Mathematics

We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.

We consider the best constant in the Rellich-Hardy inequality (with a radial power weight) for curl-free vector fields on R N , originally found by Tertikas-Zographopoulos \cite{Tertikas-Z} for unconstrained fields. This inequality is considered as an intermediate between Hardy-Leray and Rellich-Leray inequalities. Under the curl-free condition, we compute the new explicit best constant in the inequality and prove the non-attainability of the constant. This paper is a sequel to \cite{CF_MAAN,CF_Re}.

In this work we investigate a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that the system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions. We conclude, moreover, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.

Derived from a biophysical model for the motion of a crawling cell, the system (?? { u t =?u?��???u?�v) 0=?v?�kv+u is investigated in a finite domain Ω??R n , n?? , with k?? . While a comprehensive literature is available for cases with (?? describing chemotaxis systems and hence being accompanied by homogeneous Neumann-type boundary conditions, the presently considered modeling context, besides yet requiring the flux ??ν u?�u ??ν v to vanish on ?��?, inherently involves homogeneous Dirichlet conditions for the attractant v , which in the current setting corresponds to the cell's cytoskeleton being free of pressure at the boundary. This modification in the boundary setting is shown to go along with a substantial change with respect to the potential to support the emergence of singular structures: It is, inter alia, revealed that in contexts of radial solutions in balls there exist two critical mass levels, distinct from each other whenever k>0 or n?? , that separate ranges within which (i) all solutions are global in time and remain bounded, (ii) both global bounded and exploding solutions exist, or (iii) all nontrivial solutions blow up in finite time. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of (?? when posed under classical no-flux boundary conditions in planar domains, the discovery of a distinct secondary critical mass level related to the occurrence of (iii) seems to have no nearby precedent. In the planar case with the domain being a disk, the analytical results are supplemented with some numerical illustrations, and it is discussed how the findings can be interpreted biophysically for the situation of a cell on a flat substrate.

We introduce a flow approach to the generalized Loewner-Nirenberg problem (1.5)??1.7) of the ? k -Ricci equation on a compact manifold ( M n ,g) with boundary. We prove that for initial data u 0 ??C 4,α (M) which is a subsolution to the ? k -Ricci equation (1.5) , the Cauchy-Dirichlet problem (3.1)??3.3) has a unique solution u which converges in C 4 loc ( M ??) to the solution u ??of the problem (1.5)??1.7) , as t?��? .

We prove that for antisymmetric vectorfield Ω with small L 2 -norm there exists a gauge A??L ????W ? 1/2,2 ( R 1 ,GL(N)) such that div 1 2 (AΩ??d 1 2 A)=0 . This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.

In many industrial applications, rubber-based materials are routinely used in conjunction with various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theoretically the penetration depth as well as the amount of diffusants stored inside the material. In this framework, we prove the global solvability and explore the large time-behavior of solutions to a one-phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber. The key idea in the proof of the large time behavior is to benefit of a contradiction argument, since it is difficult to obtain uniform estimates for the growth rate of the free boundary due to the use of a Robin boundary condition posed at the fixed boundary.

This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows. We focus on the spherical ambient space. The flows are designed to preserve a quermassintegral and to de-/increase the other quermassintegrals. The convergence of this flow to a round sphere would settle the full set of quermassintegral inequalities for convex domains of the sphere, but a full proof is still missing. Here we collect what is known and hope to attract wide attention to this interesting problem.

The overreaching goal of this paper is to investigate the existence and uniqueness of weak solutions to a semilinear parabolic equation involving symmetric integrodifferential operators of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. The existence and uniqueness of a weak solution of the nonlocal complement value problem is proven under fair conditions on the interaction potential.

We prove short-time existence for the negative L 2 -gradient flow of the p -elastic energy of curves via a minimising movement scheme. In order to account for the degeneracy caused by the energy's invariance under curve reparametrisations, we write the evolving curves as approximate normal graphs over a fixed smooth curve. This enables us to establish short-time existence and give a lower bound on the solution's lifetime that depends only on the W 2,p -Sobolev norm of the initial data.