The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector fields have been now extended and moved in a self-contained paper available at: arXiv:1708.0139

Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients

This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients and general initial data in the whole space $\mathbb{R}^d$ $(d=2$ or 3). It is rigorously showed that, as the Mach number, the shear viscosity coefficient and the magnetic diffusion coefficient simultaneously go to zero, the weak solution of the compressible magnetohydrodynamic equations converges to the strong solution of the ideal incompressible magnetohydrodynamic equations as long as the latter exists.Comment: 17pages. We have improved our paper according to the referees' suggestion

A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy

A Calculus of Chemical Systems

In recent years various calculi have been proposed for modelling biological systems, typically intracellular pathways. These calculi generally fall into one of two camps: ones based on process calculi, such as Milner’s pi-calculus [24], and rule-based ones. Examples of the former include [31, 32, 30]; examples of the latter include BIOCHAM, κ, BioNet

Fluctuating Asymmetry of Craniological Features of Small Mammals as a Reflection of Heterogeneity of Natural Populations

Fluctuating asymmetry (FA) in nine species of small mammals (Insectivora and Rodentia) was estimated using 10 cranial features (foramina for nerves and blood vessels). The main criterion was the occurrence of the fluctuating asymmetry manifestations (OFAM). A total of 2300 skulls collected in the taiga and forest-tundra of Yakutia (Northeast Asia) were examined. The examined species are characterized by comparable OFAM values in the vast territories of the taiga zone; on the ecological periphery of the range an increased FA level is registered. Asymmetric manifestations in analyzed features are equally likely to occur in males and females. OFAM values in juveniles are higher than in adults; this difference is more pronounced on the periphery of the geographic range. Among juveniles, lower FA levels are observed in individuals that have bred. It can be surmised that the risk of elimination of individuals with high FA levels increases in stressful periods (active reproduction and winter). In conditions that are close to optimal, populations demonstrate relatively homogeneous FA levels, while on the periphery of the area an increase in occurrence of disturbances in developmental stability is observed, which leads, on one hand, to higher average FA for the population and, on the other hand, to heterogeneity of the population in this parameter

Differential Equations on Graphs

The differential equations encountered in various applications may be treated as equations on graphs. In the paper it is shown that the structure of the graph allows us to investigate the properties of the solutions of such equations