Maria Pia Gualdani

Exponential decay in time of solutions of the viscous quantum hydrodynamic equations

The long-time asymptotics of solutions of the viscous quantum hydrodynamic model in one space dimension is studied. This model consists of continuity equations for the particle density and the current density, coupled to the Poisson equation for the electrostatic potential. The equations are a dispersive and viscous regularization of the Euler equations. It is shown that the solutions converge exponentially fast to the (unique) thermal equilibrium state as the time tends to infinity. For the proof, we employ the entropy dissipation method, applied for the first time to a third-order differential equation.

A Nonlinear Fourth‐order Parabolic Equation with Nonhomogeneous Boundary Conditions

A nonlinear fourth‐order parabolic equation with nonhomogeneous Dirichlet–Neumann boundary conditions in one space dimension is analyzed. This equation appears, for instance, in quantum semiconductor modeling. The existence and uniqueness of strictly positive classical solutions to the stationary problem are shown. Furthermore, the existence of global nonnegative weak solutions to the transient problem is proved. The proof is based on an exponential transformation of variables and new “entropy” estimates. Moreover, it is proved by the entropy–entropy production method that the transient solution converges exponentially fast to its steady state in the

THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

L^1

Global Existence and Uniqueness of Solutions to a Model of Price Formation

norm as time goes to infinity, under the condition that the logarithm of the steady state is concave. Numerical examples show that this condition seems to be purely technical.

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

This paper has been withdrawn by the authors due to a crucial error in the proof of the main result. In a new manuscript (with two new authors) available at arXiv:1010.2791v1 this problem has been resolved.

Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential

We study a model, due to J. M. Lasry and P. L. Lions, describing the evolution of a scalar price which is realized as a free boundary in a one-dimensional diffusion equation with dynamically evolving, nonstandard sources. We establish global existence and uniqueness.

Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in R^d

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior.

Global Existence of Weak Even Solutions for an Isotropic Landau Equation with Coulomb Potential

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the

Factorization for non-symmetric operators and exponential H-theorem

L^\infty

Finite speed of propagation in porous media by mass transportation methods

-norm at a finite time