参考文献

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参考文献

$\bullet$ 成書、論説

1

浅倉史興, 双曲型保存則系の初期値問題 - 基本結果と近年の話題 -, 「数学」第 52 巻 3 号 (日本数学会) 2000, 257-278.

2

西田孝明- 川島秀一, 気体の運動方程式, 「非線形の現象と解析」, 山口昌也編 (日本評論社) 1979, 135-160.

3

J.Smoller. Shock waves and reaction-diffusion equations. (2nd edition), Springer, 1991.

4

C.M.Dafermous. Hyperbolic conservation laws in continuum physics. Springer, 2000.

5

D.Serre. Systems of conservation laws, 1, 2. Cambridge, 1999.

6

A.Bressan. Hyperbolic systems of conservation laws. Oxford, 2000.

7

吉川敦, 非線形保存則系入門 - 保存則の理論 -, 上智大学数学講究録 No.21 (上智大学数学教室), 1985.

8

L.H $\mbox{\uml {o}}$ rmander. Lectures on nonlinear hyperbolic differential equations. Springer, 1997.

9

R.Courant and K.O.Friedrichs. Supersonic flow and shock waves. Springer, 1991 (original edition: Interscience, 1948)

$\bullet$ 代表的な諸結果

$\circ$ 単独保存則

10

O.A.Oleinik. Discontinuous solutions of non-linear differential equations. Uspekhi Mat.Nauk 12 (1957), 3-73.

$\circ$ Glimm の差分

11

J.Glimm. Solution in the large for nonlinear hyperbolic systems of equations. Comm.Pure Appl.Math. 18 (1965), 697-715.

12

T.Nishida. Global solutions for an initial boundary value problem of a quasilinear hyperbolic system. Proc.Japan Acad. 44 (1968), 642-646.

$\circ$ 補完測度法

13

L.Tartar. Compensated compactness and applications to partial differential equations. 136-211, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol.4, ed. R.J.Knops, Research Notes in Mathematics 39, Pitman, London 1979.

14

R.J.DiPerna. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27-70.

15

R.J.DiPerna. Convergence of viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91 (1983), 1-30.

$\circ$ 波面追跡法

16

R.J.DiPerna. Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J. Differential Equations 20 (1976), 187-212.

17

A.Bressan. Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170 (1992), 414-432.

18

N.H.Risebro. A front-tracking alternative to the random choice method.
Proc.Amer.Math.Soc. 117 (1993), 1125-1139.

$\circ$ 動力学近似

19

P.L.Lions, B.Perthame, and E.Tadmor. Kinetic formulation of the isentropic gas dynamics and p-systems. Comm.Math.Phys. 163 (1994), 415-431.

20

P.L.Lions, B.Perthame, and P.E.Souganidis. Existence and stability of entropy solutions for the hyperbolic system of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm.Pure Appl.Math. 49 (1996), 599-638.

$\bullet$ 補完測度法のための参考資料

21

L.C.Evans. Weak convergence methods for nonlinear partial differential equations. CBMS regional conference ser. in Math. 74, AMS, 1990.

22

G.-Q.Chen. The compensated compactness method and the system of isentropic gas dynamics. Math.Sci.Res.Institute 00527-91, Berkeley. 1990.

23

K.N.Chueh, C.C.Conley and J.A.Smoller. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26(1977), 373-392.

24

G.-Q.Chen and Y.G.Lu. The study on application way of the compensated compactness theory. Chinese Sci.Bull. 34 (1989) 15-19.

25

竹野茂治, 気体の運動方程式の初期値境界値問題について, 新潟大学大学院修士論文, (1990), 1-65.

$\bullet$ 補完測度法に関する論文

$\circ$ 気体のバロトロピックモデル

26

X.Ding, G.-Q.Chen, and P.Luo. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I)-(I I). Acta Mathematica Scientia 5(1985), 415-432, 433-472 (Chinese edition: 7(1987), 467-481; 8(1988), 61-94).

27

G.-Q.Chen. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I I I). Acta Mathematica Scientia 6(1986), 75-120 (Chinese edition: 8(1988), 101-134).

28

X.Ding, G.-Q.Chen and P.Luo. A supplement to the papers 'Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I I)-(I I I)'. Acta Mathematica Scientia 9(1989), 43-44.

29

X.Ding, G.-Q.Chen and P.Luo. Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Comm. Math. Phys. 121(1989), 63-84.

30

T.Makino. Weak solutions to the compressible Euler equation with an asymptotic gamma-law. (2000), 1-42.

31

G.-Q.Chen and P.G.LeFloch. Compressible Euler equations with general pressure law. Arch.Rational Mech.Anal. 153 (2000), 221-259.

$\circ$ 補完測度法

32

M.E.Schonbek. Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982), 959-1000.

33

J.Shearer. Global existence and compactness in

for the quasi-linear wave equation. Comm. Partial Differential Equations 19(1994), 1829-1877.

34

P.Lin. Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics. Trans. Amer. Math. Soc. 329 (1992), 377-413.

35

M.Kru $\mbox{\u{z}}$ $\mbox{\'{\i}}$ k. Explicit characterization of

-Young measures. J. Math. Anal. Appl. 198 (1996), 830-843.

36

Y.Zhou. An

theorem for compensated compactness. Proc. Royal Soc. Edinburgh 122A (1992), 177-189.

