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The global existence of a non-negative weak solution to a multi-dimensional parabolic strongly coupled model for two competing species is proved. The main feature of the model is that the diffusion matrix is non-symmetric and generally not positive definite and that the non-diagonal matrix elements (the cross-diffusion terms) are allowed to be " large ".… (More)

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a… (More)

A cross-diffusion system of parabolic equations for the relative concentration and the dynamic repose angle of a mixture of two different granular materials in a long rotating drum is studied. The main feature of the system is the ability to describe the axial segregation of the two granu-lar components. The existence of global-in-time weak solutions is… (More)

A (Patlak-) Keller-Segel model in two space dimensions with an additional cross-diffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross… (More)

The logarithmic fourth-order equation ∂tu + 1 2 d X i,j=1 ∂ 2 ij (u∂ 2 ij log u) = 0, u(0, ·) = u0, called the Derrida-Lebowitz-Speer-Spohn equation, with periodic boundary conditions is analyzed. The global-in-time existence of weak nonnega-tive solutions in space dimensions d ≤ 3 is shown. Furthermore, a family of entropy–entropy dissipation inequalities… (More)

Uniform lower and upper bounds for positive finite-element approximations to semilinear elliptic equations in several space dimensions subject to mixed Dirichlet-Neumann boundary conditions are derived. The main feature is that the non-linearity may be non-monotone and unbounded. The discrete minimum principle provides a positivity-preserving approximation… (More)