TY - JOUR
T1 - Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs
AU - Funaki, Tadahisa
AU - Xie, Bin
N1 - Funding Information:
T. Funaki was supported in part by JSPS KAKENHI, Grant-in-Aid for Scientific Researches (A) 18H03672 and (S) 16H06338, and B. Xie was supported in part by JSPS KAKENHI, Grant-in-Aid for Scientific Research (C) 20K03627.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2022/9
Y1 - 2022/9
N2 - We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021), which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as t→ ∞. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021) except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.
AB - We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021), which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as t→ ∞. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021) except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.
KW - Energy inequality
KW - Global solvability
KW - Paracontrolled calculus
KW - Quasilinear SPDE
KW - Singular SPDE
KW - Stationary solution
UR - http://www.scopus.com/inward/record.url?scp=85125620628&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85125620628&partnerID=8YFLogxK
U2 - 10.1007/s40072-022-00243-z
DO - 10.1007/s40072-022-00243-z
M3 - Article
AN - SCOPUS:85125620628
SN - 2194-0401
VL - 10
SP - 858
EP - 897
JO - Stochastics and Partial Differential Equations: Analysis and Computations
JF - Stochastics and Partial Differential Equations: Analysis and Computations
IS - 3
ER -