arXiv subjects

J. Dolbeault

Publications and source records attributed to J. Dolbeault.

4 recordsLinked to original sources

arXiv2004

A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities

A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives.

math.AP↗

arXiv2003

Oscillating minimizers of a fourth order problem invariant under scaling

By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\mathbb{R}} u'' u^2 dx\geq I \int_{\mathbb{R}} u^4 dx\quad \forall u\in L^4({\mathbb{R}}) {such that} u''\in L^2({\mathbb{R}}) $$ for some constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but does not depend on the period. Moreover, we completely characterize the minimizers of the periodic problem.

math.AP↗

arXiv2003

An analytical proof of Hardy-like inequalities related to the Dirac operator

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.

math-ph↗

arXiv2002

On the Long Time Behavior of the Quantum Fokker-Planck equation

We analyze the long time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type scattering operator, subject to confining potentials of harmonic oscillator type. We establish the conditions under which there exists a thermal equilibrium state and prove exponential decay towards it, using (classical) entropy-methods. Additionally, we give precise dispersion estimates in the cases were no equilibrium state exists.

math-ph↗