Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

Abstract

In this paper, we consider the Cauchy problem for a nonlinear parabolic system $${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$$ in $${\mathbb {R}^3 \times (0,\infty)}$$ with initial data in Lebesgue spaces $${L^2(\mathbb {R}^3)}$$ or $${L^3(\mathbb {R}^3)}$$ . We analyze the convergence of its solutions to a solution of the incompressible Navier–Stokes system as $${\epsilon \to 0}$$ .