arXiv subjects

Giulio Schimperna

Publications and source records attributed to Giulio Schimperna.

At least 37 records · Page 2Linked to original sources

arXiv2017

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions

We consider a thermodynamically consistent model for the evolution of thermally conducting two-phase incompressible fluids. Complementing previous results, we prove additional regularity properties of solutions in the case when the evolution takes place in the two-dimensional flat torus with periodic boundary conditions. Thanks to improved regularity, we can also prove uniqueness and characterize the long-time behavior of trajectories showing existence of the global attractor in a suitable phase-space.

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arXiv2016

Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.

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arXiv2016

On the viscous Cahn-Hilliard equation with singular potential and inertial term

We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term~$u_{tt}$. The equation also contains a semilinear term $f(u)$ of "singular" type. Namely, the function $f$ is defined only on a bounded interval of ${\mathbb R}$ corresponding to the physically admissible values of the unknown $u$, and diverges as $u$ approaches the extrema of that interval. In view of its interaction with the inertial term $u_{tt}$, the term $f(u)$ is difficult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.

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arXiv2015

On asymptotic isotropy for a hydrodynamic model of liquid crystals

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 $(1 + t)^{-3/2}$ as $t \to \infty$.

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arXiv2015

On a singular heat equation with dynamic boundary conditions

In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.

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arXiv2015

Analysis of a diffuse interface model of multispecies tumor growth

We consider a diffuse interface model for tumor growth recently proposed in [Y. Chen, S.M. Wise, V.B. Shenoy, J.S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 726-754]. 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 ${\bf u}$ satisfies ${\bf u}\cdotν>0$, where $ν$ is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.

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arXiv2015

A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints

We consider a viscoelastic body occupying a smooth bounded domain of $R^3$ under the effects of volumic traction forces. Inertial effects are considered: hence, the equation describing the evolution of displacements is of the second order in time. On a part of the boundary of the domain, the body is anchored to a support and no displacement may occur; on a second part, the body can move freely; on a third portion of the boundary, the body is in adhesive contact with a solid support. The boundary forces acting there as a byproduct of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a boundary variable that represents the local fraction of active bonds and is assumed to satisfy a doubly nonlinear ODE. Following the lines of a new approach based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solution to the resulting system of equations. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.

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arXiv2015

On the strongly damped wave equation with constraint

A 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.

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arXiv2015

Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $Ω$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $Ω$). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.

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arXiv2014

Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids

We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. This model was recently introduced in a previous paper of ours, where we proved existence of weak solutions in three space dimensions. Here, we aim at studying the mathematical properties of the model in the two-dimensional case. In particular, we can show existence of global in time strong solutions. Moreover, we can admit slightly more general conditions on some material coefficients of the system.

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arXiv2014

On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids

We 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 macroscopic velocity is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter 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.

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arXiv2013

Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy

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 $\vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $\QQ$, 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 $\vt$ 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 $\QQ$ and degenerate in $\vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

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arXiv2013

Subdifferential calculus and doubly nonlinear evolution equations in L^p spaces with variable exponents

This paper is concerned with the Cauchy-Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we also analyze the relations occurring between Lebesgue spaces of space-time variables and Lebesgue-Bochner spaces of vector-valued functions, with a special emphasis on measurability issues and particularly referring to the case of space-dependent variable exponents. Moreover, we establish a chain rule for (possibly nonsmooth) convex functionals defined on variable exponent spaces. Actually, in such a peculiar functional setting the proof of this integration formula is nontrivial and requires a proper reformulation of some basic concepts of convex analysis, like those of resolvent, of Yosida approximation, and of Moreau-Yosida regularization.

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arXiv2013

The Penrose-Fife phase-field model with dynamic boundary conditions

In this paper we derive, starting from the basic principles of Thermodynamics, an extended version of the nonconserved Penrose-Fife phase transition model, in which dynamic boundary conditions are considered in order to take into account interactions with walls. Moreover, we study the well-posedness and the asymptotic behavior of the Cauchy problem for the PDE system associated to the model, allowing the phase configuration of the material to be described by a singular function.

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arXiv2012

Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential

We discuss a 3D model describing the time evolution of nematic liquid crystals in the framework of Landau-de Gennes theory, where the natural physical constraints are enforced by a singular free energy bulk potential proposed by J.M. Ball and A. Majumdar. The thermal effects are present through the component of the free energy that accounts for intermolecular interactions. The model is consistent with the general principle of thermodynamics and mathematically tractable. We identify the a priori estimates for the associated system of evolutionary partial differential equations and construct global-in-time weak solutions for arbitrary physically relevant initial data.

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arXiv2012

On a class of Cahn-Hilliard models with nonlinear diffusion

In the present work, we address a class of Cahn-Hilliard equations characterized by a nonlinear diffusive dynamics and possibly containing an additional sixth order term. This model describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases (a surfactant) is added. However, the model is also closely connected with other Cahn-Hilliard-type equations relevant in different types of applications. We first discuss the existence of a weak solution to the sixth-order model in the case when the configuration potential of the system is of singular (e.g., logarithmic) type. Then, we study the behavior of the solutions in the case when the sixth order term is let tend to 0, proving convergence to solutions of the fourth order system in a special case. The fourth order system is then investigated by a direct approach and existence of a weak solution is shown under very general conditions by means of a fixed point argument. Finally, additional properties of the solutions, like uniqueness and parabolic regularization are discussed, both for the sixth order and for the fourth order model, under more restrictive assumptions on the nonlinear diffusion term.

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arXiv2012

A Cahn-Hilliard equation with singular diffusion

In the present work, we address a class of Cahn-Hilliard equations characterized by a singular diffusion term. The problem is a simplified version with constant mobility of the Cahn-Hilliard-de Gennes model of phase separation in binary, incompressible, isothermal mixtures of polymer molecules. It is proved that, for any final time T, the problem admits a unique energy type weak solution, defined over (0,T). For any s > 0 such solution is classical in the sense of belonging to a suitable Hoelder class over (s,T), and enjoys the property of being separated from the singular values corresponding to pure phases.

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arXiv2011

Nonlocal phase-field systems with general potentials

We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.

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