Luca Scarpa

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 evaluation 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. AMS Subject Classification: 60H15; 47H06; 46N30.

Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type ☆

Abstract We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form d X t − div γ ( ∇ X t ) d t + β ( X t ) d t ∋ B ( t , X t ) d W t , where γ and β are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on R d and R respectively, and W is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray–Lions conditions on γ and with no restrictive smoothness or growth assumptions on β. The operator B is assumed to be Hilbert–Schmidt and to satisfy some classical Lipschitz conditions in the second variable.

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.

From the viscous Cahn–Hilliard equation to a regularized forward-backward parabolic equation

A rigorous proof is given for the convergence of the solutions of a viscous Cahn– Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential and the subdifferential of a possible non-smooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence. AMS Subject Classification: 35K55, 35K50, 35B25, 35D30, 74N25.

On the Well-Posedness of SPDEs with Singular Drift in Divergence Form

We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form \({\mathrm {div}}\gamma (\nabla \cdot )\), where \(\gamma \) is a maximal monotone graph in \(\mathbb {R}^n \times \mathbb {R}^n\) obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that \(\gamma \) is single-valued, a solution with better integrability and regularity properties is constructed. The proof given here, however, is self-contained.

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.

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

Abstract 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.

Existence of solutions for a model of microwave heating

This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RLC circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.

Singular stochastic Allen–Cahn equations with dynamic boundary conditions

Abstract 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.