Exterior Domain Research Articles

On the Regularity of Weak Solutions to Time-Periodic Navier–Stokes Equations in Exterior Domains

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.

Global strong solutions to the viscous liquid–gas two-phase flow model with slip boundary conditions in 3D exterior domains

This paper is concerned with the viscous liquid–gas two-phase flow model in a three-dimensional exterior domain with the slip boundary conditions. We are able to prove the existence of a global strong solution when the initial total energy is suitably small. The initial masses of liquid and gas are allowed to contain a vacuum, with no extra restriction between the initial masses of liquid and gas. It should be noted here that we consider the slip boundary condition in a three dimensional (3D) exterior domain which is in sharp contrast to result of Yu (2021) where they consider the Cauchy problem.

On radial solutions for some elliptic equations involving operators with unbounded coefficients in exterior domains

We study existence and multiplicity of radial solutions for some quasilinear elliptic problems involving the operator $L_N=\Delta - x\cdot \nabla$ on $\mathbb{R}^N\setminus B_1$, where $\Delta$ is the Laplacian, $x\cdot \nabla$ is an unbounded drift term, $N\geq 3$ and $B_1$ is the unit ball centered at the origin. We consider: (i) Eigenvalue problems, and (ii) Problems involving a nonlinearity of concave and convex type. On the first class of problems we get a compact embedding result, whereas on the second, we address the well-known question of Ambrosetti, Brezis and Cerami from 1993 concerning the existence of two positive solutions for some problems involving the supercritical Sobolev exponent in symmetric domains for the Laplacian. Specifically, we provide\linebreak a new approach of answering the ABC-question for elliptic problems with unbounded coefficients in exterior domains and we find asymptotic properties of the radial solutions. Furthermore, we study the limit case, namely when nonlinearity involves a sublinear term and a linear term. As far as we know, this is the first work that deals with such a case, even for the Laplacian. In our approach, we use both topological and variational arguments.

Effective seismic force retrieval from surface measurement for SH-wave reconstruction

We present a new method to obtain dynamic body force at virtual interfaces to reconstruct shear wave motions induced by a source outside a truncated computational domain. Specifically, a partial differential equation (PDE)-constrained optimization method is used to minimize the misfit between measured motions at a limited number of sensors on the ground surface and their counterparts reconstructed from optimized forces. Numerical results show that the optimized forces accurately reconstruct the targeted ground motions in the surface and the interior of the domain. The proposed optimization framework yields a particular force vector among other valid solutions allowed by the domain reduction method (DRM). Per this optimized or inverted force vector, the reconstructed wave field is identical to its reference counterpart in the domain of interest but may differ in the exterior domain from the reference one. However, we remark that the inverted solution is valid and introduce a simple post-process that can modify the solution to achieve an alternative force vector corresponding to the reference wave field. We also study the desired sensor spacing to accurately reconstruct the wave responses for a given dominant frequency of interest. We remark that the presented method is omnidirectionally applicable in terms of the incident angle of an incoming wave and is effective for any given material heterogeneity and geometry of layering of a reduced domain. The presented inversion method requires information on the wave speeds and dimensions of only a reduced domain. Namely, it does not need any information on the geophysical profile of an enlarged domain or a seismic source profile outside a reduced domain. Thus, the computational cost of the method is compact even though it leads to the high-fidelity reconstruction of wave response in the reduced domain, allowing for studying and predicting ground and structural responses using real seismic measurements.

