Nonhomogeneous Boundary Data Research Articles

Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE’s models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω-limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.

The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic systems, while the other one uses the theory of operator semigroups. Both methods have in common that known results for the Dirichlet problem for the second-order wave equation are instrumental. The obtained interior and boundary regularity estimates are optimal.

A Gradient Estimate Related Fractional Maximal Operators for a $p$-Laplace Problem in Morrey Spaces

In the present paper, we deal with the global regularity estimates for the $p$-Laplace equations with data in divergence form \[ \operatorname{div}(|\nabla u|^{p-2} \nabla u) = \operatorname{div}(|F|^{p-2} F) \quad \textrm{in $\Omega$}, \] in Morrey spaces with natural data $F \in L^p(\Omega;\mathbb{R}^n)$ and nonhomogeneous boundary data belongs to $W^{1,p}(\Omega)$. Motivated by the work of [M.-P. Tran, T.-N. Nguyen, New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data, J. Differential Equations 268 (2020), no. 4, 1427–1462], this paper extends that of global Lorentz–Morrey gradient estimates in which the `good-$\lambda$' technique was undertaken for a class of more general equations, and further, global regularity of weak solutions will be given in terms of fractional maximal operators.

Global weak solutions for a Vlasov–Fokker–Planck/Navier–Stokes system with nonhomogeneous boundary data

In this paper, we consider a kinetic-fluid model with nonhomogeneous Dirichlet boundary data in a 3D bounded domain. This model consists of a Vlasov–Fokker–Planck equation coupled with the compressible Navier–Stokes equations via a friction force. We establish the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient $$\gamma > \frac{3}{2}$$ ) with large initial data, and large velocity and density at the inflow boundary.

Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

Abstract A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside $(H^1(\varOmega ))^3$, with nonhomogeneous singular boundary data in the tangential trace space of $H({\mathbf{curl}}; \varOmega )$ and a singular right-hand-side source term in $(H_0({\mathbf{curl}}; \varOmega ))^{\prime}$ (the dual space of $H_0({\mathbf{curl}}; \varOmega ))$. Numerical results are presented to illustrate performance and the theoretical results.

Non-homogeneous initial-boundary-value problem of the fifth-order Korteweg-de Vries equation with a nonlinear dispersive term

We study the initial-boundary-value problem of the fifth-order KdV equation∂tu−∂x5u=c1u∂xu+c2u2∂xu+b1∂xu∂x2u+b2u∂x3u,b2≠0 with initial data u0∈Hs(0,L). The main analysis difficulty of this model is caused by the nonlinear dispersive term u∂x3u since the Kato smoothing effect of the solution to the linear fifth-order KdV equation with free source term can only improve two orders concerning the initial data. The Cauchy problem of this model has recently been proved to be globally well-posed in H−1(R) by Bringmann-Killip-Visan (2019). Under appropriate non-homogeneous boundary data, we prove that it is locally well-posed in Hs(0,L) with s∈[0,32) and admits a unique local solution as s≥32. As far as we know, this is the latest well-posedness result of the fifth-order KdV type equation with the nonlinear dispersive term when posed in any finite domain. The main innovation in this paper is that we construct a source term space which makes that the solution to the linear fifth-order KdV equation improves three orders on the source term.

An initial-boundary-value problem of 2D nonlinear Schrödinger equation in a strip

The paper studies the initial-boundary-value problem (IBVP) of a nonlinear Schrödinger equation posed on a strip domain R×[0,1] with non-homogeneous Dirichlet boundary conditions. For 0≤s<1/2, if the initial condition φ(x,y) is in Sobolev space Hs(R×[0,1]) and the boundary condition h(x,t) is inHs(R2)={h(x,t)∈L2(R2)|(1+|λ|+|ξ|)12(1+|λ|+|ξ|2)s2hˆ(λ,ξ)∈L2(R2)} where hˆ is the Fourier transform of h with respect to t and x, the local well-posedness of the IBVP in C([0,T];Hs(R×[0,1])) is proved. For 1/2<s<5/2, because of compatibility conditions, the same local well-posedness result holds if an extra condition h(x,0)∈Hs+12(R) is added. Moreover, the global well-posedness is obtained for s=1. The basic idea used here relies on the derivation of an integral operator for the non-homogeneous boundary data and the proof of Strichartz's estimates for this operator. After the problem is transformed to finding a fixed point of an integral operator, the contraction mapping argument then yields a fixed point using the Strichartz's estimates for initial and boundary operators. The global well-posedness is proved using a-priori estimates of the solutions.

Initial and boundary value problem of the unsteady Navier–Stokes system in the half-space with Hölder continuous boundary data

We study the initial–boundary value problem of the Navier–Stokes system in the half-space. We prove the unique solvability of the weak solution for some short time interval (0,T) with the velocity in Cα,α2(R+n×(0,T)), 0<α<1, when the given initial data in Cα(R+n) and the given boundary data in Cα,α2(Rn−1×(0,T)) satisfy the compatibility conditions. Our result generalizes the result in [29] considering nonhomogeneous Dirichlet boundary data.

Very weak solutions of the stationary Stokes equations in unbounded domains of half space type

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as $\mathbb R^n_+$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of $\mathbb R^n_+$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous Sobolev spaces. In addition to very weak solutions we also construct corresponding pressure functions in negative homogeneous Sobolev spaces.

