| Abstract |
In this presentation I survey several recent results on balance-law systems. First, I discuss the existence of classical solutions and then the constructive framework for weak solutions, using two fluid-dynamics models as guiding examples.
- Hot-Jupiter hydrodynamic escape.
We analyse the Euler equations with gravity, tidal forces and heating and obtain transonic solutions describing atmospheric outflow. Sharp estimates in the construction yield an explicit relation between (i) the lower velocity bound in the sub-sonic regime and (ii) the specific-heat (adiabatic) exponent of the gas.
- Time-dependent Fanno–Rayleigh flow.
For the compressible Euler equations with friction and heat exchange in a variable-area nozzle we treat an initial–boundary-value problem. A critical threshold, formulated in terms of the friction coefficient and the nozzle geometry, guarantees the existence of transonic solutions. In addition, we derive explicit ratio for both the Mach number and the power flux. These bounds determine whether an admissible flow remains purely supersonic or can pass through a transonic state, and they provide a quantitative link between nozzle shape, friction and heat-transfer parameters. |
