A regularity theory for stochastic generalized Burgers' equation driven by a multiplicative space-time white noise

2021/11/05 by Beom-Seok Han, Han, Beom-Seok · 1 citation
Computer Science · Economics, Econometrics and Finance · Engineering · #35R60 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2111.03659

openalex publication_date 2021/11/05 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We introduce the uniqueness, existence, L_p-regularity, and maximal Hölder 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 + b|u|^λu_x + σ(u) W, (t,x)∈(0,∞)×ℝ; u(0,⋅) = u₀, where λ> 0. The function σ(u) is either bounded Lipschitz or super-linear in u. The noise W is a space-time white noise. The coefficients a,b,c depend on (ω,t,x), and b depends on (ω,t). The coefficients a,b,c,b are uniformly bounded, and a satisfies ellipticity condition. The random initial data u₀ = u₀(ω,x) is nonnegative. We have the maximal Hölder regularity by employing the Hölder embedding theorem. For example, if λ∈(0,1] and σ(u) has Lipschitz continuity, linear growth, and boundedness in u, for T0, u ∈ C^{1/4 - ε,1/2 - ε}_t,x([0,T]×ℝ) (a.s.). On the other hand, if λ∈(0,1) and σ(u) = |u|^1+λ₀ with λ₀∈[0,1/2), for T0, u ∈ C^{(1/2-(λ-1/2) \vee λ₀)/(2) - ε,1/2-(λ-1/2) \vee λ₀ - ε}_t,x([0,T]×ℝ) (a.s.). It should be noted that if σ(u) is bounded Lipschitz in u, the Hölder regularity of the solution is independent of λ. However, if σ(u) is super-linear in u, the Hölder regularities of the solution are affected by nonlinearities, λ and λ₀.

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