Generalized fronts in reaction-diffusion equations with mono-stable nonlinearity

We consider transition fronts (generalized traveling fronts) of mono-stable reaction-diffusion equations with spatially inhomogeneous nonlinearity. By constructing a cutoff function and using an approximate method, we establish the existence of transition fronts of the equation. Furthermore, we give the uniform non-degeneracy estimates of the solutions, such as a lower bound on the time derivative on some level sets, as well as an upper bound on the spatial derivative.     

Reducing the feasible region to a scheduling problem

A special class of scheduling problems is considered. We consider cycle-free sets of fronts correspond to the orderings of a network. If the project is recourse-constrained, the same cycle-free set of fronts can correspond to different orderings. Some cycle-free sets of fronts can be subsets of others. The goal of the paper is to characterize maximal cycle-free sets of fronts because only those are essential for obtaining an optimal schedule.     

Scalars convected by a two‐dimensional incompressible flow

We provide a test for numerical simulations, for several two dimensional incompressible flows, that appear to develop sharp fronts. We show that in order to have a front the velocity has to have uncontrolled velocity growth. © 2001 John Wiley & Sons, Inc.     

On the evolution of sharp fronts for the quasi‐geostrophic equation

We consider the problem of the evolution of sharp fronts for the surface quasi‐geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two‐dimensional Euler equation. The special interest of the quasi‐geostrophic equation lies in its strong similarities with the three‐dimensional Euler equation, while being a two‐dimen‐sional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the three‐dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot‐Savart law still needs to be understood. We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve. Finally, using a Nash‐Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C^∞ case. © 2004 Wiley Periodicals, Inc.     

A refined Brunn–Minkowski inequality for convex sets

Starting from a mass transportation proof of the Brunn–Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincaré-type trace inequality on convex sets that is also proved in sharp form.     

Measure-Theoretic Properties of Level Sets of Distance Functions

We consider the level sets of distance functions from the point of view of geometric measure theory. This lays the foundation for further research that can be applied, among other uses, to the derivation of a shape calculus based on the level-set method. Particular focus is put on the \((n-1)\)-dimensional Hausdorff measure of these level sets. We show that, starting from a bounded set, all sub-level sets of its distance function have finite perimeter. Furthermore, if a uniform-density condition is satisfied for the initial set, one can even show an upper bound for the perimeter that is uniform for all level sets. Our results are similar to existing results in the literature, with the important distinction that they hold for all level sets and not just almost all. We also present an example demonstrating that our results are sharp in the sense that no uniform upper bound can exist if our uniform-density condition is not satisfied. This is even true if the initial set is otherwise very regular (i.e., a bounded Caccioppoli set with smooth boundary).     

Sharp convex Lorentz–Sobolev inequalities

New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L ^p Minkowski problem. New L ^p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.     

Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters

This paper deals with a nonlinear diffusion equation with double free boundaries possessing different moving parameters. We present the spreading–vanishing dichotomy and threshold between spreading and vanishing. Moreover, when spreading happens, using the zero number argument we provide sharp estimates of spreading speeds of expanding fronts, and describe how the solution approaches the semi-wave.     

A modified method of characteristics incorporating streamline diffusion

A hybrid finite-element method, combining ideas from a modified method of characteristics and the streamline diffusion method, delivers accurate solutions to the advection–diffusion equation. An error analysis for the case of tensorial diffusion shows that the lowest-order version of the scheme, which allows one to use a symmetric linear solvers at each time step, possesses first-order accuracy in time and space. Numerical experiments demonstrate the scheme's ability to model advection-dominated transport of solute plumes without distorting sharp fronts. © 1995 John Wiley & Sons, Inc.