Uniqueness and classification conjecture for the heated Riemann problem

Under the double CRPs frame, let $\gamma$ and $\kappa$ be the given parameters, let $M_1$ be the parameter determining the self-similar solution, let $Y(M_1)$ be the function whose root is considered, and let $M_{**}$ denote the threshold Mach number. The self-similar solution has one of the types specified below.

Uniqueness and classification conjecture. The solution of the Riemann problem consisting of the governing equations and initial condition is unique for any given $\gamma$ and $\kappa$, and its structure is determined by $M_1$ as follows:

• If $Y(M_1)\geq 0$, the structure of the solution is Type 1.
• If $Y(M_1)\geq 0$ and $M_1\leq M_{**}$ with $k(\gamma^2-1)<1$, the structure of the solution is Type 2.
• If $k(\gamma^2-1)<1$ and $M_1\geq M_{**}$, the structure of the solution is Type 3.

The preceding theorem establishes uniqueness when $k(\gamma^2-1)\geq 1$, and also when $k(\gamma^2-1)<1$ under the assumption that the root of $Y(M_1)$ is not greater than $M_{**}$. The conjecture asserts that the latter assumption holds generally and completes the uniqueness and structural classification of the self-similar solution.