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- W4308291310 abstract "We introduce the uniqueness, existence, $L_p$-regularity, and maximal Holder regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$ u_t = au_{xx} + bu_{x} + cu + bar b|u|^lambda u_{x} + sigma(u)dot W,quad (t,x)in(0,infty)timesmathbb{R}; quad u(0,cdot) = u_0, $$ where $lambda > 0$. The function $sigma(u)$ is either bounded Lipschitz or super-linear in $u$. The noise $dot W$ is a space-time white noise. The coefficients $a,b,c$ depend on $(omega,t,x)$, and $bar b$ depends on $(omega,t)$. The coefficients $a,b,c,bar{b}$ are uniformly bounded, and $a$ satisfies ellipticity condition. The random initial data $u_0 = u_0(omega,x)$ is nonnegative. We have the maximal Holder regularity by employing the Holder embedding theorem. For example, if $lambda in(0,1]$ and $sigma(u)$ has Lipschitz continuity, linear growth, and boundedness in $u$, for $T<infty$ and $varepsilon>0$, $$u in C^{1/4 - varepsilon,1/2 - varepsilon}_{t,x}([0,T]timesmathbb{R})quad(a.s.). $$ On the other hand, if $lambdain(0,1)$ and $sigma(u) = |u|^{1+lambda_0}$ with $lambda_0in[0,1/2)$, for $T<infty$ and $varepsilon>0$, $$u in C^{frac{1/2-(lambda -1/2) vee lambda_0}{2} - varepsilon,1/2-(lambda -1/2) vee lambda_0 - varepsilon}_{t,x}([0,T]timesmathbb{R})quad (a.s.).$$ It should be noted that if $sigma(u)$ is bounded Lipschitz in $u$, the Holder regularity of the solution is independent of $lambda$. However, if $sigma(u)$ is super-linear in $u$, the Holder regularities of the solution are affected by nonlinearities, $lambda$ and $lambda_0.$" @default.
- W4308291310 created "2022-11-10" @default.
- W4308291310 creator A5021139793 @default.
- W4308291310 date "2021-11-05" @default.
- W4308291310 modified "2023-09-23" @default.
- W4308291310 title "A regularity theory for stochastic generalized Burgers' equation driven by a multiplicative space-time white noise" @default.
- W4308291310 doi "https://doi.org/10.48550/arxiv.2111.03659" @default.
- W4308291310 hasPublicationYear "2021" @default.
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