Efficient computational framework for image-based micromechanical analysis of additively manufactured Ti-6Al-4V alloy

The increase in additively manufactured (AM) Ti-6Al-4V alloys in high-performance industrial applications has necessitated the development of robust computational models that can aid in their qualification and certification. Physics-based micromechanical models, relating the AM-processed material microstructure and defect state with the overall material response and life, can play an important role in reducing uncertainty in component behavior and increasing acceptance. Motivated by this need, the present paper develops a novel image-based crystal plasticity finite element model (CPFEM) for efficient micromechanical simulation of the additively manufactured Ti-6Al-4V alloy, whose Widmanstätten microstructure is characterized by 12 HCP α lath variants in the parent β grain. A unique feature of this work is the creation of an efficient crystal plasticity framework for the parent β grain polycrystalline ensembles with parametric representation of the α lath statistics of size, shape, orientation, and crystallography. This statistical representation is expected to significantly enhance its efficiency over models that represent each α lath explicitly in the microstructure. Defects in the form of voids are represented at two scales. The smaller voids in the microstructure are manifested as porosity or void volume fraction distribution in the crystal plasticity model. Larger voids are represented explicitly in the statistically equivalent microstructural volume element (SEMVE) model. The models are built from experimentally acquired electron back scatter diffraction (EBSD) and micro-focus X-ray computed tomography (XCT) images and calibrated and validated with mechanical testing data. This paper extends the developments in Pinz et al. (2022) through the development of a special self-consistent boundary condition in the context of a concurrent model to overcome limitations of periodicity boundary conditions. The concurrent model embeds the SEMVE in a homogenized exterior domain represented by a rate-dependent isotropic plasticity model. Parametric studies are conducted to comprehend the effect of void size, shape and orientation on the overall material response.

Global Well-Posedness of Second-Grade Fluid Equations in 2D Exterior Domain

In this article, we consider 2D second grade fluid equations in exterior domain with Dirichlet boundary conditions. For initial data $\boldsymbol{u}_0 \in \boldsymbol{H}^3(\Omega)$, the system is shown to be global well-posed. Furthermore, for arbitrary $T > 0$ and $s \geq 3$, we prove that the solution belongs to $L^\infty([0, T]; \boldsymbol{H}^s(\Omega))$ provided that $\boldsymbol{u}_0$ is in $\boldsymbol{H}^s(\Omega)$.

Ground state solutions for electromagnetic Schrödinger equations on unbounded domains

In this paper, we study the existence of ground state solutions for a electromagnetic Schrödinger equation in exterior domains. The main technical approach is based on the Nehari manifold method jointly with variational and topological methods.

Self-consistent homogenization-based parametrically upscaled continuum damage mechanics model for composites subjected to high strain-rate loading

This paper develops a Parametrically Upscaled Continuum Damage Mechanics (PUCDM) model for carbon fiber/epoxy matrix composites subjected to high strain-rate loading. The PUCDM model for predicting high strain-rate induced deformation and damage evolution at the macroscopic scale, explicitly incorporates microstructural morphology and micro-inertia in its constitutive coefficients. It is developed using self-consistent homogenization that uses a concurrent multiscale model to account for the effect of micro-inertia and stress wave interaction with the microstructure. The concurrent model embeds a statistically equivalent representative volume element (SERVE) of the microstructure in an exterior domain, whose constitutive and damage response are described by the PUCDM model. Micromechanical modeling for the composite microstructure involves a unified cohesive zone enhanced phase-field damage model for fiber-matrix interface debonding, fiber breakage, and matrix cracking. The PUCDM model parameters for the upscaled domain are determined through micro-macro-scale energy equivalence, along with traction reciprocity and displacement continuity at the SERVE-exterior domain interface that are enforced using a Lagrange multiplier-based approach. Simulations using the concurrent model analyze stress wave propagation and damage evolution in composite microstructures with three different fiber volume fractions at multiple strain rates in the range of 102 − 105 s−1. The fiber volume fraction and spatial distributions affect the overall stiffness and damage evolution in the PUCDM model. Results of the analysis demonstrate the need for representation of micro-inertia, strain rates and microstructure morphology in the PUCDM-based stiffness and damage model parameters for high strain-rates above [Formula: see text].

Monotonic sequences of minimal solutions for semipositone elliptic systems

The main goal of this paper is to formulate sufficient conditions guaranteeing the existence of decaying solutions of the system of elliptic equations in a certain exterior domain. The asymptotic behavior of these solutions and their gradients will be characterized. We focus on the construction of uncountable sets of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. Our method can be applied for nonlinearities which are negative at the origin (semipositone problem).

