Adrian Tudorascu

Pressureless Euler/Euler–Poisson Systems via Adhesion Dynamics and Scalar Conservation Laws

The “sticky particles” model at the discrete level is employed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler and the pressureless attractive/repulsive Euler–Poisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for the pressureless Euler system and obtain that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness. Our approach is motivated by earlier work of Brenier and Grenier, who showed that one-dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.

Lubrication Approximation for Thin Viscous Films: Asymptotic Behavior of Nonnegative Solutions

We use standard regularized equations and adapted entropy functionals to prove exponential asymptotic decay in the H 1 norm for nonnegative weak solutions of fourth-order nonlinear degenerate parabolic equations of lubrication approximation for thin viscous film type. The weak solutions considered arise as limits of solutions for the regularized problems. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means. The problems investigated here are one-dimensional in space, with power-law nonlinearities. Our approach is direct and natural, as it is adapted to deal with the more complex nonlinear terms occurring in the regularized, approximating problems.

On Lagrangian Solutions for the Semi-geostrophic System with Singular Initial Data

We show that weak (Eulerian) solutions for the semi-geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in dual space. However, such solutions are physically relevant to the model. Thus, we discuss a natural generalization of weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We prove existence of such solutions in the case of discrete measures in dual space. We also prove that weak Lagrangian solutions in physical space determine solutions in the dual space. This implies conservation of geostrophic energy along the Lagrangian trajectories in the physical space.

On a Nonlinear, Nonlocal Parabolic Problem with Conservation of Mass, Mean and Variance

In this paper we prove that the steepest descent of certain porous-medium type functionals with respect to the quadratic Wasserstein distance over a constrained (but not weakly closed) manifold gives rise to a nonlinear, nonlocal parabolic partial differential equation connected to the study of the asymptotic behavior of solutions for filtration problems. The result by Carlen and Gangbo on constrained optimization for steepest descent of the negative Boltzmann entropy in the Wasserstein space is generalized to porous-medium type functionals. An interesting feature of the resulting Fokker-Planck equation is the nonlocality of its drift term occurring at the same time as its nonlinearity.

One-Dimensional Pressureless Gas Systems with/without Viscosity

A general global existence result for one-dimensional pressureless Euler/Euler-Poisson systems with or without viscosity is obtained by employing the “sticky particles” model. We first construct entropy solutions for some appropriate scalar conservation laws, then we show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. Another novel contribution is the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric.

Chemical reaction-diffusion networks: convergence of the method of lines

We show that solutions of the chemical reaction-diffusion system associated to

Euler-Poisson Systems as Action-Minimizing Paths in the Wasserstein Space

A+B\rightleftharpoons C

Variational Principle for General Diffusion Problems

A+B⇌C in one spatial dimension can be approximated in