小参数条件下随机 Burgers 方程解的适定性
Well-Posedness of Solutions to the Stochastic Burgers Equation under Small Parameter Conditions
摘要: 本文研究乘性时空白噪声驱动的随机 Burgers 方程。为控制非线性漂移项在 L2(R) 空间中的增长，构造了闭球投影截断算子。利用 Young 卷积不等式与连续性论证建立先验估计，证明了在小参数条件下，该方程以任意接近 1 的概率存在唯一温和解。
Abstract: This paper studies the stochastic Burgers equation driven by multiplicative noise. To control the growth of the nonlinear drift term in the L2(R) space, a closed-ball projection truncation operator is constructed. By combining Young’s convolution inequality with a continuity argument to establish a priori estimates, it is proved that under small parameter conditions, the equation admits a unique mild solution with probability arbitrarily close to 1.
文章引用：韦思婷, 田琳琳, 李永康, 闫理坦. 小参数条件下随机 Burgers 方程解的适定性[J]. 应用数学进展, 2026, 15(5): 147-155. https://doi.org/10.12677/AAM.2026.155215

参考文献

[1] Bertini, L., Cancrini, N. and Jona-Lasinio, G. (1994) The Stochastic Burgers equation. Com- munications in Mathematical Physics, 165, 211-232. [Google Scholar] [CrossRef] 
[2] Da Prato, G., Debussche, A. and Temam, R. (1994) Stochastic Burgers’ Equation. Nonlinear Differential Equations and Applications NoDEA, 1, 389-402. [Google Scholar] [CrossRef] 
[3] Gyöngy, I. (1998) Existence and Uniqueness Results for Semilinear Stochastic Partial Differ- ential Equations. Stochastic Processes and their Applications, 73, 271-299. [Google Scholar] [CrossRef] 
[4] Gyöngy, I. and Nualart, D. (1999) On the Stochastic Burgers’ Equation in the Real Line. The Annals of Probability, 27, 782-802. [ [Google Scholar] [CrossRef] 
[5] Manthey, R. and Zausinger, T. (1999) Stochastic Evolution Equations in L2ν . Stochastics and Stochastic Reports, 66, 37-85. [Google Scholar] [CrossRef] 
[6] Liu, W. and Röckner, M. (2015) Stochastic Partial Differential Equations: An Introduction. Springer.
[7] Hairer, M. and Weber, H. (2013) Rough Burgers-Like Equations with Multiplicative Noise. Probability Theory and Related Fields, 155, 71-126. [Google Scholar] [CrossRef] 
[8] Cerrai, S. (2003) Stochastic Reaction-Diffusion Systems with Multiplicative Noise and Non- Lipschitz Reaction Term. Probability Theory and Related Fields, 125, 271-304. [Google Scholar] [CrossRef] 
[9] Liu, Z. and Shi, Z. (2025) Invariant Measures for Atochastic Burgers Equation on Unbounded Domains.
https://arxiv.org/abs/2506.07119
[10] Da Prato, G. and Zabczyk, J. (2014) Stochastic Equations in Infinite Dimensions. 2nd Edition, Cambridge University Press.
[11] Henry, D. (1981) Geometric Theory of Semilinear Parabolic Equations. In: Lecture Notes in Mathematics, Vol. 840, Springer-Verlag. [Google Scholar] [CrossRef] 

为你推荐