Abstract
We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations.
Highlights
This paper is concerned with stochastic quasilinear partial differential equations of the form du À div ðD/ðt, ruÞÞ dt þ Dwðt, uÞ dt ʯ f ðtÞ dt þ BðtÞ dW, (1)
@tud À div ðD/ðÁ, ruÞÞ þ DwðÁ, uÞ ʯ f, us 1⁄4 B, udð0Þ 1⁄4 u0: The aim of this paper is to tackle the weak solvability of equation (1) via the Weighted Energy-Dissipation (WED) variational approach
In the abstract setting of (4)-(6), letting e > 0 we introduce the WED
Summary
This paper is concerned with stochastic quasilinear partial differential equations of the form du À div ðD/ðt, ruÞÞ dt þ Dwðt, uÞ dt ʯ f ðtÞ dt þ BðtÞ dW,. @tud À div ðD/ðÁ, ruÞÞ þ DwðÁ, uÞ ʯ f , us 1⁄4 B, udð0Þ 1⁄4 u0: The aim of this paper is to tackle the weak solvability of equation (1) via the Weighted Energy-Dissipation (WED) variational approach This hinges upon the minimization of the parameter-dependent functional Ie on entire trajectories, the so-called. In the context of stochastic PDEs, the application of tools from calculus of variations in order to characterize variational solutions is much less developed, and has been employed so far mainly in connection with the Brezis-Ekeland principle In this direction, we mention the pioneering works by Barbu and Ro€ckner [46, 47] dealing with SPDEs with additive and linear multiplicative noise, and by Krylov [48]. This e-regularized problem consist of a forward-backward system of SPDEs. The identification of the Euler-Lagrange equation is more delicate compared to the deterministic framework. The proof of Theorem 2.1 is split into Section 3 (Euler-Lagrange problem), Section 4 (convergence as e ! 0), and Section 5 (existence of minimizers)
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