Large eddy simulation turbulence model with Young measures (Q812776)

The authors deal with the equations describing (stationary) turbulent flows in the three-dimensional torus \(\mathbb T^3\): \[ v\cdot\nabla v- \text{div\,}A(y, \nabla^s v)-\nu\delta v+\nabla q= f,\quad \text{div\,}v= 0, \] where \(v: \mathbb T^3\to\mathbb R^3\) is the velocity, \(q:\mathbb T^3\to \mathbb R^3\) is the velocity, \(q:\mathbb T^3\to \mathbb R\) is the pressure. The operator \(A\) is a nonlocal operator. Most of the difficulties in the existence proof are concentrated in passing to the limit in the nonlinear term \(A\), given by \(A(y,\nabla^s v)= c(y)|\nabla^s v|\nabla^sv\) with the function \(c(y)\) continuous with respect to all variables and satisfying \(0<\alpha\leq c(y)\leq\beta<\infty\). Here, by \(\nabla^s\) is denoted the symmetric part of the gradient. The authors present an alternative way of passing to this limit. It is much shorter than the previous one by the second author; however it uses Young measures technique.