In this paper, we answer the question under which conditions the
porous-medium equation with convection and with periodic boundary conditions
possesses gradient-type Lyapunov functionals (first-order entropies).
It is shown that the weighted sum of first-order and zeroth-order
entropies are Lyapunov functionals if the weight for the zeroth-order entropy
is sufficiently large, depending on the strength of the convection.
This provides new a priori estimates for the convective porous-medium equation.
The proof is based on an extension of the algorithmic entropy construction method
which is based on systematic integration by parts, formulated as a polynomial
Navier-Stokes equations for compressible quantum fluids, including the energy equation,
are derived from a collisional Wigner equation, using the
quantum entropy maximization method of Degond and Ringhofer. The viscous
corrections are obtained from a Chapman-Enskog expansion around the quantum
equilibrium distribution and correspond to the classical viscous stress tensor
with particular viscosity coefficients depending on the particle density
and temperature. The energy and entropy dissipations are computed and discussed.
Numerical simulations of a one-dimensional tunneling diode show the stabilizing
effect of the viscous correction and the impact of the relaxation terms on the
The exponential decay of the relative entropy associated to a fully discrete porous-medium equation in one space dimension is shown by means of a discrete Bakry-Emery approach. The first ingredient of the proof is an abstract discrete Bakry-Emery method, which states conditions on a sequence under which the exponential decay of the discrete entropy follows. The second ingredient is a new nonlinear summation-by-parts formula which is inspired by systematic integration by parts developed by Matthes and the first author. Numerical simulations illustrate the exponential decay of the entropy for various time and space step sizes.
A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is analyzed. Using a new semi-discrete approximation in time, a first-order entropy–entropy dissipation inequality is proved. Passing to the limit of vanishing time discretization parameter, some regularity results are deduced. Moreover, it is shown that the solution is strictly positive for large time if it does so initially.
In this paper, we prove new functional inequalities of Poincaré type
on the one-dimensional torus $S^1$ and explore their implications
for the long-time asymptotics of periodic solutions of nonlinear
singular or degenerate parabolic equations of second and fourth
order. We generically prove a global algebraic decay of an
entropy functional, faster than exponential for short times, and
an asymptotically exponential convergence of positive solutions
towards their average. The asymptotically exponential regime is
valid for a larger range of parameters for all relevant cases of
application: porous medium/fast diffusion, thin film and logarithmic
fourth order nonlinear diffusion equations. The techniques are
inspired by direct entropy-entropy production methods and based on
appropriate Poincaré type inequalities.
higher-order nonlinear PDEs
entropy-entropy production method
fast diffusion equation
Logarithmic Sobolev inequality
thin film equation.
porous media equation
A finite-difference scheme with positivity-preserving and
entropy-decreasing properties for a nonlinear fourth-order
parabolic equation arising in quantum systems and interface
fluctuations is derived. Initial-boundary value problems, the
Cauchy problem and a rescaled equation are discussed. Based on
this scheme we elucidate properties of the long-time asymptotics
for this equation.
A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.
The semiclassical limit in a quantum energy-transport model for semiconductors
is proved. The system consists of a nonlinear parabolic fourth-order equation
for the electron density, including temperature gradients; a degenerate elliptic
heat equation for the electron temperature; and the Poisson equation for the
electric potential. The equations are solved in a bounded domain with periodic
boundary conditions. The asymptotic limit is based on a priori estimates independent
of the scaled Planck constant, obtained from entropy functionals, on the
use of Gagliardo-Nirenberg inequalities, and weak compactness methods.
Some quantum fluid models are written as the Lagrangian flow of mass distributions
and their geometric properties are explored. The first model includes magnetic effects
and leads, via the Madelung transform,
to the electromagnetic Schrödinger equation in the Madelung representation.
It is shown that the Madelung transform is a symplectic map between Hamiltonian
systems. The second model is obtained from the Euler-Lagrange equations with
friction induced from a quadratic dissipative potential. This model corresponds to the
quantum Navier-Stokes equations with density-dependent viscosity. The fact that
this model possesses two different energy-dissipation identities is explained
by the definition of the Noether currents.
A small Knuden number analysis of a kinetic equation in the diffusive scaling
is performed. The collision kernel is of BGK type with a general local Gibbs state.
Assuming that the flow velocity is of the order of the Knudsen number,
a Hilbert expansion yields a macroscopic model with finite temperature
variations, whose complexity lies in between
the hydrodynamic and the energy-transport equations.
Its mathematical structure is explored and macroscopic models for
specific examples of the global Gibbs state are presented.