The critical variational setting for stochastic evolution equations. (English) Zbl 1540.60136
The authors introduce a critical variational framework for the stochastic evolution equations \[
du(t) + A(t,u(t))\, dt = B(t,u(t))\, dW(t), \quad u(0)=u_0,
\]
where \(A:\mathbb R_+ \times \Omega \times V \to V^\ast\) and \(B:\mathbb R_+ \times \Omega \times V \to \mathcal L_2(U,H)\), \(V \hookrightarrow H \hookrightarrow V^\ast\) is a triple of Hilbert spaces, and \(W\) a cylindrical Brownian motion on another Hilbert space \(U\). Assume that the nonlinearities \((A,B)\) are of quasi-linear type:
\[
A(t,v)= A_0(t,v) v - F(t,v), \quad B(t,v)= B_0(t,v) v + G(t,v),
\]
where \((A_0,B_0)\) and \((F,G)\) are the quasi-linear and semi-linear parts of \((A,B)\). Then, under a coercivity condition and a local Lipschitz condition on the coefficients, the authors prove global existence and uniqueness for the above abstract equation, as well as continuous dependence of solutions on initial data. The abstract results are applied to stochastic Cahn-Hilliard equation, stochastic tamed Navier-Stokes equations, generalized Burgers equations and so on.
Reviewer: Dejun Luo (Beijing)
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic evolution equations; variational methods; quasi- and semi-linear; coercivity; Cahn-Hilliard equation; tamed Navier-Stokes; generalized Burgers equationReferences:
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