An $$L_p$$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations
نویسندگان
چکیده
We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: 1 $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); u(0,\cdot )=0, \end{aligned}$$ where $$T (0,\infty )$$ , $$w^k$$ $$(k=1,2,\ldots are independent Wiener processes, $$({\bar{a}}^{ij}(\omega ,t))$$ is (predictable) nonnegative symmetric matrix valued process such that \kappa |\xi |^2 \le ,t) \xi ^i ^j K \forall \;(\omega ,t,\xi ) \Omega \times (0,T) {\mathbf {R}}^d for some $$\kappa L_p\left( {R}}^d, dx ; L_r(\Omega {\mathscr {F}} ,dP) g, g_x ,dP; l_2) with $$2 r p < \infty $$ appropriate measurable conditions. Moreover, u, we maximal regularity moment estimate 2 $$\begin{aligned}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right] ^{p/r} |u_{xx}(t,x)|^r\right] \nonumber \\&\le N \bigg (\int |f(t,x)|^r\right] |g(t,x)|_{l_2}^r\right] \\&\quad |g_x(t,x)|_{l_2}^r\right] ), positive constant depending only on d, p, r, K, T. As an application, (1), rth $$m^r(t,x):=\mathbb {E}|u(t,x)|^r$$ in parabolic Sobolev space $$W_{p/r}^{1,2}\left( \mathbf {R}^d\right) .
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ژورنال
عنوان ژورنال: Stochastics And Partial Differential Equations: Analysis And Computations
سال: 2021
ISSN: ['2194-0401', '2194-041X']
DOI: https://doi.org/10.1007/s40072-021-00201-1