Solvability in the large of the Cauchy problem for the system of gas dynamics
Consider the system of gas dynamics (in Lagrangian coordinates) \[ v_t- u_x=0,\quad u_t+ p_x=0,\quad E_t+(up)_x=0,\tag{1} \] where \(v\), \(u\), \(p\), \(E\), \(w\), \(\rho=v^{-1}\), and \(\theta\) are specific volume, velocity, pressure, total energy, intrinsic energy, density, and gas temperature, respectively; \(R\) is the gas constant; \(c_v(\theta)\) is the specific heat of gas; \[ p= R\rho\theta;\quad E=w+{u^2\over 2};\quad\text{and} \quad w=\int^\theta_0 c_v(\sigma)d\sigma. \] Under the conditions \[ 0< m^0_v\leq v^0(x)\leq M^0_v,\quad |u^0(x)|\leq M^0_{|u|},\quad 0<m^0_\theta\leq\theta^0(x)\leq M^0_\theta \] on every segment \([-L,L]\), where \(m^0_v\), \(m^0_\theta\), \(M^0_v\), \(M^0_{|u|}\), and \(M^0_\theta\), generally speaking, depend on \(L\), the existence of a functional solution to the Cauchy problem for (1) for \(c_v(\theta)= c^0_v=\text{const}\) is established in the strip \(\Pi=\{(t,x):0<t<T,\;|x|<\infty\}\) for any \(T\) that is independent of the data of the problem.
scientific article; zbMATH DE number 1037726