arXiv subjects

Giulio Schimperna

Publications and source records attributed to Giulio Schimperna.

At least 19 recordsLinked to original sources

arXiv2026

Weak solutions and weak-strong uniqueness for a Cahn-Hilliard type model with chemotaxis

We prove existence of weak solutions and weak-strong uniqueness for a mathematical model which couples the evolution of a phase-parameter $φ$ satisfying a Cahn-Hilliard type relation with the one of an additional variable $σ$ influencing the phase separation process. The main application of the model refers to cancer growth processes, where $σ$ may represent the concentration of a chemical substance affecting the evolution of the tumor, and is governed by a nonlinear parabolic equation characterized by a cross-diffusion term alike that occurring in the Keller-Segel model for chemotaxis. This term is also responsible for the most relevant difficulties in the mathematical analysis of the system. Complementing previous results on the model, we prove here global in time existence for a very weak notion of solution to which a suitable energy imbalance and a logarithmic inequality for the nutrient are added. Noting that the system also admits local in time "strong" solutions, we can also exhibit a weak-strong uniqueness result whose proof exploits in an essential way the entropy-type inequality satisfied by weak solutions.

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arXiv2025

Global attractor for a Cahn-Hilliard-chemotaxis model with logistic degradation

We consider a mathematical model coupling the Cahn-Hilliard system for phase separation with an additional equation describing the diffusion process of a chemical quantity whose concentration influences the physical process. The main application of the model refers to tumor progression, where the phase variable denotes the local proportion of active cancer cells and the chemical concentration may refer to a nutrient transported by the blood flow or to a drug administered to the patient. The resulting system is characterized by cross-diffusion effects similar to those appearing in the Keller-Segel model for chemotaxis; in particular, the nutrient tends to be attracted towards the regions where more active tumor cells are present (and consume it in a quickier way). Complementing various recent results on related models, we investigate here the long-time behavior of solutions under the perspective of infinite-dimensional dynamical systems. To this aim, we first identify a regularity setting in which the system is well posed and generates a closed semigroup according to the terminology introduced by Pata and Zelik. Then, partly based on the approach introduced by Rocca and the first author for the Cahn-Hilliard system with singular potential, we prove that the semigroup is strongly dissipative and asymptotically compact so guaranteeing the existence of the global attractor in a suitable phase space. Finally, we discuss the sign properties of the nutrient and prove that, under additional assumptions on the initial data, its concentration is uniformly larger than some strictly positive constant at least on finite time intervals.

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arXiv2025

On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

We investigate the Cahn-Hilliard equation with nonlinear diffusion and non-degenerate mobility modeling phase separation phenomena in complex systems (e.g., crystals and polymers). Previous results in the literature on this model relied on the strong convexity assumption of the gradient part of the energy, which excludes relevant cases. In this work, we remove the convexity condition and establish new qualitative properties of solutions under general assumptions on the diffusion and mobility functions. In two spatial dimensions, we prove uniqueness of weak solutions, their smoothing effect for positive times, and convergence to equilibrium as time tends to infinity. In three dimensions, we show local well-posedness of strong solutions for arbitrary initial data and global existence for data close to energy minimizers, yielding a Lyapunov stability principle. A key ingredient of our analysis is a Lojasiewicz-Simon inequality tailored to the nonlinear diffusion case, which enables us to characterize the longtime dynamics.

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arXiv2025

Two-phase flows through porous media described by a Cahn--Hilliard--Brinkman model with dynamic boundary conditions

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field as well as a convective Cahn--Hilliard equation with dynamic boundary conditions for the phase-field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and also for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn--Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. Moreover, we further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme.

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arXiv2024

On a modified Cahn-Hilliard-Brinkman model with chemotaxis and nonlinear sensitivity

We consider an evolutionary PDE system coupling the Cahn-Hilliard equation with singular potential, mass source and transport effects, to a Brinkman-type relation for the macroscopic velocity field and to a further equation describing the evolution of the concentration of a chemical substance affecting the phase separation process. The main application we have in mind refers to tumor growth models: in particular, the equation for the chemical prescribes that such a substance tends to migrate towards the regions where the tumor cells are more dense and consume it more actively. The cross-diffusion effects characterizing the system are similar to those occurring in the Keller-Segel model for chemotaxis. There is, however, a profound difference between the two settings: actually, the Cahn-Hilliard system prescribes a fourth-order dynamics with respect to space variables, whereas most models for chemotaxis are of the second order in space. This fact has a number of specific consequences regarding regularity properties of solutions and conditions ensuring existence. Our main results are devoted to proving existence of weak solutions in the case when the chemotactic sensitivity function depends nonlinearly on the chemical species concentration, and, more precisely, has a slow growth at infinity so to avoid finite-time blowup. We also analyze the asymptotic problem obtained by letting the viscosity go to zero so to get a Darcy flow regime in the limit.

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arXiv2024

The Oberbeck--Boussinesq approximation and Rayleigh--Benard convection revisited

We consider the Oberbeck--Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non--local term arising in the incompressible limit of the Navier--Stokes--Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.

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arXiv2023

On a class of doubly nonlinear evolution equations in Musielak-Orlicz spaces

This paper is concerned with a parabolic evolution equation of the form $A(u_t) + B(u) = f$, settled in a smooth bounded domain of ${\bf R}^d$, $d \geq 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, $-B$ stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the $m$-Laplacian for suitable $m\in(1,\infty)$), the "variable-exponent" $m(x)$-Laplacian, or even some fractional order operators. The operator $A$ is assumed to be in the form $[A(v)](x, t) = α(x, v(x, t))$ with $α$ being measurable in $x$ and maximal monotone in $v$. The main results are devoted to proving existence of weak solutions for a wide class of functions $α$ that extends the setting considered in previous results related to the variable exponent case where $α(x, v) = |v(x)|^{p(x)-2} v(x)$. To this end, a theory of subdifferential operators will be established in Musielak-Orlicz spaces satisfying structure conditions of the so-called $Δ_2$-type and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators $A$, $B$) to which the result can be applied.

