On Two-Dimensional Sonic-Subsonic Flow

A compensated compactness framework is established for sonic-subsonic approximate solutions to the two-dimensional Euler equations for steady irrotational flows that may contain stagnation points. Only crude estimates are required for establishing compactness. It follows that the set of subsonic irrotational solutions to the Euler equations is compact; thus flows with sonic points over an obstacle, such as an airfoil, may be realized as limits of sequences of strictly subsonic flows. Furthermore, sonic-subsonic flows may be constructed from approximate solutions. The compactness framework is then extended to self-similar solutions of the Euler equations for unsteady irrotational flows.

[1]  Peizhu Luo,et al.  CONVERGENCE OF THE LAX–FRIEDRICHS SCHEME FOR ISENTROPIC GAS DYNAMICS (III) , 1985 .

[2]  D. Gilbarg,et al.  UNIQUENESS AND THE FORCE FORMULAS FOR PLANE SUBSONIC FLOWS , 1958 .

[3]  Gui-Qiang G. Chen,et al.  Compressible Euler Equations¶with General Pressure Law , 2000 .

[4]  P. Souganidis,et al.  Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates , 1998 .

[5]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[6]  Gui-Qiang G. Chen.  Euler Equations and Related Hyperbolic Conservation Laws , 2005 .

[7]  D. Gilbarg,et al.  On Bodies Achieving Extreme Values of the Critical Mach Number, I , 1954 .

[8]  Gui-Qiang G. Chen,et al.  Existence Theory for the Isentropic Euler Equations , 2003 .

[9]  Zhouping Xin,et al.  Compactness methods and nonlinear hyperbolic conservation laws , 2000 .

[10]  R. Finn.  ON THE FLOW OF A PERFECT FLUID THROUGH A POLYGONAL NOZZLE. I. , 1954, Proceedings of the National Academy of Sciences of the United States of America.

[11]  R. Finn.  On a Problem of Type, with Application to Elliptic Partial Differential Equations , 1954 .

[12]  R. Perna.  Compensated compactness and general systems of conservation laws , 1985 .

[13]  Lipman Bers,et al.  Existence and uniqueness of a subsonic flow past a given profile , 1954 .

[14]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[15]  Robert Finn,et al.  ASYMPTOTIC BEHAVIOR AND UNIQUENESS OF PLANE SUBSONIC FLOWS , 1957 .

[16]  Guangchang Dong,et al.  Nonlinear partial differential equations of second order , 1991 .

[17]  Luc Tartar,et al.  Compensated compactness and applications to partial differential equations , 1979 .

[18]  Cathleen S. Morawetz,et al.  The mathematical approach to the sonic barrier , 1982 .

[19]  D. Gilbarg.  Comparison Methods in the Theory of Subsonic Flows , 1953 .

[20]  Cathleen S. Morawetz,et al.  On a weak solution for a transonic flow problem , 1985 .

[21]  Cathleen S. Morawetz,et al.  MIXED EQUATIONS AND TRANSONIC FLOW , 2004 .

[22]  On steady transonic flow by compensated compactness , 1995 .

[23]  F. Murat,et al.  Compacité par compensation , 1978 .

[24]  R. J. Diperna,et al.  Convergence of the viscosity method for isentropic gas dynamics , 1983 .

[25]  M. Lighthill.  Supersonic Flow and Shock Waves , 1949, Nature.

[26]  R. Finn.  ON THE FLOW OF A PERFECT FLUID THROUGH A POLYGONAL NOZZLE. II. , 1954, Proceedings of the National Academy of Sciences of the United States of America.

[27]  J. Ball.  A version of the fundamental theorem for young measures , 1989 .

[28]  M. Shiffman.  On the Existence of Subsonic Flows of a Compressible Fluid. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[29]  丹生 慶四郎,et al.  R. Courant and K. O. Friedrichs: Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976, 464ページ, 23.5×16cm, 7,200円. , 1978 .

[30]  Lawrence C. Evans,et al.  Weak convergence methods for nonlinear partial differential equations , 1990 .

[31]  D. Gilbarg,et al.  Uniqueness of Axially Symmetric Subsonic Flow past a Finite Body , 1955 .

[32]  David Gilbarg,et al.  Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations , 1957 .

[33]  C. Crissman.  THE PROBLEM OF TYPE , 2004 .

[34]  Michael Westdickenberg,et al.  Convergence of Approximate Solutions of Conservation Laws , 2015, 1502.00798.

[35]  Hermano Frid,et al.  Divergence‐Measure Fields and Hyperbolic Conservation Laws , 1999 .

[36]  P. Lax,et al.  Systems of conservation laws , 1960 .

[37]  D. Serre.  Systems of conservation laws , 1999 .

[38]  B. Perthame,et al.  Kinetic formulation of the isentropic gas dynamics andp-systems , 1994 .

[39]  Gui-Qiang G. Chen,et al.  Compactness Methods and Nonlinear Hyperbolic Conservation Laws , 2022 .

[40]  L Howarth,et al.  Mathematical Aspects of Subsonic and Transonic Gas Dynamics , 1959 .

[41]  L. Bers.  Results and conjectures in the mathematical theory of subsonic and transonic gas flows , 1954 .

[42]  R. J. Diperna.  Compensated compactness and general systems of conservation laws , 1985 .