Search Results
Regularity and energy conservation for the compressible Euler equations
2016, Feireisl, Eduard, Gwiazda, Piotr, Świerczewska-Gwiazda, Agnieszka, Wiedemann, Emil
We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by P. Constantin et al. for the homogeneous incompressible Euler equations.
On asymptotic isotropy for a hydrodynamic model of liquid crystals
2014, Dai, Mimi, Feireisl, Eduard, Rocca, Elisabetta, Schimperna, Giulio, Schonbek, Maria E.
We study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)-β as t → ∞ for a certain β > 1/2 .
Analysis of a diffuse interface model of multispecies tumor growth
2015, Dai, Mimi, Feireisl, Eduard, Rocca, Elisabetta, Schimperna, Giulio
We consider a diffuse interface model for tumor growth recently proposed in [3]. In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity u satisfies u ypsilon 0, where ypsilon is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.
Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy
2013, Feireisl, Eduard, Rocca, Elisabetta, Schimperna, Giulio, Zarnescu, Arghir
In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity u is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor Q, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature theta are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential f introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in Q and degenerate in theta. To treat it a careful analysis of the properties of f, particularly of its blow-up rate, is carried out.