arXiv subjects

Chiara Gavioli

Publications and source records attributed to Chiara Gavioli.

13 recordsLinked to original sources

arXiv2026

Analytic description of the moving moisture front in soils

The fact that moisture propagates in soils at a finite speed is confirmed by natural everyday experience as well as by controlled laboratory tests. In this text, we rigorously derive analytical upper bounds for the speed of moisture front propagation under gravity for the solution to the Richards equation with compactly supported initial data. The main result is an explicit criterion describing a competition between gravity and capillarity, where the dominant effect is determined by the characteristics of the soil. If capillarity prevails, the initially wet regions remain wet for all times, while if gravity is dominant, moisture travels downward at a speed that is asymptotically bounded from below and above. As a by-product, we prove the existence and uniqueness of a solution to an initial value problem for the degenerate Richards equation on the whole space. Numerical simulations based on the proposed model confirm the theoretical predictions, with results that closely match experimental observations.

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arXiv2025

On the existence of extensions for manifold-valued Sobolev maps on perforated domains

Motivated by manifold-constrained homogenization problems, we construct suitable extensions for Sobolev functions defined on a perforated domain and taking values in a compact, connected $C^2$-manifold without boundary. The proof combines a by now classical extension result for the unconstrained case with a retraction argument that heavily relies on the topological properties of the manifold. With the ultimate goal of providing necessary conditions for the existence of extensions for Sobolev maps between manifolds, we additionally investigate the relationship between this problem and the surjectivity of the trace operator for such functions.

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arXiv2025

Some remarks on a mathematical model for water flow in porous media with competition between transport and diffusion

The contribution deals with the mathematical modelling of fluid flow in porous media, in particular water flow in soils, with the aim of describing the competition between transport and diffusion. The analysis is based on a mathematical model developed by B. Detmann, C. Gavioli, and P. Krejčí, in which the effects of gravity are included in a novel way. The model consists of a nonlinear partial differential equation describing both the diffusion and the gravitational transport of water. Although analytical solutions can be obtained for some special cases, only numerical solutions are available in more general situations. The solving algorithm is based on a time discretisation and the finite element method, and is written in Matlab. The results of the numerical simulations are shown and the behaviour of the model is discussed.

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arXiv2025

Diffusion in porous media with hysteresis and bounded speed of propagation

It is shown that the problem of moisture propagation in porous media with a nonlinear relation between the mass flux and the pressure gradient as a counterpart of the Darcy law exhibits the property of bounded speed of propagation even in the case of a hysteresis relation between the capillary pressure and the moisture content. The paper specifies conditions for existence and uniqueness of solutions, and provides an upper bound for the moisture propagation speed.

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arXiv2025

Deformable porous media with degenerate hysteresis in gravity field

Hysteresis in the pressure-saturation relation in unsaturated porous media, which is due to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. Solutions to such degenerate equations have been recently constructed by the method of convexification even if the permeability coefficient depends on the hysteretic saturation. The model is extended here to the case that the solid matrix material is viscoelastic and that the system is coupled with a gravity driven moisture flux. The existence of a solution is proved by compact anisotropic embedding involving Orlicz spaces with respect to the time variable.

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arXiv2024

Degenerate diffusion in porous media with hysteresis-dependent permeability

Hysteresis in the relation between the capillary pressure and the moisture content in unsaturated porous media, which is due to surface tension at the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. Solutions to such degenerate equations have been recently constructed by the method of convexification. We show here that the convexification argument works even if the permeability coefficient depends on the hysteretic saturation. The problem of uniqueness remains open in this case.

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arXiv2024

Homogenization of high-contrast media in finite-strain elastoplasticity

This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. We study the $Γ$-convergence of the related energy functionals as the periodicity tends to zero. The main challenge is posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the $Γ$-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately. Eventually, convergence of the energy minimizing configurations is obtained.

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arXiv2024

Existence results for Cahn-Hilliard type systems driven by nonlocal integrodifferential operators with singular kernels

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as \textit{minimizing-movements scheme}, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the $q$-Laplacian. In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.

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arXiv2024

Long-time behaviour of a porous medium model with degenerate hysteresis

Hysteresis in the pressure-saturation relation in unsaturated porous media, owing to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. As an extension of previous existence and uniqueness results, we prove that under physically admissible initial conditions and without mass exchange with the exterior, the unique global solution of the fluid diffusion problem exists and asymptotically converges as time tends to infinity to a possibly non-homogeneous mass distribution and an a priori unknown constant pressure.

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arXiv2024

A homogenization result in finite plasticity

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the $Γ$-convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to $\mathsf{SL}(3)$ poses the biggest hurdle to the analysis, and we address it by regarding $\mathsf{SL}(3)$ as a Finsler manifold.

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arXiv2023

Degenerate diffusion with Preisach hysteresis

Fluid diffusion in unsaturated porous media manifests strong hysteresis effects due to surface tension on the liquid-gas interface. We describe hysteresis in the pressure-saturation relation by means of the Preisach operator, which makes the resulting evolutionary PDE strongly degenerate. We prove the existence and uniqueness of a strong global solution in arbitrary space dimension using a special weak convexity concept.

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arXiv2022

Phase transitions in porous media

The full quasistatic thermomechanical system of PDEs, describing water diffusion with the possibility of freezing and melting in a visco-elasto-plastic porous solid, is studied in detail under the hypothesis that the pressure-saturation hysteresis relation is given in terms of the Preisach hysteresis operator. The resulting system of balance equations for mass, momentum, and energy coupled with the phase dynamics equation is shown to admit a global solution under general assumptions on the data.

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arXiv2021

A model for lime consolidation of porous solids

We propose a mathematical model describing the process of filling the pores of a building material with lime water solution with the goal to improve the consistency of the porous solid. Chemical reactions produce calcium carbonate which glues the solid particles together at some distance from the boundary and strengthens the whole structure. The model consists of a 3D convection-diffusion system with a nonlinear boundary condition for the liquid and for calcium hydroxide, coupled with the mass balance equations for the chemical reaction. The main result consists in proving that the system has a solution for each initial data from a physically relevant class. A 1D numerical test shows a qualitative agreement with experimental observations.

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