65 research outputs found
Higher regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations
We show that Lp vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation 'in H' of the problem, in a reference geometrical setting allowing for material heterogeneities
Analysis of improved Nernst--Planck--Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix
We continue our investigations of the improved Nernst-Planck-Poisson model introduced by Dreyer, Guhlke
and Müller 2013. In the paper by Dreyer, Druet, Gajewski and Guhlke 2016, the analysis relies on the hypothesis that the mobility matrix has maximal rank
under the constraint of mass conservation (rank N-1 for the mixture of N species).
In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero
along with the partial mass densities of certain species. In this approach the mobility matrix has a
variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of \emph{differences of chemical potentials} that supports the modelling
and the analysis in Dreyer, Druet, Gajewski and Guhlke 2016. We prove the global-in-time existence in this solution class
On weak solutions to the stationary MHD-equations coupled to heat transfer with nonlocal radiation boundary conditions
We study the coupling of the stationary system of magnetohydrodynamics to the heat equation. Coupling occurs on the one hand from temperature-dependent coefficients and from a temperature-dependent force term in the Navier-Stokes equations. On the other hand, the heat sources are given by the dissipation of current in the electrical conductors, and of viscous stresses in the fluid. We consider a domain occupied by several different materials, and have to take into account interface conditions for the electromagnetic fields. Since we additionally want to treat high-temperatures applications, we also take into account the effect of heat radiation, which results in nonlocal boundary conditions for the heat flux. We prove the existence of weak solutions for the coupled system, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small. We prove a uniqueness result in the case of constant coefficients and small data. Finally, we discuss the regularity issue in a simplified setting
Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface
We investigate the regularity of the weak solution to elliptic
transmission problems that involve two layered anisotropic materials
separated by a boundary intersecting interface. Under a compatibility
condition for the angle of contact of the two surfaces and the boundary data,
we prove the existence of square-integrable second derivatives, and the
global Lipschitz continuity of the solution. We show that the second weak
derivatives remain integrable to a certain power less than two if the
compatibility condition is violated
Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact
We investigate the regularity of the weak solution to elliptic transmission problems that involve several materials intersecting at a closed interior line of contact. We prove that local weak solutions possess second order generalized derivatives up to the contact line, mainly exploiting their higher regularity in the direction tangential to the line. Moreover we are thus able to characterize the higher regularity of the gradient and the Hoelder exponent by means of explicit estimates known in the literature for two dimensional problems. They show that strong regularity properties, for instance the integrability of the gradient to a power larger than the space dimension d =3, are to expect if the oscillations of the diffusion coefficient are moderate (that is for far larger a range than what a theory of small perturbations would allow), or if the number of involved materials does not exceed three
Global-in-time solvability of thermodynamically motivated parabolic systems
In this paper, doubly non linear parabolic systems in divergence form
are investigated form the point of view of their global-in-time weak
solvability. The non-linearity under the time derivative is given by the
gradient of a strictly convex, globally Lipschitz continuous potential,
multiplied by a position-dependent weight. This weight admits singular
values. The flux under the spatial divergence is also of monotone gradient
type, but it is not restricted to polynomial growth. It is assumed that the
elliptic operator generates some equi-coercivity on the spatial derivatives
of the unknowns. The paper introduces some original techniques to deal with
the case of nonlinear purely Neumann boundary conditions. In this respect, it
generalises or complements the results by Alt and Luckhaus (1983) and Alt
(2012). A field of application of the theory are the multi species diffusion
systems driven by entropy
Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient
Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries
We consider the stationary Maxwell system in a domain filled with different materials. The magnetic permeability being only piecewise smooth, we have to take into account the natural interface conditions for the electromagnetic fields. We present two sets of hypotheses under which we can prove the existence of weak solutions to the Maxwell system such that the Lorentz force jxB is integrable to a power larger than 6/5. This property is important for the investigation of problems in magnetohydrodynamics, with many industrial applications such as crystal growth
Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact
We investigate the regularity of the weak solution to elliptic
transmission problems that involve several materials intersecting at a closed
interior line of contact.We prove that local weak solutions possess second
order generalized derivatives up to the contact line, mainly exploiting their
higher regularity in the direction tangential to the line. Moreover we are
thus able to characterize the higher regularity of the gradient and the
Hölder exponent by means of explicit estimates known in the literature for
two dimensional problems. They show that strong regularity properties, for
instance the integrability of the gradient to a power larger than the space
dimension d = 3, are to expect if the oscillations of the diffusion
coefficient are moderate (that is for far larger a range than what a theory
of small perturbations would allow), or if the number of involved materials
does not exceed three
A curvature estimate for open surfaces subject to a general mean curvature operator and natural contact conditions at their boundary
In the seventies, L. Simon showed that the main curvatures of two-dimensional hypersurfaces obeying a general equation of mean curvature type are a priori bounded by the Hölder norm of the coefficients of the surface differential operator. This was an essentially interior estimate. In this paper, we provide a complement to the theory, proving a global curvature estimate for open surfaces that satisfy natural contact conditions at the intersection with a given boundary
- …