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A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion
1. | Laboratoire de Mathématique, Analyse Numérique et EDP, University of Paris-Sud, F-91405 Orsay Cedex, France |
2. | CNRS and Laboratoire de Mathématique, Analyse Numérique et EDP, University of Paris-Sud, F-91405 Orsay Cedex, France |
3. | Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan |
In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diffusion equation with nonlinear diffusion together with a nul-flux boundary condition in an open bounded domain of $\mathbb{R}^n$ with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.
References:
[1] |
D. C. Antonopoulou, P. W. Bates, D. Blömker and G. D. Karali,
Motion of a droplet for the stochastic mass-conserving allen-cahn equation, SIAM Journal on Mathematical Analysis, 48 (2016), 670-708.
doi: 10.1137/151005105. |
[2] |
C. Bauzet, G. Vallet and P. Wittbold,
The cauchy problem for conservation laws with a multiplicative stochastic perturbation, Journal of Hyperbolic Differential Equations, 9 (2012), 661-709.
doi: 10.1142/S0219891612500221. |
[3] |
C. Bennett and R. C Sharpley, Interpolation of Operators, volume 129. Academic press, 1988. |
[4] |
S. Boussaïd, D. Hilhorst and T. N. Nguyen,
Convergence to steady states for solutions of a reaction-diffusion equation, Evol. Equ. Control Theory, 4 (2015), 39-59.
doi: 10.3934/eect.2015.4.39. |
[5] |
W. Cheney, Analysis for Applied Mathematics, Springer, 2001.
doi: 10.1007/978-1-4757-3559-8. |
[6] |
G. Da Prato and A. Debussche,
Stochastic cahn-hilliard equation, Nonlinear Analysis: Theory, Methods & Applications, 26 (1996), 241-263.
doi: 10.1016/0362-546X(94)00277-O. |
[7] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, 2014.
doi: 10.1017/CBO9781107295513. |
[8] |
T. Funaki and S. Yokoyama, Sharp interface limit for stochastically perturbed mass conserving allen-cahn equation, arXiv preprint, arXiv: 1610.01263, 2016. |
[9] |
L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Springer, 2011.
doi: 10.1007/978-3-642-16194-0. |
[10] |
B. Gess,
Strong solutions for stochastic partial differential equations of gradient type, Journal of Functional Analysis, 263 (2012), 2355-2383.
doi: 10.1016/j.jfa.2012.07.001. |
[11] |
I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, volume 113. Springer Science & Business Media, 2013.
doi: 10.1007/978-3-642-31898-6. |
[12] |
N. V. Krylov and B. L. Rozovskii,
Stochastic evolution equations. stochastic differential equations: Theory and applications, Journal of Soviet Mathematics, 14 (1981), 1233-1277.
|
[13] |
H. H. Kuo, Introduction to Stochastic Integration, Springer Science & Business Media, 2006. |
[14] |
M. Marion,
Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Applicable Analysis, 25 (1987), 101-147.
doi: 10.1080/00036818708839678. |
[15] |
C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905. Springer, 2007. |
[16] |
M. Reiß, Stochastic Differential Equations, Lecture Notes, Humboldt University Berlin, 2003. |
[17] |
J. Rubinstein and P. Sternberg,
Nonlocal reaction-diffusion equations and nucleation, IMA Journal of Applied Mathematics, 48 (1992), 249-264.
doi: 10.1093/imamat/48.3.249. |
[18] |
R. Temam, Navier-stokes Equations, volume 2. North-Holland Amsterdam, revised edition 1979. |
show all references
References:
[1] |
D. C. Antonopoulou, P. W. Bates, D. Blömker and G. D. Karali,
Motion of a droplet for the stochastic mass-conserving allen-cahn equation, SIAM Journal on Mathematical Analysis, 48 (2016), 670-708.
doi: 10.1137/151005105. |
[2] |
C. Bauzet, G. Vallet and P. Wittbold,
The cauchy problem for conservation laws with a multiplicative stochastic perturbation, Journal of Hyperbolic Differential Equations, 9 (2012), 661-709.
doi: 10.1142/S0219891612500221. |
[3] |
C. Bennett and R. C Sharpley, Interpolation of Operators, volume 129. Academic press, 1988. |
[4] |
S. Boussaïd, D. Hilhorst and T. N. Nguyen,
Convergence to steady states for solutions of a reaction-diffusion equation, Evol. Equ. Control Theory, 4 (2015), 39-59.
doi: 10.3934/eect.2015.4.39. |
[5] |
W. Cheney, Analysis for Applied Mathematics, Springer, 2001.
doi: 10.1007/978-1-4757-3559-8. |
[6] |
G. Da Prato and A. Debussche,
Stochastic cahn-hilliard equation, Nonlinear Analysis: Theory, Methods & Applications, 26 (1996), 241-263.
doi: 10.1016/0362-546X(94)00277-O. |
[7] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, 2014.
doi: 10.1017/CBO9781107295513. |
[8] |
T. Funaki and S. Yokoyama, Sharp interface limit for stochastically perturbed mass conserving allen-cahn equation, arXiv preprint, arXiv: 1610.01263, 2016. |
[9] |
L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Springer, 2011.
doi: 10.1007/978-3-642-16194-0. |
[10] |
B. Gess,
Strong solutions for stochastic partial differential equations of gradient type, Journal of Functional Analysis, 263 (2012), 2355-2383.
doi: 10.1016/j.jfa.2012.07.001. |
[11] |
I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, volume 113. Springer Science & Business Media, 2013.
doi: 10.1007/978-3-642-31898-6. |
[12] |
N. V. Krylov and B. L. Rozovskii,
Stochastic evolution equations. stochastic differential equations: Theory and applications, Journal of Soviet Mathematics, 14 (1981), 1233-1277.
|
[13] |
H. H. Kuo, Introduction to Stochastic Integration, Springer Science & Business Media, 2006. |
[14] |
M. Marion,
Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Applicable Analysis, 25 (1987), 101-147.
doi: 10.1080/00036818708839678. |
[15] |
C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905. Springer, 2007. |
[16] |
M. Reiß, Stochastic Differential Equations, Lecture Notes, Humboldt University Berlin, 2003. |
[17] |
J. Rubinstein and P. Sternberg,
Nonlocal reaction-diffusion equations and nucleation, IMA Journal of Applied Mathematics, 48 (1992), 249-264.
doi: 10.1093/imamat/48.3.249. |
[18] |
R. Temam, Navier-stokes Equations, volume 2. North-Holland Amsterdam, revised edition 1979. |
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