An Lp -maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations
- Authors
- Kim, I.
- Issue Date
- 3월-2022
- Publisher
- Springer
- Keywords
- Maximal regularity moment estimate; Stochastic partial differential equations; Zero initial evolution equation
- Citation
- Stochastics and Partial Differential Equations: Analysis and Computations, v.10, no.1, pp.278 - 316
- Indexed
- SCIE
SCOPUS
- Journal Title
- Stochastics and Partial Differential Equations: Analysis and Computations
- Volume
- 10
- Number
- 1
- Start Page
- 278
- End Page
- 316
- URI
- https://scholar.korea.ac.kr/handle/2021.sw.korea/137612
- DOI
- 10.1007/s40072-021-00201-1
- ISSN
- 2194-0401
- Abstract
- We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: du=(a¯ij(ω,t)uxixj+f)dt+gkdwtk,t∈(0,T);u(0,·)=0,where T∈ (0 , ∞) , wk(k= 1 , 2 , …) are independent Wiener processes, (a¯ ij(ω, t)) is a (predictable) nonnegative symmetric matrix valued stochastic process such that κ|ξ|2≤a¯ij(ω,t)ξiξj≤K|ξ|2∀(ω,t,ξ)∈Ω×(0,T)×Rdfor some κ, K∈ (0 , ∞) , f∈Lp((0,T)×Rd,dt×dx;Lr(Ω,F,dP)),and g,gx∈Lp((0,T)×Rd,dt×dx;Lr(Ω,F,dP;l2))with 2 ≤ r≤ p< ∞ and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate ∫0T∫Rd(E[|u(t,x)|r])p/rdxdt+∫0T∫Rd(E[|uxx(t,x)|r])p/rdxdt≤N(∫0T∫Rd(E[|f(t,x)|r])p/rdxdt+∫0T∫Rd(E[|g(t,x)|l2r])p/rdxdt+∫0T∫Rd(E[|gx(t,x)|l2r])p/rdxdt),where N is a positive constant depending only on d, p, r, κ, K, and T. As an application, for the solution u to (1), the rth moment mr(t, x) : = E| u(t, x) | r is in the parabolic Sobolev space Wp/r1,2((0,T)×Rd). © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Files in This Item
- There are no files associated with this item.
- Appears in
Collections - College of Science > Department of Mathematics > 1. Journal Articles
Items in ScholarWorks are protected by copyright, with all rights reserved, unless otherwise indicated.