Acta mathematica scientia, Series B >
A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION
Received date: 2011-11-03
Online published: 2011-11-20
Supported by
The paper is partially supported by National Natural Science Foundation of China (10531020) and the Doctorial Foundation of National Educational Ministry (20090071110002).
In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of themixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method
developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.
Chen Shuxing . A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION[J]. Acta mathematica scientia, Series B, 2011 , 31(6) : 2378 -2388 . DOI: 10.1016/S0252-9602(11)60406-2
[1] Bitsadze A V. Differential Equations of Mixed Type. New York: Macmillan Co, 1964
[2] Bitsadze A V. Some classes of partial differential equations. Steklov Mathematics Institute, USSR Academy of Sciences, Moscow, 4, Translated by Gordon and Breach Science Publishers, 1988
[3] Bony J M. Calcul symbolique et propagation des singularit’es poue équation aux dérivées partilles non-lináires. Ann Scient R N S, 1981, 14: 209–246
[4] Chen S X. Existence of stationary supersonic flow past a pointed body. Archive Rat Mech Anal, 2001, 156: 141–181
[5] Chen S X. Stability of a Mach Configuration. Comm Pure Appl Math, 2006, 59: 1–35
[6] Chen S X. Mach configuration in pseudo-stationary compressible flow. Jour Amer Math Soc, 2008, 21: 63–100
[7] Chen S X. Mixed type equation in gas dynamics. Quat Appl Math, 2010, 68: 487–511
[8] Cole J D, Cook L P. Transonic Aerodynamics. Amsterdam, New York, Oxford, Tokyo: North-Holland, 1986
[9] Chen S X, Qiu Q J, Li C Z. Introduction to Paradifferential Operators. Beijing: Science Press, 1997
[10] Grisvard P. Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. London: Pitman, 1985
[11] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equation of Second Order. Second ed. Grundlehren der Mathematischen Wissenschaften, 224. Berlin-New York: Springer, 1983
[12] Hua L K. On Lavrentiev partial differential equation of the mixed type. Scientia Sinica, 1964, 13: 1755–1762
[13] Lavrentiev M A, Bitsadze A V. On problem of mixed type equations. Dokl Akad Nauk, SSSR, 1950, 20
[14] Morawetz C S. Potential theory for regular and Mach reflection of a shock at a wedge. Comm Pure Appl Math, 1994, 47: 593–624
[15] Rassias J M. Mixed Type Equations. Teubner-Texte zur Mathematics, Band 90, Leipzig, 1986
[16] Smirnov M M. Equations of Mixed type. Translations of Mathematical Monographs, 51. Providence: Amer Math Soc, 1978
[17] Tricomi F. Sulle equazioni lineari alle derivate parziali di secondo ordino, di tipo misto. Rend Accad Naz Lincei, 1923, 14: 134–247
/
| 〈 |
|
〉 |