2019/05/07 by Ionuţ Munteanu, Munteanu, Ionut, Michael Roeckner +1
Economics, Econometrics and Finance · Computer Science · Mathematics · #Stochastic processes and financial applications #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1905.02437
The aim of this work is to prove an existence and uniqueness result of\nKato-Fujita type for the Navier-Stokes equations, in vorticity form, in 2-D\nand 3-D, perturbed by a gradient type multiplicative Gaussian noise (for\nsufficiently small initial vorticity). These equations are considered in order\nto model hydrodynamic turbulence. The approach was motivated by a recent result\nby V. Barbu and the second named author in citeb1, that treats the\nstochastic 3D-Navier-Stokes equations, in vorticity form, perturbed by linear\nmultiplicative Gaussian noise. More precisely, the equation is transformed to a\nrandom nonlinear parabolic equation, as in citeb1, but the transformation is\ndifferent and adapted to our gradient type noise. Then global unique existence\nresults are proved for the transformed equation, while for the original\nstochastic Navier-Stokes equations, existence of a solution adapted to the\nBrownian filtration is obtained up to some stopping time.\n