arXiv:1305.7086v2 [math.PR] 29 Jan 2014 ON A 2D STOCHASTIC EULER EQUATION OF TRANSPORT TYPE: EXISTENCE AND ENERGY TRANSFER ANA BELA CRUZEIRO AND IVÁN TORRECILLA Abstract. We prove weak existence of Euler equation (or Navier-Stokes equation) per- turbed by a multiplicative noise on bounded domains of R 2 with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are H 1 regular. The equa- tions are of transport type. The transfer of energy between Fourier modes is discussed. 1. Introduction We consider a stochastic partial differential equation which can be regarded as a random perturbation of the Euler as well as the Navier-Stokes equation on a two-dimensional bounded domain where we consider Dirichlet boundary conditions (or periodic boundary conditions). The noise is chosen in a natural way: the equations model the transport of an initial velocity and an initial random dispersion along the Lagrangian flow. It is therefore a direct generalization of the deterministic transport equations. In the second section we formulate the problem and state the weak existence of the stochastic p.d.e. in the space H 1 . We define in section 3 the finite-dimensional approxi- mations of the solution and complete the proof in section 4. Our next result (section 5) is the definition of a Girsanov transformation: under the new measure solutions of the non-linear s.p.d.e. become solutions of a linear stochastic transport equation. This linear equation is actually a stochastic parallel transport over Brownian paths and, as such, can be characterized in terms of the geometry defined by the L 2 metric in the space of measure-preserving diffeomorphisms of the underlying two-dimensional domain. Section 6 is devoted to the periodic boundary conditions case. We explain the geometric formulation of our equations in Section 7. Finally, in the last section, we discuss the energy transfer between Fourier modes for solutions of the stochastic p.d.e’s. considered before in the periodic case. 2. Euler equation perturbed by a multiplicative noise We consider the following stochastic Euler equation in dimension 2:                    du(t,θ)= −(u(t,θ) ·∇)u(t,θ) dt −∇p(t,θ) dt + ∑ 2 l=1 ∂ l u(t,θ) ◦ dB l (t) in ]0,T [×Θ, div u(t,θ)=0 in ]0,T [×Θ, u(t,θ)=0 on ]0,T [×Γ, u(0,θ)= u 0 (θ) in Θ, (1) Date : January 30, 2014. 1