On variational inequalities on exterior domains with multivalued convection terms

In this paper, we study variational inequalities of the form 〈−ΔNu,v−u〉+〈F(u),v−u〉≥0,∀v∈Ku∈K,where ΔNu=div|∇u|N−2∇u is the N-Laplacian, K is a closed convex set in the Beppo–Levi space X=D01,N(Ω), where Ω is the exterior of the closed unit ball B(0,1)¯ in RN (N≥2).We are interested here in the case where F is given by a multivalued function f=f(x,u,∇u) that depends also on the gradient ∇u of the unknown function. We consider a functional analytic framework of the above problem, including conditions on the lower order term f such that the problem can be appropriately formulated in X and a related weighted Orlicz space, and the involved mappings have certain useful monotonicity-continuity properties.Existence of solutions in coercive and noncoercive cases are studied. In the noncoercive case, we follow a sub-supersolution approach and prove the existence of solutions of the above inequality under a multivalued local growth condition of Bernstein–Nagumo type.

A new approach for solving heat conduction under zero and non-zero initial conditions

This paper provides a new approach for the analysis of transient heat diffusion arising from conduction. The diffusion is subject to zero and non-zero initial conditions in a two-dimensional domain, containing confined heterogeneous subdomains. This system can be subjected to a heat source. A new formulation is presented, in the frequency domain. It allows coupling between the boundary element method (BEM) and other methods (such as the method of fundamental solutions (MFS), the finite element method (FEM), the finite difference method (FDM), the meshless Petrov-Galerkin (MLGP) method, the virtual element method (VEM), and analytical formulations). The BEM is used for the homogeneous exterior domain and the other method is used to evaluate the solutions for the non-homogeneous confined subdomains, to overcome the specific limitations of each of these methods.

Global solutions to an initial boundary problem for the compressible 3D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

An initial boundary value problem for compressible magnetohydrodynamics (MHD) is considered on an exterior domain (with the vanishing first Betti number) in in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier–Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which was not available in literature prior to the work in this paper, to the best of knowledge of the authors’.

Shape optimization for the strong routing of light in periodic diffraction gratings

In the quest for the development of faster and more reliable technologies, the ability to control the propagation, confinement, and emission of light has become crucial. The design of guide mode resonators and perfect absorbers has proven to be of fundamental importance. In this project, we consider the shape optimization of a periodic dielectric slab aiming at efficient directional routing of light to reproduce similar features of a guide mode resonator. For this, the design objective is to maximize the routing efficiency of an incoming wave. That is, the goal is to promote wave propagation along the periodic slab. A Helmholtz problem with a piecewise constant and periodic refractive index medium models the wave propagation, and an accurate Robin-to-Robin map models an exterior domain. We propose an optimal design strategy that consists of representing the dielectric interface by a finite Fourier formula and using its coefficients as the design variables. Moreover, we use a high order finite element (FE) discretization combined with a bilinear Transfinite Interpolation formula. This setting admits explicit differentiation with respect to the design variables, from where an exact discrete adjoint method computes the sensitivities. We show in detail how the sensitivities are obtained in the quasi-periodic discrete setting. The design strategy employs gradient-based numerical optimization, which consists of a BFGS quasi-Newton method with backtracking line search. As a test case example, we present results for the optimization of a so-called single port perfect absorber. We test our strategy for a variety of incoming wave angles and different polarizations. In all cases, we efficiently reach designs featuring high routing efficiencies that satisfy the required criteria.

Global well-posedness of 2D Euler-α equation in exterior domain

After casting Euler-α equations into vorticity-stream function formula, we obtain some very useful estimates from the properties of the vorticity formula in exterior domain. Basing on these estimates, one can have got the global existence and uniqueness of the solutions to Euler-α equations in 2D exterior domain provided that the initial data is regular enough.

