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$L_p$-solvability and H"older regularity for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise | | Beomseok Han
; | | Date: |
2 Jan 2023 | | Abstract: | We present the L subscript p-solvability for stochastic time fractional
Burgers’ equations driven by multiplicative space-time white noise:
$$ partial_t^alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + ar b^i u
u_{x^i} + partial_t^etaint_0^t sigma(u)dW_t,,t>0;,,u(0,cdot) = u_0 $$
where $alphain(0,1)$, $eta < 3alpha/4+1/2$, and $d< 4 -
2(2eta-1)_+/alpha$. The operators $partial_t^alpha$ and $partial_t^eta$
are the Caputo fractional derivatives of order $alpha$ and $eta$,
respectively. The process $W_t$ is an $L_2(R^d)$-valued cylindrical Wiener
process, and the coefficients $a^{ij}, b^i, c$ and $sigma(u)$ are random.
In addition to the existence and uniqueness of a solution, I also suggest the
H"older regularity of the solution. For example, for any constant $T<infty$,
small $varepsilon,delta>0$, and, almost sure $omegainOmega$, we have $$
sup_{xin R^d}|u(omega,cdot,x)|_{C^{left[ frac{alpha}{2}left( left(
2-(2eta-1)_+/alpha-d/2
ight)wedge1
ight)+frac{(2eta-1)_{-}}{2}
ight]wedge 1-varepsilon}([delta,T])}<infty $$ and $$ sup_{tleq
T}|u(omega,t,cdot)|_{C^{left( 2-(2eta-1)_+/alpha-d/2
ight)wedge1 -
varepsilon}(R^d)} < infty. $$ Moreover, $delta$ can be $0$ if the initial
data $u_0 = 0$. Additionally, the H"older regularity of the solution in time
changes behavior at $eta = 1/2$. Furthermore, if $etageq1/2$, then the
H"older regularity of the solution in time is $alpha/2$ times the one in
space. | | Source: | arXiv, 2301.00536 | | Services: | Forum | Review | PDF | Favorites | |
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