Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations. (English) Zbl 1527.35241
Summary: This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.
MSC:
| 35Q31 | Euler equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76H05 | Transonic flows |
| 76J20 | Supersonic flows |
| 76N15 | Gas dynamics (general theory) |
| 35M33 | Initial-boundary value problems for mixed-type systems of PDEs |
| 35A09 | Classical solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
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