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Luca Scarpa

Publications and source records attributed to Luca Scarpa.

At least 55 records · Page 3Linked to original sources
arXiv2019

Well-posedness of monotone semilinear SPDEs with semimartingale noise

We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and of the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path-dependent. The solution is obtained via a priori estimates on solutions to regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and ensuing compactness properties. A key role is played by an infinite-dimensional Doob-type inequality due to Métivier and Pellaumail.

math.PR↗
arXiv2019

Fréchet differentiability of mild solutions to SPDEs with respect to the initial datum

We establish n-th order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial datum, extending previous results in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.

math.PR↗
arXiv2019

An order approach to SPDEs with antimonotone terms

We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.

math.AP↗
arXiv2019

Bounded solutions and their asymptotics for a doubly nonlinear Cahn-Hilliard system

In this paper we deal with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscosity and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard maximum-principle argument for time-discretizations of doubly nonlinear equations. Possibly including singular potentials, this novel result brings improvements over previous approaches to this problem. Secondly, under suitable assumptions on the data, we show the convergence of solutions to the respective limit problems once either of the two regularization parameters vanishes.

math.AP↗
arXiv2019

Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms

We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type.

math.AP↗
arXiv2019

Optimal control of stochastic phase-field models related to tumor growth

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

math.AP↗
arXiv2019

Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

math.AP↗
arXiv2019

Optimal distributed control of a stochastic Cahn-Hilliard equation

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.

math.OC↗
arXiv2019

The stochastic viscous Cahn-Hilliard equation: well-posedness, regularity and vanishing viscosity limit

Well-posedness is proved for the stochastic viscous Cahn-Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular, also super-polynomial) provided that it is everywhere defined on the real line. A vanishing viscosity argument is carried out and the convergence of the solutions to the ones of the pure Cahn-Hilliard equation is shown. Some refined regularity results are also deduced for both the viscous and the non-viscous case.

math.AP↗
arXiv2019

From nonlocal to local Cahn-Hilliard equation

In this paper we prove the convergence of a nonlocal version of the Cahn-Hilliard equation to its local counterpart as the nonlocal convolution kernel is scaled using suitable approximations of a Dirac delta in a periodic boundary conditions setting. This convergence result strongly relies on the dynamics of the problem. More precisely, the $H^{-1}$-gradient flow structure of the equation allows to deduce uniform $H^1$ estimates for solutions of the nonlocal Cahn-Hilliard equation and, together with a Poincaré type inequality by Ponce, provides the compactness argument that allows to prove the convergence result.

math.AP↗
arXiv2018

Existence and uniqueness of solutions to singular Cahn-Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn-Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the the diffusion operator on the boundary vanishes is also shown.

math.AP↗
arXiv2018

Strong solutions to SPDEs with monotone drift in divergence form

We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and weak compactness arguments. The result is essentially sharp, in the sense that no extra hypotheses are needed, bar continuity of the nonlinear function in the drift, with respect to the deterministic theory.

math.AP↗
arXiv2018

On the stochastic Cahn-Hilliard equation with a singular double-well potential

We prove well-posedness and regularity for the stochastic pure Cahn-Hilliard equation under homogeneous Neumann boundary conditions, with both additive and multiplicative Wiener noise. In contrast with great part of the literature, the double-well potential is treated as generally as possible, its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed. The regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem. The proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework.

math.AP↗
arXiv2017

A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity

In this paper we discuss a family of viscous Cahn-Hilliard equations with a non-smooth viscosity term. This system may be viewed as an approximation of a "forward-backward" parabolic equation. The resulting problem is highly nonlinear, coupling in the same equation two nonlinearities with the diffusion term. In particular, we prove existence of solutions for the related initial and boundary value problem. Under suitable assumptions, we also state uniqueness and continuous dependence on data.

math.AP↗
arXiv2017

Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: a variational approach

We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures $μ$, solvability of the associated Kolmogorov equation in $L^1(μ)$ is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.

math.AP↗
arXiv2017

Singular Stochastic Allen-Cahn equations with dynamic boundary conditions

We prove a well-posedness result for stochastic Allen-Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented.

math.AP↗
arXiv2017

A variational approach to dissipative SPDEs with singular drift

We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.

math.AP↗