Justification of the Courant-Friedrichs Conjecture for the Problem About Flow Around a Wedge

[BOOK DESCRIPTION]
The classical problem about a steady-state supersonic flow of an inviscid n-heat-conductive gas around an infinite plane wedge under the assumption that the angle at the vertex of the wedge is less than some limit value is considered. The gas is supposed to be in the state of thermodynamical equilibrium and admits the existence of a state equation. As is well-kwn, the problem has two discontinuous solutions, one of which is associated with a strong shock wave (the gas velocity behind the shock wave is less than the sound speed) and the second one corresponds to the weak shock wave (the gas velocity behind the shock wave is, in general, larger than the sound speed) (Courant R, Friedrichs K.O. Supersonic flow and shock waves. N. Y.: Interscience Publ. Inc., 1948). One of the possible explanations of this phemen was given by Courant and Friedrichs. They conjectured that the solution corresponding to the strong shock wave is instable in the sense of Lyapuv, whereas the solution corresponding to the weak shock wave is stable. This conjecture has been confirmed in a number of studies in which either particular cases were considered or the proposed argumentation was given at the qualitative (mostly, physical) level of rigor. In this mograph, the Courant-Friedrichs conjecture is strictly mathematically justified at the linear level. The mechanism of generating the instability for the case of a strong shock is explained. The smoothness of the solution essentially depends on the peculiarity of the boundary at the vertex of the wedge. The situation with a weak shock drastically differs from the previous one. It is amazing but for the compactly supported initial data the solution to the linear problem reaches the steady state regime infinite time.

[TABLE OF CONTENTS]

Preface                                            1  (2)
Introduction                                       3  (20)
    Chapter 1 Instability of Strong Shock Wave.    23 (62)
    Case of Small Vertex Angle
      1 Preliminaries. Statement of Classical      23 (7)
      and Generalized Problems. The Main Results
      2 Boundary Value Problem for Traces of       30 (8)
      Solutions
      3 Partition of Roots for a Polynomial by     38 (14)
      the Unit Circle. The Cohn Algorithm.
      Verification of Eq. (2.30)
      4 The Carleman Problem. Finding              52 (23)
      Z((λ), s). Proof of Theorem 1.1
      5 Representation of the Boundary Function    75 (10)
      V(y, t) in the Cartesian Coordinates and
      the Asymptotic Behavior of V(y, t) as t
      → ∞
    Chapter 2 Instability of Strong Shock Wave.    85 (28)
    General Case
      1 Reduction to the Problem in Equations      87 (12)
      (1.2.34) and (1.2.35) for the Riemann
      Problem on the Half-Line. Representation
      of the Trace V(y, t) in the Cartesian
      Coordinates
      2 Solvability Condition in. Equation         99 (7)
      (1.34) for Cartesian Coordinates
      3 Trace Solution of V(y, t) on the Shock     106(7)
      Wave with no Compactly Supported Initial
      Data in R2+. The Lyapunov Instability to
      Solutions as t → +∞
    Chapter 3 Stability of Weak Shock Wave         113(26)
      1 Statement of the Main and Auxiliary        114(4)
      Problems. The Main Results
      2 Proof of Theorem 1.1                       118(11)
      3 Boundary Values for the Solution to the    129(10)
      Problem in Equations (1.16)--- (1.20) and
      Its Derivatives. Asymptotics
Conclusion                                         139(2)
Bibliography                                       141(8)
Index                                              149