TY - JOUR

T1 - An Lp -maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

AU - Kim, Ildoo

N1 - Funding Information:
The author has been supported by the National Research Foundation of Korea grant funded by the Korea government (NRF-2020R1A2C1A01003959).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021

Y1 - 2021

N2 - 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).

AB - 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).

KW - Maximal regularity moment estimate

KW - Stochastic partial differential equations

KW - Zero initial evolution equation

UR - http://www.scopus.com/inward/record.url?scp=85107805227&partnerID=8YFLogxK

U2 - 10.1007/s40072-021-00201-1

DO - 10.1007/s40072-021-00201-1

M3 - Article

AN - SCOPUS:85107805227

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

SN - 2194-0401

ER -