On the existence and regularity of solutions of a quasilinear mixed equation of Leray-Lions type. (English) Zbl 0669.35080

Michaux, B.; Rakotoson, J. M.; Shen, J.

Acta Appl. Math. 12, No. 3, 287-316 (1988).

The main object of this article is to study the existence and regularity of solutions of quasilinear equations. One specific application is the equation governing the flow of a perfect and isentropic fluid in a rectangle domain with the deviation as well as the Mach number distributions prescribed along the channel walls. The authors analyze this equation by setting it in a more general framework \[ (1)\text{ Find }u\in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\text{ such that } Au+F(u,\nabla u)=T\text{ in \(\Omega\) and }u-g\in W_ 0^{1,p}(\Omega). \] Here \(\Omega\) denotes a bounded set of \({\mathbb{R}}^ n\), \(Au=-(\partial /\partial x_ i)a_ i(x,u,\nabla u)\) is an elliptic-hyperbolic operator which maps \(W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) into \(W^{- 1,p'}(\Omega)\) \((1/p+1/p'=1)\) and such that for almost every \(x\in \Omega\), all \(u\in I\subset {\mathbb{R}}^ 1\) and all \(\xi \in {\mathbb{R}}^ n\) \(a_ i(x,u,\xi)\xi_ i\geq \nu (u)| \xi |^ p,\) g is given in \(W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) and F is in \(W^{- 1,p'}(\Omega).\)
It is not assumed that the operator A is elliptic on the real axis but, that there exists an interval \(I=(0,u_ 0)\), \(u_ 0>0\) in which the function \(\nu\) (u) is continuous and strictly positive.
The authors give assumptions such that a Hölder continuous weak solution on the problem (1) exists. They show how the physical model case is represented by (1) and propose a numerical scheme based on a finite difference method.

Reviewer: M.Schneider

MSC:

35M99	Partial differential equations of mixed type and mixed-type systems of partial differential equations
35R30	Inverse problems for PDEs
76N15	Gas dynamics (general theory)
35J20	Variational methods for second-order elliptic equations
35J60	Nonlinear elliptic equations
76G25	General aerodynamics and subsonic flows
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

existence; regularity; quasilinear; perfect and isentropic fluid; rectangle domain; elliptic-hyperbolic operator; Hölder continuous weak solution; numerical scheme; finite difference method

References:

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