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We construct a two-parameter family of self-similar solutions to both the compressible and incompressible two-dimensional Euler equations with axisymmetry. The equations can be reduced under the situation to two systems of ordinary differential equations. In the compressible and polytropic case, the system in autonomous form consists of four ordinary differential equations with a two-dimensional set of stationary points, one of which is degenerate up to order four. Through asymptotic analysis and computations of numerical solutions, we are fortunate to be able to recognize a one-parameter family of exact solutions in explicit form. All the solutions (exact or numerical) are globally bounded and continuous, have finite local energy and vorticity, and have well-defined initial and boundary values at time zero and spatial infinity respectively. Particle trajectories of some of these solutions are spiral-like. In the incompressible case, we also find explicit self-similar axisymmetric spiral solutions, which are, however, somewhat less physical due to unbounded pressures or infinite local energy near their swirling centers.
We explore the reflection off a sonic curve and the domain of determinacy, via the method of characteristics, of self-similar solutions to the two dimensional isentropic Euler system through several examples with axially symmetric initial data. We find that characteristics in some cases can be completely absorbed by the sonic curve so that the characteristics vanish tangentially into the sonic boundary, exemplifying a classical scenario of the Keldysh type; however, the characteristics can wrap around the closed sonic curve unboundedly many times, so that the domain of determinacy of the hyperbolic characteristic boundary value problem or the Goursat problem exhibit layered annulus structures. As the number of layers increases, the layers become thinner, and the solution at an interior point of the domain depends eventually on the entire boundary data.
We construct patches of self-similar solutions, in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves, to the two-dimensional pressure gradient system. This type of solutions is common in the solutions of two-dimensional Riemann problems, as seen from numerical experiments. They are not determined by the hyperbolic domain of determinacy in the traditional sense. They are middle-way between the fully hyperbolic (supersonic) and elliptic region, which we call semi-hyperbolic or partially hyperbolic. Our intention is to use the patches as building tiles to construct global solutions to general Riemann problems.
We establish the global existence of solutions to the Cauchy problem for a system of hyperbolic partial differential equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and twist coefficients. Our results have no restrictions on the angles of the director, as we use the director in its natural three-component form, rather than the two-component form of spherical angles.
We present two new types of self-similar solutions to the Chaplygin gas model in two space dimensions: Simple waves and pressure delta waves, which are absent in one space dimension, but appear in the solutions to the two-dimensional Riemann problems. A simple wave is a flow in a physical region whose image in the state space is a one-dimensional curve. The solutions to the interaction of two rarefaction simple waves are constructed. Comparisons with polytropic gases are made. Pressure delta waves are Dirac type concentration in the pressure variable, or impulses of the pressure on discontinuities. They appear in the study of Riemann problems of four rarefaction shocks. This type of discontinuities and concentrations are different from delta waves for the pressureless gas flow model, for which the delta waves are associated with convection and concentration of mass. By re-interpreting the terms in the Chaplygin gas system into new forms we are able to define distributional solutions that include the pressure delta waves. Generalized Rankine-Hugoniot conditions for pressure delta waves are derived.
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