We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ǫ. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit ε = 0 is always no-slip. We give in this paper error estimates for this homo...

We study the Lp Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is sufficient for the solvability of the Dirichlet problem with boundary data in LpN (∂Ω, R d). Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability of the Lp Dirichlet prob...

We investigate the condition number of the matrices that appear in the boundary element method. In particular we consider the Laplace equation with mixed boundary conditions. For Dirichlet boundary conditions, the condition number of the system matrix increases linearly with the number of boundary elements. We extend the research and search for a relation between the condition number and the nu...

In this paper we investigate well-posedness for the problem ut + divφ(u) = f on (0, T )×Ω, Ω ⊂ R , with initial condition u(0, ·) = u0 on Ω and with general dissipative boundary conditions φ(u) · ν ∈ β(t,x)(u) on (0, T )×∂Ω. Here for a.e. (t, x) ∈ (0, T )×∂Ω, β(t,x)(·) is a maximal monotone graph on R. This includes, as particular cases, Dirichlet, Neumann, Robin, obstacle boundary conditions a...

In this paper, we consider the controllability of the Korteweg-de Vries equation in a bounded interval when the control operates via the right Dirichlet boundary condition, while the left Dirichlet and the right Neumann boundary conditions are kept to zero. We prove that the linearized equation is controllable if and only if the length of the spatial domain does not belong to some countable cri...

Abstract In this note we study the controllability of the Generalized BenjaminBona-Mahony equation (BBM) with homogeneous Dirichlet boundary conditions. Under some conditions we shall prove the system is approximately controllable on [0, t1] if and only if the following algebraic condition holds Rank[Bj ] = γj , where Bj acts from IR m to R(Ej), λj ’ s are the eigenvalues of −∆ with Dirichlet b...

We prove a Yau’s type gradient estimate for positive f-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces compact when infinite dimensional Bakry–Emery Ricci tensor and weighted mean curvature are bounded below. As an application, we give Liouville result condition. Our results do not depend any assumption potential function f.