Gas flows with shock waves which diverge from an axis or centre of symmetry with finite velocity

The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

“…Under some constraints due to the physical meaning of the problems, convergence of series was proven. In [15,16] S. P. Bautin and the author, using the GCP, studied the gas flow with shock waves in a neighborhood of a symmetry axis or symmetry center where the system of equations of gas dynamics has a singularity of the form u/r (u is the gas velocity and r is the distance to the symmetry axis or center).In this article we construct a solution to the GCP with data on two surfaces for a second-order quasilinear analytic system. Our approach bases on the methods by S. P. Bautin [9]; we supplement and generalize them to the case when the system of partial differential equations also includes the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes.…”

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

“…V. M. Teshukov [10-13] studied the following problems: destruction of an arbitrary quasi-one-dimensional discontinuity; abrupt insertion of an impenetrable piston into a gas; reflection of a curvilinear shock wave from a rigid wall; and spatial interaction of strong discontinuities in a gas. In [14,15] the gas flows with shock waves were studied when the system of equations of gas dynamics has a singularity of the form u/r (u is the gas velocity and r is the distance to the axis or center of symmetry).We consider the following quasilinear system of partial differential equations in the case of three unknown functions depending on three independent variables:(1) Here U = {u, v, w} is the vector of unknown functions; x = {x, y, z} is the vector of independent variables; D 1 , D 2 , and D 3 are (3 × 3)-matrices; f ∈ R 3 ; the entries of D 1 , D 2 , and D 3 and the components of f are functions of x, y, z, u, v, and w. …”

“…V. M. Teshukov [10][11][12][13] studied the following problems: destruction of an arbitrary quasi-one-dimensional discontinuity; abrupt insertion of an impenetrable piston into a gas; reflection of a curvilinear shock wave from a rigid wall; and spatial interaction of strong discontinuities in a gas. In [14,15] the gas flows with shock waves were studied when the system of equations of gas dynamics has a singularity of the form u/r (u is the gas velocity and r is the distance to the axis or center of symmetry).…”

“…Moreover, (14) and (15) are necessary and sufficient conditions for unique existence of a solution presented as formal series in the powers of x, y, and z. Remark 2.…”

“…(23) By (14), the inequality 1 − B 6 C 4 = 0 is valid; therefore, from (22) we obtain v l,k+1,n−l−k = 1 If n = l then (22) yields the expressions v n,0,1 = 0, w n,0,1 = C 1 u n,1,0 + C 2 u n,0,1 + r n,0,0 , w n,1,0 = 0, v n,1,0 = B 1 u n,1,0 + B 2 u n,0,1 + q n,0,0 .…”