Diego Rial

LOCAL EXISTENCE OF SOLUTIONS TO THE TRANSIENT QUANTUM HYDRODYNAMIC EQUATIONS

The existence of weak solutions locally in time to the quantum hydrodynamic equations in bounded domains is shown. These Madelung-type equations consist of the Euler equations, including the quantum Bohm potential term, for the particle density and the particle current density and are coupled to the Poisson equation for the electrostatic potential. This model has been used in the modeling of quantum semiconductors and superfluids. The proof of the existence result is based on a formulation of the problem as a nonlinear Schrodinger–Poisson system and uses semigroup theory and fixed-point techniques.

Existence and regularity of weak solutions to the prescribed mean curvature equation for a nonparametric surface

It is known that for the parametric Plateau’s problem, weak solutions can be obtained as critical points of a functional (see [2, 6, 7, 8, 10, 11]). The nonparametric case has been studied for H = H(x,y) (and generally H = H(x1, . . . ,xn) for hypersurfaces in Rn+1) by Gilbarg, Trudinger, Simon, and Serrin, among other authors. It has been proved [5] that there exists a solution for any smooth boundary data if the mean curvature H ′ of ∂ satisfies H ′ ( x1, . . . ,xn )≥ n n−1 ∣∣H (x1, . . . ,xn)∣∣ (1.3)

LIMB-DARKENED RADIATION-DRIVEN WINDS FROM MASSIVE STARS

We calculated the influence of the limb-darkened finite-disk correction factor in the theory of radiation-driven winds from massive stars. We solved the one-dimensional m-CAK hydrodynamical equation of rotating radiation-driven winds for all three known solutions, i.e., fast, Ω-slow, and δ-slow. We found that for the fast solution, the mass-loss rate is increased by a factor of ~10%, while the terminal velocity is reduced about 10%, when compared with the solution using a finite-disk correction factor from a uniformly bright star. For the other two slow solutions, the changes are almost negligible. Although we found that the limb darkening has no effects on the wind-momentum-luminosity relationship, it would affect the calculation of synthetic line profiles and the derivation of accurate wind parameters.

Optimal distributed control problem for cubic nonlinear Schrödinger equation

We consider an optimal internal control problem for the cubic nonlinear Schrödinger (NLS) equation on the line. We prove well-posedness of the problem and existence of an optimal control. In addition, we show first-order optimality conditions. Also, the paper includes the proof of a smoothing effect for the non-homogeneous NLS, which is necessary to obtain the existence of an optimal control.

Bifurcation due rotation in in radiation driven wind from hot stars

The theory of radiative-line driven wind including stellar rotation is re-examined. After a suitable change of variables a new set of non-linear equations for the position of the critical (singular) point is obtained. For a constant massloss rate, these equations suggests the existence of many additional critical points, besides the standard m-CAK critical point. The number and location of these points depends strongly on the values of the rotational velocity of the star.

Existence and Multiplicity Results for the Nonlinear Klein-Gordon Equation

In this work, we study the multiplicity of solutions for a stationary nonhomogeneous problem associated to the nonlinear one-dimensional Klein-Gordon Equation. We prove that the existence of positive solutions is equivalent to the solvability of a scalar equation 2F(M) = 1, where F is a real function depending on V. Moreover, we prove some existence and multiplicity results for the Dirichlet problem in the superlinear case.