Asymptotic Analysis

We consider quasi-neutral limits in two-fluid isentropic Euler–Poisson equations arising in the modeling of unmagnetized plasmas and semiconductors. For periodic smooth solutions, we justify an asymptotic expansion in a time interval independent of the Debye length. This implies the convergence of the equations to compressible Euler equations. The proof is based on energy estimates for symmetrizable hyperbolic equations and on the exploration of the coupling between the Euler equations and the Poisson equation.

In this article, we prove the existence of weak solutions of variational parabolic inequalities with pseudo-monotonicity and nonzero initial data using finite differences in time.

We study the long-time behavior of bounded solutions of certain systems of nonlinear integro-differential equations, including differential equations of fractional order between 1 and 2. We obtain appropriate Lyapunov functions for this system and prove that any bounded global solution converges to a steady state if the nonlinear potential E occurring in the system satisfies the Łojasiewicz inequality.

We give an approach to exponential stability within the framework of evolutionary equations due to Picard [Math. Methods Appl. Sci. 32(14) (2009), 1768–1803]. We derive sufficient conditions for exponential stability in terms of the material law operator which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.

In this paper, we derive a semiclassical asymptotic expansion of the Wick symbol, for the product of two operators, when the first one is a Weyl pseudodifferential operator in the class of Calderón–Vaillancourt and the second one is trace class in L²(Rⁿ). This result is extended in the direction of Schatten classes. We also report a corresponding result in the context of anti-Wick operators. The Wick expansion is used to derive an equation satisfied, up to an arbitrary small error term, by the Wick symbol of density operators governed by the time dependent Hartree–Fock (TDHF) equation. Still in the situation of the TDHF dynamics, we prove the finiteness of the Ehrenfest time.