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Malliavin calculus for the stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion. (English) Zbl 1391.35193

The authors describe properties of the solution to a stochastic partial differential equation which is a combination of a Cahn-Hilliard and a Allen-Cahn equation and written as \(u_{t}=-\rho \Delta (\Delta u-f(u))+\Delta u-f(u)+\sigma (u)\overset{.}{W}\) posed in \(\mathcal{D}\times (0,\infty )\) where \(\mathcal{D}\) is a bounded domain of \(\mathbb{R}^{d}\), \( d=1,2,3\). Here \(f(u)=u^{3}-u,\rho >0\), \(\overset{.}{W}\) is a space-time white noise in the sense of Walsh and \(\sigma (u)\) is the noise diffusion. \( (\Omega ,\mathcal{F},P)\) is the underlying probability space where \(\mathcal{ F}\) is the \(\sigma \)-algebra generated by \(W\). The solution starts from \( u_{0}\) at \(t=0\) and satisfies the homogeneous Neumann boundary conditions \( \frac{\partial u}{\partial \nu }=\frac{\partial \Delta u}{\partial \nu }=0\) on \(\partial \mathcal{D}\times (0,T)\). The authors will assume \(d=1\). The main result of the paper proves that the law of the solution \(u\) is absolutely continuous with respect to the Lebesgue measure on \(\mathbb{R}\), under hypotheses on the data. The authors first observe that the solution to this problem may be represented as \(u(x,t)=\int_{\mathcal{D} }u_{0}(y)G(x,y,t)dy+\int_{0}^{t}\int_{\mathcal{D}}[\Delta G(x,y,t-s)-G(x,y,t-s)]f(u(y,s))dyds+\int_{0}^{t}\int_{\mathcal{D} }G(x,y,t-s)]\sigma (u(y,s))W(dy,ds)\) where \(G\) is the Green function which is given as \(G(x,y,t)=\sum_{k=0}^{+\infty }e^{-(\lambda _{k}^{2}+\lambda _{k})t}\alpha _{k}(x)\alpha _{k}(y)\). The main tool of the proof consists to introduce a \(\mathcal{C}^{1}\)-cut-off function \(H_{n}:\mathbb{R} ^{+}\rightarrow \mathbb{R}^{+}\) which satisfies \(\left| H_{n}\right| \leq 1\), \(\left| H_{n}^{\prime }\right| \leq 2\), \(H_{n}(x)=1\) if \( \left| x\right| < n\) and \(H_{n}(x)=0\) if \(\left| x\right| \geq n+1\), the function \(f_{n}(x)=H_{n}(\left| x\right| )f(x)\) and the subdomain \(\Omega _{n}=\{\omega \in \Omega :\sup_{t\in [ 0,T]}\sup_{x\in \mathcal{D}}\left| u(x,t,\omega )\right| \leq n\}\). The authors then define a problem similar to the preceding integral representation but with the function \(f_{n}\) instead of \(f\), for which they prove the existence of a unique solution \(u_{n}\) which further satisfies the uniform bound \(\sup_{t\in [ 0,T]}\mathbf{E}(\left\| u_{n}\right\| _{L^{\infty }(\mathcal{D})}^{p})<\infty \). They establish the equation satisfied by the Malliavin derivative of \(u_{n}\) and deduced from the integral representation of \(u_{n}\) from which they deduce the absolute continuity of \(u_{n}\) and finally that of \(u\).

MSC:

35K55 Nonlinear parabolic equations
35K40 Second-order parabolic systems
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:

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