Three-dimensional Exterior Domain Research Articles

Global strong solutions to full compressible magnetohydrodynamic system with vacuum in 3D exterior domains

This paper investigates the full compressible magnetohydrodynamic system in three-dimensional exterior domains. For the initial-boundary-value problem of this system with slip boundary condition for the velocity, adiabatic one for the temperature, and perfect one for the magnetic field, we establish the global existence and uniqueness of strong solutions, under the condition that the initial data are of small energy but possibly large oscillations, where the initial density and temperature are both allowed to vanish. Moreover, the large-time behavior of the strong solutions is also shown.

Global well-posedness to the compressible nematic liquid crystal flows with large oscillations and vacuum in 3D exterior domains

This paper aims at extending our previous work (Y. Liu and X. Zhong, arXiv:2204.06227) to the exterior problem. We show the global well-posedness of strong solutions to a simplified Ericksen–Leslie model of compressible nematic liquid crystal flows in a three-dimensional exterior domain when the initial total energy is sufficiently small. In order to obtain necessary a priori estimates on the boundary, some new analytic techniques are used.

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.

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. "

Low Frequency Asymptotics and Electro-Magneto-Statics for Time-Harmonic Maxwell's Equations in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

We prove that the time-harmonic solutions to Maxwell's equations in a three-dimensional exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sob...

On time-periodic solutions to the Boussinesq equations in exterior domains

We construct time-periodic solutions to the Boussinesq equations in a three-dimensional exterior domain. To this end, we estimate the L1 norm of buoyancy, and give some estimates of the heat semigroup in the Lorentz space.

Solutions to a nonlinear Neumann problem in three-dimensional exterior domains

We prove the existence of multipeak solutions to a nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three-dimensional exterior domains.

The existence theorem for the steady Navier–Stokes problem in exterior axially symmetric 3D domains

We study the nonhomogeneous boundary value problem for the Navier–Stokes equations of steady motion of a viscous incompressible fluid in a three-dimensional exterior domain with multiply connected boundary. We prove that this problem has a solution for axially symmetric domains and data (without any smallness restrictions on the fluxes). Our main tool is a recent version of the Morse–Sard theorem for Sobolev functions obtained by Bourgain et al. (Rev Mat Iberoam 29(1):1–23, 2013).

Strong solutions of the Boussinesq system in exterior domains

Abstract We consider the instationary Boussinesq equations in a smooth three-dimensional exterior domain. A strong solution is a weak solution such that the velocity field additionally satisfies Serrin's condition. The crucial point in this concept of a strong solution is the fact that we have required no additional integrability condition for the temperature. We present a sufficient criterion for the existence of such a strong solution. Further we will characterize the class of initial values that allow the existence of such a strong solution in a sufficiently small interval. Finally, we will obtain a uniqueness criterion for weak solutions of the Boussinesq equations which is based on the identification of a weak solution with a strong solution.

On weak $$D$$ D -solutions to the non-stationary Navier–Stokes equations in a three-dimensional exterior domain

In this note we study the Navier–Stokes initial boundary value problem in exterior domains. We assume that the initial data has just finite Dirichlet norm. We call the solution \(D\)-solution. It is well known that the analogous steady problem is solved in Galdi (An Introduction to the Mathematical Theory of the Navier–Stokes Equations II. Springer, Berlin, 1994), as well as the existence of time periodic solutions in Maremonti et al. (J Math Sci 93(5):719–746, 1999, Zap. Nauchn. Semin. POMI 233:142–182, 1996). So it is natural to inquire about the case of the nonstationary problem.

Spatial Decay of Time-Dependent Incompressible Navier--Stokes Flows with Nonzero Velocity at Infinity

We consider the time-dependent Navier--Stokes system in a three-dimensional exterior domain with nonzero velocity at infinity. Under suitable assumptions on the data, it is shown that the velocity part of strong solutions, after subtraction of the far-field velocity, decays as $\bigl( |x| \cdot (1+|x|-x_1) \bigr) ^{-1}$, and its spatial gradient as $\bigl( |x| \cdot (1+|x|-x_1) ,\bigr) ^{-3/2}$, for $|x|\to \infty $. The solution class in question includes solutions obtained by $L^2$-variational methods and characterized by the fact that the velocity $u$ and its spatial gradient $ \nabla _xu$ are $L ^{ \infty } (L^2)$-functions, and $\nabla _xu$ is additionally $L^2$-integrable in time and in space.

