Local existence theory for the generalized Schrödinger equation
Local existence, uniqueness and regularity for a class of degenerate parabolic systems arising in biological models
Local free boundary problem for incompressible magnetohydrodynamics [Book]
Local generalized solutions of mixed problems for quasilinear hyperbolic systems of functional partial differential equations in two independent variables
Local gradient estimates of -harmonic functions, -flow, and an entropy formula
In the first part of this paper, we prove local interior and boundary gradient estimates for -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the -harmonic...
Local ill-posedness of the 1D Zakharov system.
Local invariants of spectral asymmetry.
Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations
The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form over the functions that assume given boundary values ϕ on ∂Ω. The vector field satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...
Local minimizers with vortex filaments for a Gross-Pitaevsky functional
This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a Γ-limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence...
Local null controllability of a fluid-solid interaction problem in dimension 3
We are interested by the three-dimensional coupling between an incompressible fluid and a rigid body. The fluid is modeled by the Navier-Stokes equations, while the solid satisfies the Newton's laws. In the main result of the paper we prove that, with the help of a distributed control, we can drive the fluid and structure velocities to zero and the solid to a reference position provided that the initial velocities are small enough and the initial position of the structure is close to the reference...
Local null controllability of a two-dimensional fluid-structure interaction problem
In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given , the system can be driven at rest and the structure to its reference configuration at time . To show this result, we first consider a linearized system....
Local null controllability of a two-dimensional fluid-structure interaction problem
In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T > 0, the system can be driven at rest and the structure to its reference configuration at time T. To show this result, we first consider a linearized system....
Local one-dimensional scheme for the third boundary value problem for the heat equation.
Local properties of stationary solutions of some nonlinear singular Schrödinger equations.
We study the local behaviour of solutions of the following type of equation,-Δu - V(x)u + g(u) = 0 when V is singular at some points and g is a non-decreasing function. Emphasis is put on the case when V(x) = c|x|-2 and g has a power-like growth.
Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R k →Y is a C 2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R 1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R k and we describe the solution set of the studied equation in a small neighbourhood of this point.
Local refinement techniques for elliptic problems on cell-centered grids. III. Algebraic multilevel BEPS preconditioners.
Local regularity and local boundedness results for very weak solutions of obstacle problems.
Local regularity for the tangential Cauchy-Riemann complex.
Local regularity results for minima of anisotropic functionals and solutions of anisotropic equations.
Local Riesz transform characterization of local Hardy spaces