Existence and regularity for perturbed Stokes system with critical drift

On some mixed-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

The purpose of this paper is to study boundary value problems of Robin type for the Brinkman system and a semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system, on Lipschitz domains in Euclidean setting. In the first part of the paper, we exploit a layer potential analysis and a fixed point theorem to show the existence and uniqueness of the solution to the nonlinear Robin problem for the Darcy–Forchheimer–Brinkman system on a bounded Lipschitz domain in $${\mathbb{R}^n}$$ $${(n \in \{2,3\})}$$ with small data in L 2-based Sobolev spaces. In the second part, we show an existence result for the mixed Dirichlet–Robin problem for the same semilinear Darcy–Forchheimer-Brinkman system on a bounded creased Lipschitz domain in $${\mathbb{R}^3}$$ with small L 2-boundary data. We also study mixed Dirichlet–Robin problems and boundary value problems of mixed Dirichlet–Robin and transmission type for Brinkman systems on bounded creased Lipschitz domains in $${\mathbb{R}^n}$$ (n ≥ 3). Finally, we show the well-posedness of the Navier problem for the Brinkman system with boundary data in some L 2-based Sobolev spaces on a bounded Lipschitz domain in $${\mathbb{R}^3}$$ .

Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows “forbidden” endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schrödinger differential expression − Δ + V -\Delta +V . In turn, this boundary trace theory serves as a platform for developing a spectral theory for Schrödinger operators on bounded Lipschitz domains, along with their associated Weyl–Titchmarsh operators. Overall, this pushes the present state of knowledge a significant step further. For example, we succeed in extending the Dirichlet and Neumann trace operators in such a way that all self-adjoint extensions of a Schrödinger operator on a bounded Lipschitz domain may be described with explicit boundary conditions, thus providing a final answer to a problem that has been investigated for more than 60 years in the mathematical literature. Along the way, a number of other open problems are solved. The most general geometric and analytic setting in which the theory developed here yields satisfactory results is that of Lipschitz subdomains of Riemannian manifolds and for the corresponding Laplace–Beltrami operator (in place of the standard flat-space Laplacian). In particular, such an extension yields results for variable coefficient Schrödinger operators on bounded Lipschitz domains.

The purpose of this paper is to study the mixed Dirichlet‐Neumann boundary value problem for the semilinear Darcy‐Forchheimer‐Brinkman system in Lp‐based Besov spaces on a bounded Lipschitz domain in , with p in a neighborhood of 2. This system is obtained by adding the semilinear term |u|u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well‐posedness results for the Dirichlet and Neumann problems in Lp‐based Besov spaces on bounded Lipschitz domains in (n≥3) are also presented. Then, using integral potential operators, we show the well‐posedness in L2‐based Sobolev spaces for the mixed problem of Dirichlet‐Neumann type for the linear Brinkman system on a bounded Lipschitz domain in (n≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann‐to‐Dirichlet operator, we extend the well‐posedness property to some Lp‐based Sobolev spaces. Next, we use the well‐posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy‐Forchheimer‐Brinkman system in Lp‐based Besov spaces, with p∈(2−ε,2+ε) and some parameter ε>0.

The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a boundary value problem (BVP) that will be described in detail in the next section. Layer potential methods in smoothly bounded domains in Euclidean space have proven useful in the analysis of the Stokes system, starting with work of Odqvist and Lichtenstein, and including work of Solonnikov and many others. See the discussion in Chapter III of [10] and in [17], for the case of flow in regions with smooth boundary. A treatment based on the modern language of pseudodifferential operators can be found in [18]. In 1988, E. Fabes, C. Kenig and G. Verchota [6], extended this classical layer potential approach to cover Lipschitz domains in Euclidean space. In [6] the main result concerning the (constant coefficient) Stokes system on Lipschitz domains with connected boundary in Euclidean space, is the treatment of the L-Dirichlet boundary value problem (and its regular version). To achieve this, the authors solve certain auxiliary Neumann type problems and then exploit the duality between these and the original BVP’s at the level of boundary integral operators. P. Deuring and W. von Wahl [4] made use of the analysis in [6] to demonstrate the short-time existence of solutions to the Navier-Stokes equations in bounded Lipschitz domains in threedimensional Euclidean space. It was necessary in [4] to include the hypothesis that the boundary be connected. The hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on Lipschitz domains. It was certainly natural to speculate that this restriction was an artifact of the methods used and not ∗Partly supported by NSF grant DMS-9870018. †Partially supported by NSF grant DMS-9877077. 1991Mathematics Subject Classification. Primary 35Q30, 76D05, 35J25; Secondary 42B20, 45E05.

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi-boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kreĭn type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results by Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L2‐based Sobolev spaces. A useful intermediary result is the well‐posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L2‐based Sobolev spaces. We obtain this well‐posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well‐posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.

We give a proof that the first eigenfunction of the α-symmetric stable process on a bounded Lipschitz domain in $\mathbb {R}^{d}$ , d≥1, is superharmonic for α=2/m, where m>2 is an integer. This result was first proved by M. Kaßmann and L. Silvestre for the ball (private communication) and for C 1,1 domains in Kaßmann and Silvestre ( http:// www.math.uchicago.edu/luis/preprints/cfe.pdf ). For α=1, the result for Lipschitz domains was proved in Bañuelos and Kulczycki (J. Funct. Anal. 211, 355–423, 2004, Theorem 4.7).

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^infty -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in {mathbb {R}}^{ntimes n} with zero matrix traces. We analyze, in L^2-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in {mathbb {R}}^n, nge 2 (n=2,3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in ${\mathbb R}^3$, one of them is a bounded Lipschitz domain $\Omega $ with connected boundary, and another one is the exterior Lipschitz domain ${\mathbb R}^3\setminus \overline{\Omega }$. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces.

In this paper we study regularity properties of Green functions associated with elliptic differential operators L in Lipschitz domains. In particular, we discuss the membership of G D (x, ·) and to weak Sobolev spaces in bounded Lipschitz domains Ω, uniformly for x ∈ Ω, where G D is the Green function with Dirichlet boundary condition associated with L, δ∂Ω is the distance function to the boundary of Ω, and α ∈ [0, 1]. Our analysis includes the case of second and higher order elliptic systems with constant coefficients, the bi-Laplacian, as well as the Stokes system.

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori [Formula: see text]-estimates for any [Formula: see text] when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a [Formula: see text]-estimate and prove the solvability for any [Formula: see text] when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $$\dot{W}^1_q$$ -estimates for any $$q\in [2,\infty )$$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $$W^1_q$$ -estimate and prove the solvability for any $$q\in (1,\infty )$$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.