We present new regularity criteria involving the integrability of the pressure for the Navier-Stokes equations in bounded domains with smooth boundaries. We prove that either if the pressure belongs to $L_{x,t}^{\gamma,q}$ with $3/\gamma+2/q\leq 2$ and $3/2
When a strong external electric field is applied to a nonionized gas, an ionization process may take place leading to the appearance of patterns in the form of sparks. There are many processes involved in these phenomena, being two of the most relevant the ionization of the medium through impact with accelerated electrons and diffusion of electrons inside the medium.
Planar, cylindrical and spherical ionization fronts can exist as solutions of a minimal model taking into account both processes. We study how the combination of electron impact ionization and electron diffusion does affect the stability of planar fronts, leading to the formation of finger-like patterns, and deduce characteristic lengths for the thickness of sparks as a function of the parameters of the problem. The phenomenon can be viewed as an analog to the appearence of fingers in Hele-Shaw cells and dendritic growth.
We consider the geometric optics problem of constructing a system consisting of two reflectors which transforms a plane wave front with given intensity into an output plane wave front with prescribed output intensity.
In this talk, we describe how this problem is deeply connected to the Monge-Kantorovich mass transfer problem (MKP) with quadratic cost function. Namely, we show that the way in which the two light fronts are transformed into each other minimizes (or maximizes) an energy transportation cost.
This connection yields a new method for solving the two-reflector problem. Conversely, the connection also gives novel insights into the geometric nature of the dual formulation of the MKP.
The techniques extend to other reflector construction problems, for example, a single reflector problem which can be linked to Monge-Kantorovich problem on the sphere. We will further present some numerical computations of reflectors based on a linear programming approach to the MKP.
This talk is based on joint work with V. Oliker.
The plasmodium of the slime mold Physarum polycephalum is an amoeboid organism which consists of a single cell containing a lot of nuclei (so-called 'synthetium'). Although it looks merely like a mass of protoplasm, it can well manage its growth and movement behavior in response to different feeding conditions in the environment. The motion is basically driven by periodic contractions of the ectoplasm (mainly consisting of an actin-myosin polymer system). We will introduce some challenging ideas for understanding the observed dynamics using mathematical modeling and suitable differential equation systems.
We consider in Morrey spaces the Cauchy problem of the semilinear heat equation with an external force. Both the external force and initial data belong to suitable Morrey spaces. When the norm of the external force is small, we proved the unique existence of small solution to the corresponding stationary problem. Moreover, if the initial data is close enough to the stationary solution, we verified the time-global solvability of the Cauchy problem, which leads to the stability of the small stationary solution.
In this talk we will give some results about a system which models chemotaxis and where the chemical is non-diffusing. We will study local existence of solutions for general models. For some special cases, we will investigate sufficient conditions for the existence and the non-existence of global solutions. Moreover we see that for a special model of this type, the existence of global solutions depends sensitively on the choice of the initial data.
