Gas-dynamical effects connected with unboundedness of the solutions of quasilinear systems

Abstract

THIS paper is devoted to the problem of the unboundedness of the classical solutions of quasilinear hyperbolic systems for n ⩾ 3. An example illustrating this fact is constructed. A class of systems is isolated for which the solution remains bounded. This class includes the system of equations of gas dynamics in the non-isentropic case with some constraints on the equation of state. For model equations of states close to the absolute zero, where these constraints are not satisfied, effects connected with localised unbounded growth of density or specific volume, are assumed, which are confirmed by the results of numerical experiment. According to a statement in [1], the absence of integrating factors of the Pfaffian forms dω k = l α k ( t, x, u) du α for n ⩾ 3 (“spin of the eigenvectors”) may lead to unbounded growth of the classical solution of the Cauchy problem for the hyperbolic system of equations (1) ∂u ∂t + A (t, x, u) ∂u ∂x = F(t, x, u), u(0, x) = u 0(x), xϵ(− ∞,∞) . where t and x are the independent variables; u and F are n-dimensional vector functions; A is a matrix of order n; l k is the left eigenvector of the matrix A corresponding to the eigenvalue ξ k ; F( t, x, u), A( t, x, u), u 0( x) ϵ C 1, even, if F  0. (Here and everywhere below repetition of a Greek subscript means summation.) This paper is devoted to a study of the question of unboundedness of solutions. The main results are as follows. An example is constructed illustrating the unboundedness of solutions for n = 3. A narrow class of systems for which the classical solution remains bounded is isolated. These are systems permitting n − 1 or n − 2 “entropic” [1] variables and satisfying constraints on the growth of the coefficients of the characteristic form. For the equations of gas-dynamics in the non-isentropic case, these constraints reduce to conditions on the equation of state which are expressed by the inequalities (2) 0 < c v 0 ⩽ ¦ c v ¦ ∗, ¦ ( ∂p ∂ϱ∂s ) ( ∂p ∂ϱ ) s −1, ¦ ⩽ M 0, C V 0, M 0 = const . We will use the following notation: ϱ is the density; u is the velocity; s is the entropy of unit mass; T is the temperature; p = p( ϱ, s) is the pressure (equation of state); c v = T ( ∂s ∂T ) ϱ is the specific heat per unit mass; c = √ ( ∂p ∂ϱ ) s is the speed of sound. For such systems the solution satisfies some non-linear multi-dimensional integral equation of Volterra type where the domain of definition of the hyperbolic equations emerges as the cone of integration, and the boundedness follows from the property of the local absolute integrability of the kernel, which in terms of quasilinear equations indicates the absence of discontinuities of the entropic variables or the possibility of discontinuities with small variations, and implies another remarkable property of the gas-dynamical equations. We consider the case where the constraints (2) are not satisfied. This is the case if the temperature approaches the absolute zero, and then by Nernst's theorem the entropy and specific heat tend to zero. As models we take the equations of state of degenerate Fermi and Bose gases close to the absolute zero. For a Fermi gas ( ∂p ∂s ) ϱ  0 the system of equations of hydrodynamics may be split up and the question of unboundedness of the solution does not arise. For a Bose gas the question requires additional study. In this case we consider a Cauchy problem with the initial functions ϱ(0, x)= ϱ 0 = const, u(0, x) = 0, s(0, x) = s 0( x) where s 0 ( x) has at the point x = 0 a local minimum s min , which tends to zero. We have here parabolic degeneracy of the gas-dynamical equations, since the speed of sound tends to zero. We construct a trivial model of degeneracy which assumes the possibility of a gas-dynamical effect which consists of the unbounded increase in density at a point of minimum when s min tends to zero. This assumption is confirmed by the results of a numerical experiment.