Low frequency expansion in thermoelasticity with second sound in three dimensions

We consider the initial-boundary value problem in hyperbolic thermoelasticity with second sound in a three-dimensional exterior domain. The low frequency expansion of solutions to the corresponding stationary resolvent problem is given and the limit to the classical thermoelastic problem is investigated.

Local Energy Decay Estimate of Solutions to the Thermoelastic Plate Equations in Two- and Three-Dimensional Exterior Domains

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations: utt + ∆2\_u\_ + ∆\_θ\_ = 0 and θt − ∆\_θ\_ − ∆\_ut\_ = 0 in Ω × (0, ∞), subject to Dirichlet boundary conditions: u|Γ = Dν u|Γ = θ|Γ = 0 and initial conditions (u, ut, θ)|t=0 = (\_u\_0, \_v\_0, θ\_0). Here Ω is an exterior domain (domain with bounded complement) in ℝ\_n with n = 2 or n = 3, the boundary Γ of which is assumed to be a \_C\_4-hypersurface.

Spatial Decay of Time-Dependent Oseen Flows

We consider an initial-boundary value problem for the time-dependent Oseen system in a three-dimensional exterior domain. We show that weak solutions of this problem may be represented by a sum of volume and single layer potentials. This representation is then used in order to study spatial decay of weak solutions.

THREE-DIMENSIONAL EXTERIOR PROBLEM OF DARWIN MODEL AND ITS NUMERICAL COMPUTATION

Darwin model is a good approximation to Maxwell's equations, and this paper is concerned with the boundary value problem of the Darwin model in three-dimensional exterior domains. Firstly we establish the variational formulations of Darwin model in exterior domains and prove the existence and uniqueness. Then we prove a useful decomposition theorem that any function in unbounded exterior domains belonging to L2(Ω) can be decomposed into the sum of a function's gradient and another function's rotation, such decomposition is crucial in inducing the Darwin model. At last we spend some energy in using the infinite element method to solve the Darwin model in axis-symmetric exterior domains cases. Error estimates are obtained in weighted spaces, and numerical examples verify the convergence once again.

On an existence theorem of global strong solutions to the magnetohydrodynamic system in three-dimensional exterior domains

In this paper we study the initial-boundary-value problem for the magnetohydrodynamic system in three-dimensional exterior domains. We show an existence theorem of global in time strong solutions for small $L^3$-initial data and we also show its asymptotic behavior when time goes to infinity.

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Let Ω be a three-dimensional exterior domain of class C2,α, 0<α<1. Assume that a Navier-Stokes liquid is moving in Ω under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|−1 and its spatial gradient decays as |x|−2, both uniformly in time. In addition, the pressure p decays like |x|−2 and its gradient like |x|−3, for almost all t∈[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finn’s ‘‘physically reasonable’’ solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the ‘‘energy inequality’’ and with corresponding pressure fields verifying mild summability conditions in Ω×[0,T].

Stability of periodic solutions of Navier-Stokes equations in a three-dimensional exterior domain

We study the problem of stability of periodic solutions of Navier-Stokes equations in an exterior domain with a given constant velocity vector at infinity. We study spectral properties of the monodromy operator, using its connection with a certain Wiener-Hopf operator with an operator-valued symbol.

Numerical method for Oseen's linearized equations in three-dimensional exterior domains

In this paper, we consider Oseen's linearized equations, which appear in fluid mechanics, in three-dimensional exterior domains, and propose a new numerical method based on fundamental solution method coupled with collocation method to solve this problem. We show the effectiveness of our method by numerical experiments.

Some Estimates of Solutions for the Equations of Motion of Compressible Viscous Fluid in the Three-Dimensional Exterior Domain

We consider the equations of motion of compressible viscous fluid in an exterior domain in R 3. We give the L q − L p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem. In particular, we shall give L 1 estimates, which implies the diffusion wave of compressible Navier–Stokes flow.