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Abstract. A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk lattice model in the diffusion limit. Compared to previous results in the literature, the novelty is the combination of general degenerate diffusion and volume-filling effects. Conditions on the nonlinear diffusion coefficients are identified, which yield a formal gradient-flow or entropy structure. This structure allows for the proof of global-in-time existence of bounded weak solutions and the exponential convergence of the solutions to the constant steady state. The existence proof is based on an approximation argument, the entropy inequality, and new nonlinear Aubin-Lions compactness lemmas. The proof of the large-time behavior employs the entropy estimate and convex Sobolev inequalities. Moreover, under simplifiying assumptions on the nonlinearities, the uniqueness of weak solutions is shown by using the H −1 method, the E-monotonicity technique of Gajewski, and the subadditivity of the Fisher information.

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cut-off condition. Two proofs are given: a non-constructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the "cross-effects" coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectral-gap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multi-species Boltzmann system on the torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.

The existence of global nonnegative martingale solutions to a stochastic crossdiffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski's generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia's truncation method due to Chekroun, Park, and Temam. 1 n i=1 π i h s (u i )ds, where π i > 0 are some numbers and h s (z) = z(log z − 1) + 1 for s = 1, z s /s for s = 1.It was shown in [12] that B = (B ij ) in (4) is positive semi-definite in the two-species case n = 2 with π 1 = π 2 = 1. This property generally does not hold for the n-species system. It turns out [13] that B is symmetric, positive semi-definite if the numbers π i are chosen such that π i a ij = π j a ji for all i, j = 1, . . . , n.This condition is recognized as the detailed-balance condition for the Markov chain associated to (a ij ) and (π 1 , . . . , π n ) is the reversible measure. The detailed-balance condition

A general class of cross-diffusion systems for two population species in a bounded domain with no-flux boundary conditions and Lotka-Volterra-type source terms is analyzed. Although the diffusion coefficients are assumed to depend linearly on the population densities, the equations are strongly coupled. Generally, the diffusion matrix is neither symmetric nor positive definite. Three main results are proved: the existence of global uniformly bounded weak solutions, their convergence to the constant steady state in the weak competition case, and the uniqueness of weak solutions. The results hold under appropriate conditions on the diffusion parameters which are made explicit and which contain simplified Shigesada-Kawasaki-Teramoto population models as a special case. The proofs are based on entropy methods, which rely on convexity properties of suitable Lyapunov functionals.

A class of energy-transport equations without electric field under mixed Dirichlet-Neumann boundary conditions is analyzed. The system of degenerate and strongly coupled parabolic equations for the particle density and temperature arises in semiconductor device theory. The global-in-time existence of weak nonnegative solutions is shown. The proof consists of a variable transformation and a semi-discretization in time such that the discretized system becomes elliptic and semilinear. Positive approximate solutions are obtained by Stampacchia truncation arguments and a new cut-off test function. Nonlogarithmic entropy inequalities yield gradient estimates which allow for the limit of vanishing time step sizes. Exploiting the entropy inequality, the long-time convergence of the weak solutions to the constant steady state is proved. Because of the lack of appropriate convex Sobolev inequalities to estimate the entropy dissipation, only an algebraic decay rate is obtained. Numerical experiments indicate that the decay rate is typically exponential.

We analyze the mathematical properties of a multi-species biofilm crossdiffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volumefilling type cross-diffusion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdiffusion-type singularity when the biomass reaches its maximum cell capacity, which make the analysis extremely challenging. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.

Communicated by P. M. MarianoWe derive a fluid-dynamic model for electron transport near a Dirac point in graphene. Starting from a kinetic model, based on spinorial Wigner functions, the derivation of the fluid model is based on the minimum entropy principle, which is exploited to close the moment system deduced from the Wigner equation. To this aim we make two main approximations: the usual semiclassical approximation (h -1) and a new one, namely, the 'strongly mixed state' approximation, which allow to compute the closure explicitly. Particular solutions of the fluid-dynamic equations are discussed which are of physical interest because of their connection with the Klein paradox phenomenon.

In this manuscript we consider an isotropic modification for the Landau equation with Coulomb potential in three space dimensions. Global in time existence of weak solutions for even initial data is shown by employing a time semi-discretization of the equation, an entropy inequality and a uniform estimate for the second moment of the solution to the discretized problem. Moreover, under an additional condition that has to be satisfied uniformly over time, uniform boundedness of the solution is proved, with bounds depending solely on the mass, second moment and entropy of the solution. A byproduct of our analysis is a proof of improved regularity for weak solutions to the Landau equation with Coulomb potential.and u satisfies the following weak formulation of (1)

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