1,721,016 research outputs found
Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type
Full text linkWe prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form View the MathML source, where γ and β are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on Rd 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
Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions
Full text linkWe 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 diffusion operator on the boundary vanishes is also shown
Optimal distributed control of a stochastic Cahn-Hilliard equation
Full text linkWe 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 a relaxed optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analyzed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved
The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential
Full text linkWe prove existence of martingale solutions for the stochastic Cahn-Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or double-obstacle type. By extending to the stochastic framework a regularization procedure introduced by Elliott and Garcke in the deterministic setting, we show that a compatibility condition between the degeneracy of the mobility and the blow-up of the potential allows to confine some approximate solutions in the physically relevant domain. By using a suitable Lipschitz-continuity property of the noise, uniform energy and magnitude estimates are proved. The passage to the limit is then carried out by stochastic compactness arguments in a variational framework. Applications to stochastic phase-field modelling are also discussed
Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation
Full text linkA Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid’s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables
The Stochastic Viscous Cahn–Hilliard Equation: Well-Posedness, Regularity and Vanishing Viscosity Limit
Full text linkWell-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
On the stochastic Cahn–Hilliard equation with a singular double-well potential
Full text linkWe 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
Refined existence and regularity results for a class of semilinear dissipative SPDEs
Full text linkWe prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the pth moment of solutions depends on the integrability of the initial datum, in the whole range p∈]0,∞[. Lipschitz continuity of the solution map in pth moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range p∈[0,∞[. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures
A note on doubly nonlinear SPDEs with singular drift in divergence form
Full text linkWe prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift of the form − div γ(∇·) +
β(·), where γ is the subdifferential of a convex function on R
d
and β is a
maximal monotone graph everywhere defined on R, on which neither growth
nor continuity assumptions are imposed
Correction to: Doubly nonlinear stochastic evolution equations II (Stochastics and Partial Differential Equations: Analysis and Computations, (2022), 10.1007/s40072-021-00229-3)
No full textIn our paper [1], we show existence of martingale solutions for doubly nonlinear stochastic evolution equations of the form
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