Vacuum states and equidistribution of the random sequence for Glimm's scheme
[An identical version of this paper was published in: Int. Workshop on Applied differential equations, Beijing/China 1985, ed. by Shutie Xiao and Fuquan Pu (1986; Zbl 0604.00010); pp. 310-322] Glimm's scheme is a main tool for studying hyperbolic conservation laws. It has been shown that in order for Glimm's difference approximations to approach the generalized solution, the random sequence \(a\equiv \{a_ n\}\) should be equidistributed in the interval (-1,1) [\textit{J. Glimm}, Commun. Pure Appl. Math. 18, 697-715 (1965; Zbl 0141.289)]. However, in the standard application for Glimm's scheme, the compactness is established for all sequences and not only for the equidistributed ones. This is a source of essential difficulties for the standard application for Glimm's scheme. In this paper, we only consider the Glimm's scheme generated by equidistributed sequences. We expect that this is the proper application for Glimm's scheme. Consider the initial value problem for isentropic gas dynamics equations in Lagrangian coordinates \[ v_ t-u_ x=0,\quad u_ t+p(v)_ x=0,\quad (-\infty,\quad \infty)\times [0,\infty), \] \[ (u(x,0),v(x,0))=(u_ 0(x),v_ 0(x))\quad (-\infty,\quad \infty), \] where the pressure \(p=p(v)>0\) is a \(C^ 2\) function of the specific volume \(v>0\) and u is the velocity of the gas. We assume that \(p'(v)<0\), \(p''(v)>0\), \(\lim_{v\to \infty}p(v)=0\), and \[ (*)\quad \int^{\infty}_{1}\sqrt{-p'(v)}dv<\infty. \] Condition (*), which is dictated by physics, is a source of essential difficulties in the mathematical treatment of this problem. For instance, it is well known that vacuum may occur, that is, v may be infinite, in a solution of the Riemann problem consisting of rarefaction waves. Our main purpose is to show that the flow and their Glimm's difference approximations contain no vacuum, if they consist of rarefaction waves and the initial data are Lipschitz-continuous.
scientific article; zbMATH DE number 4055272