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### Open Access Journals

KRM

We consider the finite extensible nonlinear elasticity (FENE) dumbbell
model in viscoelastic polymeric fluids. We employ the maximum
entropy principle for FENE model to obtain the solution which
maximizes the entropy of FENE model in stationary situations. Then
we approximate the maximum entropy solution using the second order
terms in microscopic configuration field to get an probability
density function (PDF). The approximated PDF gives a solution to
avoid the difficulties caused by the nonlinearity of FENE model.
We perform the moment-closure approximation procedure with the PDF
approximated from the maximum entropy solution, and compute the
induced macroscopic stresses. We also show that the moment-closure
system satisfies the energy dissipation law. Finally, we show some
numerical simulations to verify the PDF and moment-closure system.

DCDS-B

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.
KRM

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernelsare constructed by inverting Fredholm integral operators or byemploying certain integral identities. These solutions are expected tobe the global energy stable equilibria and to characterize the generic behaviorsof stationary solutions for more general interactions.

KRM

We construct a new family of semi-discrete numerical schemes for
the approximation of the one-dimensional periodic Vlasov-Poisson
system. The methods are based on the coupling of discontinuous
Galerkin approximation to the Vlasov equation and several finite
element (conforming, non-conforming and mixed) approximations for
the Poisson problem. We show optimal error estimates for all the
proposed methods in the case of smooth compactly supported initial
data. The issue of energy conservation is also analyzed for some
of the methods.

KRM

We present a kinetic theory for swarming systems of
interacting, self-propelled discrete particles. Starting from the
Liouville equation for the many-body problem we
derive a kinetic equation for the single particle probability
distribution function and the related macroscopic hydrodynamic equations.
General solutions include flocks of constant density and fixed
velocity and other non-trivial morphologies such as compactly
supported rotating mills. The kinetic theory approach leads us to the
identification of macroscopic structures otherwise not recognized as
solutions of the hydrodynamic equations, such as double mills of two
superimposed flows. We find the conditions allowing for the
existence of such solutions and compare to the case of single
mills.

DCDS

We introduce and discuss spatially homogeneous Maxwell-type models of the
nonlinear Boltzmann equation undergoing binary collisions with a random
component. The random contribution to collisions is such that the usual
collisional invariants of mass, momentum and energy do not hold pointwise, even
if they all hold in the mean. Under this assumption it is shown that, while the
Boltzmann equation has the usual conserved quantities, it possesses a steady
state with power-like tails for certain random variables. A similar situation
occurs in kinetic models of economy recently considered by two of the authors
[24], which are conservative in the mean but possess a steady
distribution with Pareto tails. The convolution-like gain operator is
subsequently shown to have good contraction/expansion properties with respect
to different metrics in the set of probability measures. Existence and
regularity of isotropic stationary states is shown directly by constructing
converging iteration sequences as done in [8].
Uniqueness, asymptotic stability and estimates of overpopulated high energy
tails of the steady profile are derived from the basic property of
contraction/expansion of metrics. For general initial conditions the solutions
of the Boltzmann equation are then proved to converge with computable rate as
$t\to\infty$ to the steady solution in these distances, which metricizes the
weak convergence of measures. These results show that power-like tails in
Maxwell models are obtained when the point-wise conservation of momentum
and/or energy holds only globally.

DCDS-B

We investigate the confinement properties of solutions of the
aggregation equation with repulsive-attractive potentials. We show
that solutions remain compactly supported in a large fixed ball
depending on the initial data and the potential. The arguments
apply to the functional setting of probability measures with
mildly singular repulsive-attractive potentials and to the
functional setting of smooth solutions with a potential being the
sum of the Newtonian repulsion at the origin and a smooth suitably
growing at infinity attractive potential.

KRM

We are concerned with the global well-posedness of a two-phase
flow system arising in the modelling of fluid-particle
interactions. This system consists of the Vlasov-Fokker-Planck
equation for the dispersed phase (particles) coupled to the
incompressible Euler equations for a dense phase (fluid) through
the friction forcing. Global existence of classical solutions to
the Cauchy problem in the whole space is established when initial
data is a small smooth perturbation of a constant equilibrium
state, and moreover an algebraic rate of convergence of solutions
toward equilibrium is obtained under additional conditions on
initial data. The proof is based on the macro-micro decomposition
and Kawashima's hyperbolic-parabolic dissipation argument. This
result is generalized to the periodic case, when particles are in
the torus, improving the rate of convergence to exponential.

DCDS-B

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.

KRM

Solutions of the spatially inhomogeneous diffusive\linebreak Aizenmann-Bak
model for clustering within a bounded domain with homogeneous
Neumann boundary conditions are shown to stabilize, in the fast
reaction limit, towards local equilibria determined by their
monomer density. Moreover, the sequence of monomer densities
converges to the solution of a nonlinear diffusion equation whose
nonlinearity depends on the size-dependent diffusion coefficient.
Initial data are assumed to be integrable, bounded and with a
certain number of moments in size. The number density of clusters
for the solutions is assumed to verify uniform bounds away from
zero and infinity independently of the scale parameter.

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