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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)
A new approach to the construction of entropies and entropy productions for a large class of nonlinear evolutionary PDEs of even order in one space dimension is presented. The task of proving entropy dissipation is reformulated as a decision problem for polynomial systems. The method is successfully applied to the porous medium equation, the thin film(More)
The long-time asymptotics of solutions of the viscous quantum hydrodynamic model in one space dimension is studied. This model consists of continuity equations for the particle density and the current density, coupled to the Poisson equation for the electrostatic potential. The equations are a dispersive and viscous regularization of the Euler equations. It(More)
The logarithmic fourth-order equation ∂tu + 1 2 d X i,j=1 ∂ 2 ij (u∂ 2 ij log u) = 0, u(0, ·) = u 0 , called the Derrida-Lebowitz-Speer-Spohn equation, with periodic boundary conditions is analyzed. The global-in-time existence of weak nonnegative solutions in space dimensions d ≤ 3 is shown. Furthermore, a family of entropy–entropy dissipation inequalities(More)
A nonlinear fourth-order parabolic equation with nonhomogeneous Dirichlet–Neu-mann boundary conditions in one space dimension is analyzed. This equation appears, for instance, in quantum semiconductor modeling. The existence and uniqueness of strictly positive classical solutions to the stationary problem are shown. Furthermore, the existence of global(More)
Isothermal quantum hydrodynamic equations of order O(2) using the quantum entropy minimization method recently developed by Degond and Ringhofer are derived. The equations have the form of the usual quantum hydrodynamic model including a correction term of order O(2) which involves the vorticity. If the initial vorticity is of order O(), the standard model(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)