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Let $$\partial_t u + \nabla_u u = - \nabla p$$ be Euler's equation (Wikipedia) for an ideal incompressible fluid. Let

$$\partial_t u + \nabla_u u - \nu \Delta u = - \nabla p$$ be the Navier-Stokes equations for a Newtonian incompressible fluid with a real parameter $\nu$, the viscosity. Obviously this equation reduces to Euler's equation if we set $\nu = 0$ (and domains , and initial conditions are equal and suitable boundary conditions are equal)prescribed).

What is known about the convergence of solutions $u_{\nu}$ of the Navier-Stokes equation as stated above to a solution of Euler's equation in the limit $\nu \to 0?$ That is, are there topological vector spaces T and sequences or nets of solutions $u_{\nu_k}$ of the Navier-Stokes equation with viscosity $\nu_k$ in T such that the limit $\lim_{\nu_k \to 0} u_{\nu_k}$ exists and is a solution to Euler's equation?

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# Convergence of solutions to Navier-Stokes to Euler's equation for viscosity $\to$ zero

Let $$\partial_t u + \nabla_u u = - \nabla p$$ be Euler's equation (Wikipedia) for an ideal incompressible fluid. Let

$$\partial_t u + \nabla_u u - \nu \Delta u = - \nabla p$$ be the Navier-Stokes equations for a Newtonian incompressible fluid with a real parameter $\nu$, the viscosity. Obviously this equation reduces to Euler's equation if we set $\nu = 0$ (and domains, initial and boundary conditions are equal).

What is known about the convergence of solutions $u_{\nu}$ of the Navier-Stokes equation as stated above to a solution of Euler's equation in the limit $\nu \to 0?$ That is, are there topological vector spaces T and sequences or nets of solutions $u_{\nu_k}$ of the Navier-Stokes equation with viscosity $\nu_k$ in T such that the limit $\lim_{\nu_k \to 0} u_{\nu_k}$ exists and is a solution to Euler's equation?