6533b7defe1ef96bd127696c
RESEARCH PRODUCT
A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities
Jean DolbeaultAnsgar JüngelIvan Gentilsubject
Cauchy problemLogarithmApplied MathematicsGeneral Mathematics35B40Mathematical analysisNon-equilibrium thermodynamicsPoincaré inequalitySobolev inequalityNonlinear systemsymbols.namesake35K3535K55symbolsPeriodic boundary conditionsUniquenessMathematicsdescription
A logarithmic 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 and some regularity results are shown. Furthermore, we prove that the solution converges exponentially fast to its mean value in the ``entropy norm'' and in the Fisher information, using a new optimal logarithmic Sobolev inequality for higher derivatives. In particular, the rate is independent of the solution and the constant depends only on the initial value of the entropy.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2006-06-01 | Communications in Mathematical Sciences |