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- ItemWeak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert; Rocca, Elisabetta; Schimperna, GiulioIn this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.
- ItemAttractors for the semiflow associated with a class of doubly nonlinear parabolic equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Schimperna, Giulio; Segatti, AntonioA doubly nonlinear parabolic equation of the form [alpha](ut)-[delta]u+W'(u)=f, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function [alpha] and by the derivative W' of a smooth but possibly nonconvex potential W; f is a given known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for t>0. Such regularizing solutions contitute a semiflow S for which unqueness is satisfied for strictly positive times and we can study long time behaviour properties,. In particular, we can prove existence of both global and exponential attractors and investigate the structure of [omega]-limits of single trajectories.
- ItemA contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Scala, Riccardo; Schimperna, GiulioWe consider a three-dimensional viscoelastic body subjected to external forces. Inertial effects are considered; hence the equation for the displacement field is of hyperbolic type. The equation is complemented with Dirichlet and Neuman conditions on a part the boundary, while on another part the body is in adhesive contact with a solid support. The boundary forces acting on the latter part due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a nonlinear ODE which describes the evolution of the delamination order parameter z. Following the lines of a new approach introduced by the authors in a preceding paper and based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solutions to the resulting PDE system. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.
- ItemAnalysis of a diffuse interface model of multispecies tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, GiulioWe 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.
- ItemOn the strongly damped wave equation with constraint(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bonetti, Elena; Rocca, Elisabetta; Schimperna, Giulio; Scala, RiccardoA weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is defined. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable "relaxation" of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite "physical" energy.
- ItemNonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Zarnescu, ArghirIn 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.
- ItemOn asymptotic isotropy for a hydrodynamic model of liquid crystals(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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 .
- ItemOn a non-isothermal diffuse interface model for two-phase flows of incompressible fluids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eleuteri, Michaela; Rocca, Elisabetta; Schimperna, GiulioWe introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity u is ruled by the Navier-Stokes system with temperaturedependent viscosity, while the order parameter Phi representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a-priori estimates and showing weak sequential stability of families of approximating solutions.
- ItemExistence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eleuteri, Michela; Rocca, Elisabetta; Schimperna, GiulioWe consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in [12] where existence of weak solutions was proved in three space dimensions. Here, we aim at studying the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in [12]. Moreover, we can admit slightly more general conditions on some material coefficients of the system.