinear terms are power nonlinearity |u|^{p} , derivative nonlinearity |u_{t}|^{p} and combined nonlinearities |u_{t}|^{p}+ |u|^{q} , respectively. Upper bound lifespan estimates of solutions to the problem are obtained by constructing suitable test functions and utilizing the test function technique. The main novelty is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.