We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more...We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.展开更多
Consider d-dimensional magneto-hydrodynamic(MHD)equations with fractional dissipations driven by multiplicative noise.First,we prove the existence of martingale solutions for stochastic fractional MHD equations in the...Consider d-dimensional magneto-hydrodynamic(MHD)equations with fractional dissipations driven by multiplicative noise.First,we prove the existence of martingale solutions for stochastic fractional MHD equations in the case of d=2,3 andα∧β〉0,whereα,βare the parameters of the fractional dissipations in the equation.Second,for d=2,3 andα∧β≥12+d4,we show the pathwise uniqueness of solutions and then obtain the existence and uniqueness of strong solutions using the Yamada-Watanabe theorem.Furthermore,we establish the exponential mixing property for stochastic MHD equations with degenerate multiplicative noise when d=2,3 andα∧β≥12+d4.展开更多
文摘We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.
基金The research of S.Li was supported by the National Natural Science Foundation of China(Grant No.12001247)the Natural Science Foundation of Jiangsu Province(No.BK20201019)+2 种基金the Natural Science Foundation of Jiangsu Higher Education Institutions of China(No.20KJB110015)the Foundation of Jiangsu Normal University(No.19XSRX023)The research of W.Liu was supported by the National Natural Science Foundation of China(Grant Nos.11822106,11831014,12090011)and the PAPD of Jiangsu Higher Education Institutions.
文摘Consider d-dimensional magneto-hydrodynamic(MHD)equations with fractional dissipations driven by multiplicative noise.First,we prove the existence of martingale solutions for stochastic fractional MHD equations in the case of d=2,3 andα∧β〉0,whereα,βare the parameters of the fractional dissipations in the equation.Second,for d=2,3 andα∧β≥12+d4,we show the pathwise uniqueness of solutions and then obtain the existence and uniqueness of strong solutions using the Yamada-Watanabe theorem.Furthermore,we establish the exponential mixing property for stochastic MHD equations with degenerate multiplicative noise when d=2,3 andα∧β≥12+d4.