Higher order evolution inequalities with convection terms in an exterior domain of [formula omitted

We establish new blow-up results for a higher order (in time) evolution inequality involving a convection term in an exterior domain of RN. We study two types of inhomogeneous boundary conditions: Dirichlet and Neumann. Using a unified approach, we obtain optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary problems. We also investigate the effect of a nonlinear memory term on the critical behaviors of the considered problems. Notice that no restriction on the sign of solutions is imposed.

On a Hardy–Sobolev-type inequality and applications

In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition [Formula: see text] where [Formula: see text] is an exterior domain such that [Formula: see text].

Critical curve for a two-species chemotaxis model with two chemicals in * *Supported by National Natural Science Foundation of China (12171218), LiaoNing Revitalization Talents Program (XLYC2007022) and Key Project of Education Department of Liaoning Province (LJKZ0083).

This paper deals with a two-species chemotaxis model with two chemicals in . In our previous work (2019 Nonlinearity 32 4762–78), the critical mass was obtained that the solutions exist globally if m 1 m 2 − 4π(m 1 + m 2) < 0, and the finite time blow-up of solutions may occur if m 1 m 2 − 4π(m 1 + m 2) > 0, where m 1 and m 2 describe the initial mass of the two species, respectively. In the present paper we furthermore determine that the critical situation belongs to the global existence case, namely, the system admits global solutions if m 1 m 2 − 4π(m 1 + m 2) = 0. To apply the key logarithmic Hardy–Littlewood–Sobolev inequality of vector form for the critical case, we should establish a positive lower bound for L 1-norms of the two species in exterior domains uniformly for t > 0.

A numerical model with high-accuracy and high-efficiency for the soil dynamic impedance of pile groups under vertical vibration

This paper develops a novel numerical model for the calculation of dynamic impedance of soil considering end-bearing pile groups under vertical vibration. The soil layer is treated as a linear viscoelastic material with hysteretic damping. The vertical displacement is assumed to be uniform in each individual pile in the groups at the same embedment depth. Using separation of variables and applying proper boundary conditions at the surface and rigid bedrock, the governing equation of soil layer in three-dimensional (3D) is first transformed into a two-dimensional (2D) Helmholtz equation in xy plane, where the solution in the vertical direction is obtained analytically. To solve this problem efficiently and accurately, the whole 2D unbounded soil is then divided into a finite interior domain and an infinite exterior domain. The finite domain is discretized using the finite element method (FEM) while an exact circular artificial boundary condition (ABC) is developed to model the infinite domain. Finally, the effect of pile group interaction on the dynamic impedance of soil is investigated considering various factors, e.g., dimensionless frequency, pile number, pile slenderness ratio and relative distance between piles.

Boundedness and Stability of Solutions to the Non-autonomous Oseen-Navier-Stokes Equation

"We consider the motion of a viscous imcompressible fluid past a rotating rigid body in three-dimensional, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the non-autonomous Oseen-Navier-Stokes equations in a fixed exterior domains. We prove the existence and stability of bounded mild solutions in time t to ONSE in three-dimensional exterior domains when the coefficients are time dependent. Our method is based on the L^p-L^q-estimates of the evolution family (U(t,s))¬¬ and that of its gradient to prove boundedness of solution to linearized equations. After, we use fixed-point arguments to obtain the result on boundedness of solutions to non-linearized equations when the data belong to L^p-space and are sufficiently small. Finally, we prove existence and polynomial stability of bounded solutions to ONSE with the same condition. Our result is useful for the study of the time-periodic mild solution to the non-autonomous Oseen-Navier-Stokes equations in an exterior domains. "

On the Poisson equation in exterior domains

We construct a solution of the Poisson equation in exterior domains $\Omega \subset \mathbb R^n,\;n \ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(\Omega),;1 &amp;lt; q &amp;lt;\infty,$ with methods of potential theory and integral equations. We investigate the corresponding null spaces and prove that its dimensions is equal to $n+1$ independent of $q$.