Singular limits in compressible fluid dynamics
The nonstationary motion of a compressible fluid is considered in \(n\)- dimensional torus \((n \geq 2)\), \(\rho_t + U \nabla \rho + \rho \nabla U = 0\), \(\rho [U_t + (U \nabla) U]- \nu \Delta U - \mu \nabla (\nabla U) + \nabla p = 0\), \(\rho (x,0) = \rho_0 (x) + 1\), \(U(x,0) = U_0 (x)\). Here \(U\) is the velocity, \(\rho\) is the density of the fluid, \(p\) is the pressure which is considered as a function of the density \(\rho\) and the Mach number \(\lambda^{-1}\), \(p (\lambda, \rho) = \lambda^2 p(\rho)\). The behaviour of \((U,p)\) is studied as \(\lambda \to \infty\), \(\nu \to \overline \nu\) \(U_0 \to W_0\), \(\rho_0 \to 0\), \(\mu\) remains bounded. The limit equations are the equations of motion of an incompressible fluid with density 1 and viscosity \(\overline \nu \geq 0\): \(W_t - \overline \nu \Delta W + (W \nabla) W + \nabla \pi = 0\), \(\nabla W = 0\), \(W(x,0) = W_0\). The convergence of \((U, \rho)\) to \((U,1)\) is established in Sobolev spaces \(H^m\).
scientific article; zbMATH DE number 754029