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

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
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