Inhomogeneous Boundary Data Research Articles

On the Global Existence for a Class of Compressible Non-Newtonian Fluids with Inhomogeneous Boundary Data

On the Global Existence for a Class of Compressible Non-Newtonian Fluids with Inhomogeneous Boundary Data

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones [Formula: see text] in the specific scale [Formula: see text] of Besov spaces. The regularity of the solution in these spaces determines theorder of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale [Formula: see text] in all cases,which justifies the use of adaptive schemes.

Stabilization of higher order Schrödinger equations on a finite interval: Part II

<p style='text-indent:20px;'>Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [<xref ref-type="bibr" rid="b3">3</xref>] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.</p>

Maximal $L^{1}$-regularity for parabolic boundary value problems with inhomogeneous data in the half-space

End-point maximal $L^{1}$-regularity for the parabolic initial-boundary value problem is considered in the half-space. For the inhomogeneous boundary data of both the Dirichlet and the Neumann type, maximal $L^{1}$-regularity for the initial-boundary value problem of parabolic equation is established in time end-point case upon the Besov space as well as the optimal trace estimates. We derive the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity.

The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

This paper is the first attempt to give a rigorous mathematical study of Neumann initial boundary value problems for the multidimensional dispersive evolution equations considering as example famous nonlinear Schrodinger equation. We consider the inhomogeneous initial-boundary value problem for the nonlinear Schrodinger equation, formulated on upper right-quarter plane with initial data $$u({\mathbf {x}},t)\left| _{t=0}\right. =u_{0}({\mathbf {x}})$$ and Neumann boundary data $$u_{x_{1}}\left| _{\partial _{1}{\mathbf {D}}}\right. =h_{1}(x_{2},t),u_{x_{2}}\left| _{\partial _{2}{\mathbf {D}}}\right. =h_{2}(x_{1},t)$$ given in a suitable weighted Lebesgue spaces. We are interested in the study of the influence of the Neumann boundary data on the asymptotic behavior of solutions for large time. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data. To get a nonlinear theory for the multidimensional model. we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. The advantage of this method is that it can also be applied to non-integrable equations with general inhomogeneous boundary data.

Axioms of adaptivity

This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators.Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

Well-Posedness of a Fully Coupled Navier--Stokes/Q-tensor System with Inhomogeneous Boundary Data

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier--Stokes equations coupled with an evolution equation for the $Q$-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the nonlinear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.

Classical solutions for a one-phase osmosis model

For a moving boundary problem modeling the motion of a semipermeable membrane by osmotic pressure and surface tension, we prove the existence and uniqueness of classical solutions on small time intervals. Moreover, we construct solutions existing on arbitrary long time intervals, provided the initial geometry is close to an equilibrium. In both cases, our method relies on maximal regularity results for parabolic systems with inhomogeneous boundary data.

A New Approach to the Existence of Weak Solutions of the Steady Navier–Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries

We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ1, . . . , ΓM and N − M bounded components ΓM+1, . . . , ΓN. We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L3-extension to Ω with the gradient in L2(Ω)3×3. We assume that the fluxes of α through the bounded components ΓM+1, . . . , ΓN of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ1, . . . , ΓM.

Regularity results for generalized electro-magnetic problems

We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show ‘polynomially weighted regularity in exterior domains as well’.

Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data

Consider the stationary Navier–Stokes equations in a bounded domain whose boundary consists of multi-connected components. We investigate the solvability under the general flux condition which implies that the total sum of the flux of the given data on each component of the boundary is equal to zero. Based on our Helmholtz–Weyl decomposition, we prove existence of solutions if the harmonic part of the solenoidal extension of the given boundary data is sufficiently small in L 3 compared with the viscosity constant.

Convergence of Rothe's scheme for the Navier‐Stokes equations with slip conditions in 2D domains

AbstractWe investigate Rothe's scheme for the Navier‐Stokes equations in two dimensional bounded domains with slip boundary conditions admitting flow across the boundary. The structure of the model enables to reformulate it into the coupled system of the vorticity and velocity problems. The obtained convergence of our first order approximation is the same as for case in domains without boundaries although we admit inhomogeneous boundary data. The method is based on the maximum principle for the vorticity equation which delivers a new bound for the solutions. However our study can be realized only under a geometrical constraint which restricts the shape of the domain.

ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ > 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

Quasilinear parabolic integro-differential equations with nonlinear boundary conditions

We study the $L_p$-theory of a class of quasilinear parabolic partial integro-differential equations with nonlinear boundary conditions. The main objective here is to prove existence and uniqueness of local (in time) strong solutions of these problems. Our approach relies on linearization and the contraction mapping principle. To make this work we establish optimal regularity estimates of $L_p$ type for associated linear problems with inhomogeneous boundary data, using here recent results on maximal $L_p$-regularity for abstract parabolic Volterra equations.

Maximal regularity of type Lp for abstract parabolic Volterra equations

We prove maximal regularity results of type L p for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on L p -regularity of abstract Cauchy problems and abstract parabolic pde’s with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition.

Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces

Several abstract model problems of elliptic and parabolic type with inhomoge- neous initial and boundary data are discussed. By means of a variant of the Dore-Venni the- orem, real and complex interpolation, and trace theorems, optimal Lp-regularity is shown. By means of this purely operator theoretic approach, classical results on Lp-regularity of the diffusion equation with inhomogeneous Dirichlet or Neumann or Robin condition are recovered. An application to a dynamic boundary value problem with surface diffusion for the diffusion equation is included.

A hamiltonian analysis of Yang-Mills equations

We discuss the structure of the constraint set of Yang-Mills theory in Minkowski space-time, and in spatially bounded domains for the bag boundary conditions and also for inhomogeneous boundary data. In each case the reduced phase space is a union of symplectic manifolds preserved by the evolution, and the reduced evolution equations are Hamiltonian.

A symmetric positive system with nonuniformly characteristic boundary

We study linear symmetric positive systems under maximal nonnegative boundary conditions. First we consider the noncharacteristic boundary and nonhomogeneous boundary conditions; in this case we give sufficient conditions on the boundary data in order to have $L^2$ and $H^1$ solutions. The inhomogeneous boundary data are treated directly with the advantage of requiring minimal regularity assumptions. Secondly we consider a boundary value problem with boundary matrix not of constant rank. We assume that the boundary is divided in two parts by an embedded manifold which is the intersection of the reference domain and a noncharacteristic hypersurface. The boundary matrix is negative definite on one side of the boundary with respect to the embedded manifold and is positive semi-definite on the other one. Using also the results of the first part, we discuss the existence of regular solutions.