Stochastic Burgers equations with fractional derivative driven by fractional noise
DOI:
https://doi.org/10.58997/ejde.2023.49Keywords:
Stochastic Burgers equation; Caputo derivative; fractional noise; Mittag-Leffler operator; mild solutionAbstract
by fractional noise. Existence and uniqueness of a mild solution is given by a fixed point argument. Then, we explore Holder regularity of the mild solution in \(C([0,T_{*}];L^p(\Omega; \dot{H}^{\gamma}))\) for some stopping time \(T_{*}\).
For more information see https://ejde.math.txstate.edu/Volumes/2023/49/abstr.html
References
S. Albeverio, F. Flandoli, Y. G. Sinai; SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Mathematics. 1942. Springer-Verlag C.I.M.E. florence, 2008.
L. Bertini, N. Cancrini, G. Jina-Lasinio; The stochastic Burgers equation, Communications in Mathematical Physics, 165 (1994), no. 2, 211-232.
J. P. Bouchaud, M. M ́ezard; Velocity fluctuations in forced Burgers turbulence, Physical Review E, 54 (1996), no. 5, 5116-5121.
Z. Brze ́zniak, L. Debbi, B. Goldys; Ergodic properties of fractional stochastic Burgers euqations, Mathematics, 1, no. 2. 2011
J. Burgers; Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, 1939
J. M. Burgers; A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, 1 (1948), 171-199.
J. M. Burgers; The Nonlinear diffusion equation: Asymptotic solutions and Staticstical problems, Springer Netherlands, 1974.
L. Chen, Y. Hu, D. Nualart; Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd, Stochastic Analysis and Application, 129 (2019), no. 12, 5073-5112.
G. Da Prato, A. Debussche, R. Temam; Stochastic Burgers euqtion, Nodea Nonlinear Differential Equations and Applications, 1 (1998), no. 4, 389-402.
L. Debbi; Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, Joural of Mathematical Fluid Mechanics, 18 (2016), no. 1, 753-756.
P. Guasoni; No arbitrage under transaction costs, with fractional Brownian motion and beyond, Mathematical Finance, 16 (2006), 569-582.
E. Hopf; The partial differential euqation ut + uux = μuxx, Communications on Pure and Applied Mathematics, 3 (1950), 201-230.
M. Inc; The approximate and exact solutions of the space and time fractional Burgers equations with initial conditions by variational iteration method, Journal of Applied Mathematical Analysis and Appliacation, 345 (2008), no. 1, 476-484.
Y. Jiang, T. Wei, X. Zhou; Stochastic generalized Burgers equations driven by fractional noise, Journal of Differential Equations, 252 (2012), 1934-1961.
A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and applications of fractional differential equation, North-Holland Mathematics Studies, 2006.
E. T. Kolkovska; On a stochastic Burgers equation with Dirichlet boundary conditions International Journal of Mathematics and Mathematical Sciences, 43 (2003), 2735-2746.
A. N. Kolmogorov; Wienersche spiralen und einige andere interessante kurven im hilbertschenraum, C.R. (Doklady) Acad. URSS(N.S), 26 (1940), 115-118.
S. C. Kou; Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins, The Annals of Applied Statistics, 2 (2008), 501-535.
F. Mainardi; On the initial value problem for the fractional diffusion-wave equation, Wave and Stability in Continuous Media, 1994.
F. Mainardi; The fundamental solutions for the fractional diffusion-wave equation, Applied Mathematics Letters, 9 (1996), no. 6, 23-28.
A. D. Neate, A. Truman; On the stochastic Burgers equation and some applications to turbulence and astrophysics, Oxford University Press, (2008), 281-305.
D. Nualart, Y. Ouknine; Regularization of quasilinear heat equations by a fractional noise, Stochastics Dynamics, 4 (2004), no. 2, 201-221.
D. J. Odde, E. M. Tanaka, S. S. Hawkins, H. M. Buettner; Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth, Biotechnol Bioeng, 50 (1996), no. 4, 452-461.
R. Sakthivel, S. Suganya, S. M. Anthoni; Approximate controllability of fractional stochastic evolution equations, Computers and Mathematics with Applications, 63 (2012), 660-668.
A. Truman, J. L. Wu; Fractal Burgers equation driven by L ́evy noise, Lecture Notes Pure Mathematics and Applications, 245 Chapman and Hall/CRC, Boca Raton, FL, 2006.
G. Wang, M. Zeng, B. Guo; Stochastic Burgers equation driven by fractional brownian motion, Computers and Mathematics with Applications, 371 (2010), 210-222.
R. N. Wang, D. H. Chen, T. J. Xiao; Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), no. 1, 202-235.
D. Yang; m-Dissipativity for Kolmogorov operator of a fractional Burgers equation with space-time white noise, Potential Analysis, 44 (2016), no. 2, 215-227.
X. Zhou, X. Liu, S. Zhong; Stability of delayed implusive stochastic differentual equations driven by a fractional Brownian motion with time-varying delay, Advances in Difference equations, 328, 2016.
Y. Zhou, F. Jiao; Existence of mild solutions for fractional neutral evolution equations, Computers and Mathematics with Applications, 59 (2010), 1063-1077.
G. Zou, B. Wang; Stochastic Burgers equation with fractional derivative driven by white noise, Computers and Mathematics with Applications, 74 (2017), 3195-3208
Downloads
Published
License
Copyright (c) 2023 Yubo Duan, Yiming Jiang, Yang Tian, Yawei Wei
This work is licensed under a Creative Commons Attribution 4.0 International License.
