Non-Conventional and Conventional Solutions for One-Dimensional Pressureless Gas

Abstract

The paper deals with various generalized solutions to one-dimensional pressureless gas dynamics. Along with the entropy solutions other solutions are considered that include the combination of concentration and decay processes. This research is motivated by multidimensional case where the concentration mechanism seems to be not enough for the construction of rigorous theory. Thus the present paper describes in one-dimensional setting a possible behavior that could appear in multidimensional case.

ON THE OTHER SELF-SIMILAR SOLUTIONS

In the previous sections we use the self-similar solution $\varrho=(1/t)R\left(x/t\right)$, $u=x/t$. Meanwhile it is possible to construct much wider family of self-similar solutions, namely the continuous solutions to system (1) with $A^{\prime}\left(u\right)\equiv u$ in the form

$$\varrho=t^{\alpha}R\left(\zeta\right),\quad u=t^{\gamma}U\left(\zeta\right),\quad\zeta\equiv xt^{-\beta},\quad\beta>0.$$

(A1)

Theorem. Suppose $\gamma=\beta-1$ . Then there exist self-similar solutions to (1) in the form (A1) under the following relations

$$\left|\zeta-U\right|^{1-\beta}\left|U\right|^{\beta}=\bar{C}\geq 0,$$

$$\frac{R^{\prime}}{R}=\frac{\alpha+U^{\prime}}{\beta\zeta-U},$$

(A2)

where $\bar{C}$ is some constant.

Proof. Let substitute the functions $\varrho,u$ in the form (A1) to system (1) with $A^{\prime}\left(u\right)\equiv u$. Than it is easy to come to the following system

$$\gamma U+U^{\prime}\left(U-\beta\zeta\right)=0,$$

$$\alpha R+U^{\prime}R+R^{\prime}\left(U-\beta\zeta\right)=0.$$

(A3)

Now perform the change of variables $\zeta V=U$, then first equation (A3) yields $\zeta V^{\prime}\left(V-\beta\right)=V\left(1-V\right)$. Suppose $V\neq 1$, then obtained differential equation is the equation with separable variables and thus can be solved explicitly. Henceforth we come to (A2) with $\bar{C}>0$. In case $V=1$ (A2) is also valid with $\bar{C}=0$. $\Box$

Self-similar solution considered in the previous sections correspond to the case $\beta=1$. Then $\gamma=0$ and from the first equation (A3) it follows that $U=\zeta$. Similarly from the second equation (A3) it follows $\alpha=-1$ with arbitrary $R$.

So we have a variety of self-similar solutions that in principle can be used as building blocks for the construction of generalized solutions with shocks.