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An existence theorem for the multifluid Navier-Stokes problem. (English) Zbl 0842.35079

The authors study the flow of \(N\) fluids filling subdomains \(\Omega_k (t)\), \(k = 1, \dots, N\) of a fixed domain \(\Omega \subset \mathbb{R}^d\), \(d = 2,3\). With the help of variational arguments they show that the classical non-miscibility and transmission conditions on the interfaces are equivalent to transport equations for the globally defined viscosity \(\eta\) and concentration \(\rho\). This leads to the subsequent system of partial differential equations \[ \rho_t + \text{div} (\rho u) = 0, \quad \eta_t + \text{div} (\eta u) = 0, \]
\[ (\rho u)_t + \text{div} (\rho u \otimes u) - \text{div} \bigl( \eta \in (u) \bigr) = f, \] with \(u |_{\partial \Omega} = 0\), \(t > 0\) and \((\rho, \eta,u) (0) = (\rho_0, \eta_0, u_0)\) in \(\Omega\). The main result of the paper is the following: If \((\rho_0, \eta_0) \in \{(\rho_k, \eta_k)\), \(k = 1, \dots, N\}\) a.e. with \(\rho_k > 0\), \(0 < \eta_1 < \cdots < \eta_N\), \(f \in L^2 (0,T; H^{-1} (\Omega))\) and \(u_0 \in L^2 (\Omega) \), \(\text{div} u_0 = 0\), then the above system has a weak solution \((\rho, \eta, u) \in L^\infty (\Omega \times (0,T))^2 \times L^2 (0,T; H^1_0 (\Omega))\), \(\text{div} u = 0\) a.e. in \(\Omega \times (0,T)\). Moreover, \((\rho, \eta) \in \{(\rho_k, \eta_k)\), \(k = 1, \dots, N\}\) a.e.
The main ingredients of the proof are a careful study of the linearized Navier-Stokes system together with a result due to di Perna-Lions on renormalized weak solutions for transport equations \(\rho_t + \text{div} (\rho u) = 0\).

MSC:

35Q30 Navier-Stokes equations
76V05 Reaction effects in flows
35D05 Existence of generalized solutions of PDE (MSC2000)
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