The one-dimensional Darcy’s law as the limit of a compressible Euler flow. (English) Zbl 0715.35065

Marcati, Pierangelo; Milani, Albert

J. Differ. Equations 84, No. 1, 129-147 (1990).

The authors study the singular convergence of solutions to a damped compressible Euler flow in one space dimension, with a polytropic equation of state, when the inertial terms tend to zero in a suitable rescaling. More precisely, they consider for all \(\epsilon >0\) the system of equations \[ \rho^{\epsilon}_ t+(\rho^{\epsilon}u^{\epsilon})_ x=0;\quad \epsilon (\rho^{\epsilon}u^{\epsilon})_ t+(\epsilon \rho^{\epsilon}(u^{\epsilon})^ 2+p^{\epsilon})_ x=- ku^{\epsilon}, \]
\[ p^{\epsilon}=p(\rho^{\epsilon})=c(\rho^{\epsilon})^{\gamma};\quad \rho^{\epsilon}(x,0)=\rho_ 0(x)\geq 0,\quad u^{\epsilon}(x,0)=u_ 0(x), \] where \(k>0\), \(c>0\), \(\rho\) is the density, u the Eulerian velocity and \(\gamma\) is the adiabatic exponent, that is, \(\gamma =1+2/n\), n denoting the number of degrees of freedom of the molecules (n\(\geq 3).\)
They show that there exists limit functions \(\rho\) and u such that, as \(\epsilon\downarrow 0\), \(\rho^{\epsilon}\to \rho\) in \(L^ p_{loc}\), \(u^{\epsilon}\to u\) in \(L^ 2\) weak and \(\sqrt{\epsilon}u^{\epsilon}\to 0\) in \(L^ p_{loc}\), for all \(p\in (1,\infty)\). Moreover, \(\rho\) satisfies, in the sense of distributions, Darcy’s law so that \(\rho\) is a weak solution of a porous media equation.

Reviewer: A.D.Osborne

MSC:

35Q35	PDEs in connection with fluid mechanics
76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B25	Singular perturbations in context of PDEs
76S05	Flows in porous media; filtration; seepage

Keywords:

Euler flow; limit functions; Darcy’s law

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