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arXiv2022

On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential

We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter $φ$ is designed in such a way to keep $φ$ in between the reference interval $(-1,1)$ despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.

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arXiv2022

On a Cahn-Hilliard-Keller-Segel model with generalized logistic source describing tumor growth

We propose a new type of diffuse interface model describing the evolution of a tumor mass under the effects of a chemical substance (e.g., a nutrient or a drug). The process is described by utilizing the variables $φ$, an order parameter representing the local proportion of tumor cells, and $σ$, representing the concentration of the chemical. The order parameter $φ$ is assumed to satisfy a suitable form of the Cahn-Hilliard equation with mass source and logarithmic potential of Flory-Huggins type (or generalizations of it). The chemical concentration $σ$ satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated Keller-Segel model. In this respect, the model we propose represents a new coupling between the Cahn-Hilliard equation and a subsystem of the Keller-Segel model. We believe that, compared to other models, this choice is more effective in capturing the chemotactic effects that may occur in tumor growth dynamics (chemically induced tumor evolution and consumption of nutrient/drug by tumor cells). Note that, in order to prevent finite time blowup of $σ$, we assume a chemical source term of logistic type. Our main mathematical result is devoted to proving existence of weak solutions in a rather general setting that covers both the two- and three- dimensional cases. Under more restrictive assumptions on coefficients and data, and in some cases on the spatial dimension, we prove various regularity results. Finally, in a proper class of smooth solutions we show uniqueness and continuous dependence on the initial data in a number of significant cases.

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arXiv2020

On the Existence of Strong Solutions to the Cahn-Hilliard-Darcy system with mass source

We study a diffuse interface model describing the evolution of the flow of a binary fluid in a Hele-Shaw cell. The model consists of a Cahn-Hilliard-Darcy (CHD) type system with transport and mass source. A relevant physical application is related to tumor growth dynamics, which in particular justifies the occurrence of a mass inflow. We study the initial-boundary value problem for this model and prove global existence and uniqueness of strong solutions in two space dimensions as well as local existence in three space dimensions.

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arXiv2020

Local well-posedness for Frémond's model of complete damage in elastic solids

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularizing terms.

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arXiv2020

On a non-isothermal Cahn-Hilliard model based on a microforce balance

This paper is concerned with a non-isothermal Cahn-Hilliard model based on a microforce balance. The model was derived by A. Miranville and G. Schimperna starting from the two fundamental laws of Thermodynamics, following M. Gurtin's two-scale approach. The main working assumptions are made on the behaviour of the heat flux as the absolute temperature tends to zero and to infinity. A suitable Ginzburg-Landau free energy is considered. Global-in-time existence for the initial-boundary value problem associated to the entropy formulation and, in a subcase, also 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. At last, some highlights are given regarding a possible approximation scheme compatible with the a-priori estimates available for the system.

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arXiv2019

On a Class of Sixth-order Cahn-Hilliard Type Equations with Logarithmic Potential

We consider a class of six-order Cahn-Hilliard equations with logarithmic type potential. This system is closely connected with some important phase-field models relevant in different applications, for instance, the functionalized Cahn-Hilliard equation that describes phase separation in mixtures of amphiphilic molecules in solvent, and the Willmore regularization of Cahn-Hilliard equation for anisotropic crystal and epitaxial growth. The singularity of the configuration potential guarantees that the solution always stays in the physical relevant domain [-1,1]. Meanwhile, the resulting system is characterized by some highly singular diffusion terms that make the mathematical analysis more involved. We prove existence and uniqueness of global weak solutions and show their parabolic regularization property for any positive time. Besides, we investigate long-time behavior of the system, proving existence of the global attractor for the associated dynamical process in a suitable complete metric space.

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arXiv2019

Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

In 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.

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arXiv2019

On a Navier-Stokes-Allen-Cahn model with inertial effects

A mathematical model describing the flow of two-phase fluids in a bounded container $Ω$ is considered under the assumption that the phase transition process is influenced by inertial effects. The model couples a variant of the Navier-Stokes system for the velocity $u$ with an Allen-Cahn-type equation for the order parameter $φ$ relaxed in time in order to introduce inertia. The resulting model is characterized by second-order material derivatives which constitute the main difficulty in the mathematical analysis. Actually, in order to obtain a tractable problem, a viscous relaxation term is included in the phase equation. The mathematical results consist in existence of weak solutions in 3D and, under additional assumptions, existence and uniqueness of strong solutions in 2D. A partial characterization of the long-time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed.

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arXiv2018

On the long time behavior of a tumor growth model

We consider the problem of the long time dynamics for a diffuse interface model for tumor growth. The model describes the growth of a tumor surrounded by host tissues in the presence of a nutrient and consists in a Cahn-Hilliard-type equation for the tumor phase coupled with a reaction-diffusion equation for the nutrient concentration. We prove that, under physically motivated assumptions on parameters and data, the corresponding initial-boundary value problem generates a dissipative dynamical system that admits the global attractor in a proper phase space.

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arXiv2018

Convergence of solutions for the fractional Cahn-Hilliard system

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called Łojasiewicz-Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.

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arXiv2017

On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials

We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $φ_p$, $φ_d$ (proliferating and dead cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $φ_p$, $φ_d$ satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a form of the Darcy law and $n$ satisfies a quasistatic diffusion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector $(φ_p,φ_d)$ is constrained to remain in the range of physically admissible values. On the other hand, in view of the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of $φ_p$ and $φ_d$. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.

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