Combined DG--CG Time Stepping for Wave Equations

The continuous and discontinuous Galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order. This eliminates the doubling of the number of variables that results when a second order problem is written as a first order system. Stability, convergence, and accuracy, of these schemes is established in the context of the wave equation. It is shown that natural interpolation of nonhomogeneous boundary data can degrade accuracy, and that this problem can be circumvented using interpolants matched with the time stepping scheme.

A direct discontinuous Galerkin method for the generalized Korteweg–de Vries equation: Energy conservation and boundary effect

A direct discontinuous Galerkin method for solving the generalized Korteweg–de Vries (KdV) equation with both periodic data and non-homogeneous boundary data is developed. A class of numerical fluxes is identified so that the method preserves both momentum and energy for the initial value problem. The method with such a property is numerically shown robust with less error in both phase and amplitude after long time simulation. Numerical examples are given to confirm the theoretical result and the capacity of this method for capturing soliton wave phenomena and various boundary wave patterns.

Global Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Boundary Data and Divergence

Consider a smooth bounded domain \Omega\subseteq\mathbb R^3 with boundary \partial\Omega , a time interval [0,T) , with T\in(0,\infty] , and the Navier–Stokes system in [0,T) \times \Omega , with initial value u_0 \in L^2_{\sigma} (\Omega) and external force f= {\mathrm{div}}\,F , F \in L^2 (0,T;L^2(\Omega)) . Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u_{\vert{\partial \Omega}}=0 , {\mathrm{div}}\,u=0 to the more general class of Leray-Hopf type weak solutions u with general data u_{\vert{\partial \Omega}} =g , {\mathrm{div}}\,u=k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u=v+E with some vector field E in [0,T)\times \Omega satisfying the (linear) Stokes system with f=0 and nonhomogeneous data. This reduces the general system to a perturbed Navier–Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier–Stokes system we get the existence of global weak solutions for the more general system.

Asymptotic behavior for logarithmic diffusion

In this paper we prove, via the entropy dissipation method, that the solutions of the d-dimensional logarithmic diffusion equation, with nonhomogeneous Dirichlet boundary data, decay exponentially in time toward its own steady state. The result is valid not only in L1-norm (as customary when applying entropy dissipation methods) but also in any Lp-norm with p∊[1,+∞).

Large time asymptotic and numerical solution of a nonlinear diffusion model with memory

Large time behavior of solutions and finite difference approximation of a nonlinear system of integro-differential equations associated with the penetration of a magnetic field into a substance is studied. Two initial-boundary value problems are investigated: the first with homogeneous conditions on whole boundary and the second with nonhomogeneous boundary data on one side of lateral boundary. The rates of convergence are also given. Mathematical results presented show that there is a difference between stabilization rates of solutions with homogeneous and nonhomogeneous boundary conditions. The convergence of the corresponding finite difference scheme is also proved. The decay of the numerical solution is compared with the analytical results.

Large time behavior of solutions to a nonlinear integro-differential system

Asymptotic behavior of solutions as t → ∞ to the nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is studied. Initial–boundary value problems with two kinds of boundary data are considered. The first with homogeneous conditions on whole boundary and the second with non-homogeneous boundary data on one side of lateral boundary. The rates of convergence are given too.

On nonlinear diffusion problems with strong degeneracy

In this paper, we study the “triply” degenerate problem: b ( v ) t − Δ g ( v ) + div Φ ( v ) = f on Q : = ( 0 , T ) × Ω , b ( v ( 0 , ⋅ ) ) = b ( v 0 ) on Ω and “ g ( v ) = g ( a ) on some part of the boundary ( 0 , T ) × ∂ Ω ,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions b , g are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition b ( 0 ) = g ( 0 ) = 0 and the range condition R ( b + g ) = R . Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111–139].

Constraint-Preserving Boundary Conditions for the Linearized Baumgarte-Shapiro-Shibata-Nakamura Formulation

We derive two sets of explicit homogeneous algebraic constraint-preserving boundary conditions for the second-order in time reduction of the linearized Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system. Our second-order reduction involves components of the linearized extrinsic curvature only. An initial-boundary value problem for the original linearized BSSN system is formulated and the existence of the solution is proved using the properties of the reduced system. A treatment is proposed for the full nonlinear BSSN system to construct constraint-preserving boundary conditions without invoking the second order in time reduction. Energy estimates on the principal part of the BSSN system (which is first order in temporal and second order in spatial derivatives) are obtained. Generalizations to the case of nonhomogeneous boundary data are proposed.

Numerical approximation of acoustic waves by spectral element methods

In this paper we consider the approximation of acoustic wave propagation problems. Precisely, we investigate the stability and convergence of the classical Newmark schema in time and spectral element discretization in space for the wave problems. A special attention is payed to the non-homogeneous boundary data. Some detailed error estimates are obtained. From these results, the spectral accuracy and influences of the non-homogeneous boundary data are made evident. Several numerical examples are provided to confirm our theoretical analysis. The advantage of the present method is demonstrated by a numerical comparison with the finite element method.

Radial solutions of semilinear elliptic equations with broken symmetry

The aim of this paper is to prove the existence of infinitely many radial solutions of a superlinear elliptic problem with rotational symmetry and non-homogeneous boundary data.

A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain \(\Omega \subseteq \mathbb{R}^{3}\). This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.