37

J.M.Ball. A version of the fundamental theorem for Young measures. Lecture Notes in Phys. 344, (1989), 207-215

38

N.Hungerb $\mbox{\uml {u}}$ hler. A refinement of Ball's theorem on Young measures New York J.Math. 3 (1997), 48-53.

$\circ$ 緩和項を含む方程式

39

G.-Q.Chen and T.P.Liu. Zero relaxation and dissipation limits for hyperbolic conservation laws. Comm.Pure Appl.Math. 46 (1993), 755-781.

40

G.-Q.Chen, C.D.Levermore, and T.P.Liu. Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm.Pure Appl.Math. 47 (1994), 787-830.

41

P.Marcati and R.Natalini. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch.Rational Mech.Anal. 129 (1995), 129-145.

42

C.Lattanzio and P.Marcati. The zero relaxation limit for the hydrodynamic Whitham traffic flow model. J.Differential Equations 141 (1997), 150-178.

43

C.Lattanzio and P.Marcati. The zero relaxation limit for 2x2 hyperbolic systems. Nonlinear Anal. 38 (1999), 375-389.

$\circ$ その他

44

C.-H.Hsu, S.-S.Lin, and T.Makino. On the relativistic Euler equation. (2000), 1-61.

45

G.-Q.Chen and P.T.Kan Hyperbolic conservation laws with Umbilic degeneracy I. Arch.Rational Mech.Anal. 130 (1995), 231-276.

46

P.D.Serre. La compacit $\mbox{\'{e}}$ par compensation pour les syst $\mbox{\\lq {e}}$ mes hyperboliques non lin $\mbox{\'{e}}$ aires de deux $\mbox{\'{e}}$ quations a une dimension d'espace. J.Math.Pures Appl. 65 (1986), 423-468.

47

R.J.DiPerna. Compensated compactness and general systems of conservation laws Trans.Amer.Math.Soc. 292 (1985), 383-420.

48

G.-Q.Chen and Y.G.Lu Convergence of the approximate solutions to isentropic gas dynamics. Acta Mathematica Scientia 10 (1990), 39-45.

49

V.Roytburd and M.Slemrod. Positively invariant regions for a problem in phase transitions. Arch.Rational Mech.Anal. 93 (1986), 61-79.

50

T.Makino and S.Takeno. Initial boundary value problem for the spherically symmetric motion of isentropic gas. Japan J.Indust.Appl.Math. 11 (1994), 171-183.

51

G.-Q.Chen and J.Glimm. Global solutions to the compressible Euler equations with geometrical structure. Comm.Math.Phys. 180 (1996), 153-193.

$\circ$ Young 測度、補完測度法に関する本記事以外の他の話題など

52

E.J.Balder. Consequences of denseness of Dirac Young measures. J.Math.Anal.Appl. 207 (1997), 536-540.

53

S.Demoulini. Young measure solutions for nonlinear evolutionary system of mixed type. Ann.Inst.H.Poincare 14 (1997), 143-162.

54

H.Bellout, F.Bloom, and J.Ne $\mbox{\v{c}}$ as. Young measure-valued solutions for
non-Newtonian incompressible fluids. Comm.Partial Differential Equations 19 (1994) 1763-1803.

55

G.Dolzmann, N.Hungerb $\mbox{\uml {u}}$ hler, and S.M $\mbox{\uml {u}}$ ller. Non-linear elliptic systems with measure-valued right hand side. Math.Z. 226 (1997), 545-574.

56

D.Kinderlehrer and P.Pedregal. Characterizations of Young measures generated by gradients. Arch.Rational Mech.Anal. 115 (1991), 329-365.

57

D.Kinderlehrer and P.Pedregal. Weak convergence of integrands and the Young measure representation. SIAM J.Math.Anal. 23 (1992), 1-19.

58

J.Michel and R.Robert. Large deviations for Young measures and statistical
mechanics of infinite dimensional dynamical systems with conservation law.
Comm.Math.Phys. 159 (1994), 195-215.

59

P.Pedregal. Nonlocal variational principles. Nonlinear Anal. 29 (1997), 1379-1392.

60

P.Pedregal. Parametrized measures and variational principles. Birkhauser, 1997.

61

B.Dacorogna. Weak continuity and weak lower semicontinuity of Non-linear functionals. Lecture Note in Math. 922, Springer, 1982.

62

M.Sychev. Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands. Ann.Inst.H.Poincare 15 (1998), 755-782.

63

A.Szepessy. An existence result for scalar conservation laws using measure valued solutions. Comm.Partial Differential Equations 14 (1989), 1329-1350.

64

F.Theil. Young-measure solutions for a viscoelastically damped wave equation with nonmonotone stress-strain relation. Arch.Rational Mech.Anal. 144 (1998), 47-78.

$\bullet$ WWW

65

conservation law に関して
http://takeno.iee.niit.ac.jp/~shige/math/conser.html

66

compensated compactness に関する文献
http://takeno.iee.niit.ac.jp/~shige/math/bib/bib.html

67

Conservation Laws Preprint Server (Norway)
http://www.math.ntnu.no/conservation/

68

Compactness Methods and Nonlinear Hyperbolic Conservation Laws
http://www.math.nwu.edu/~gqchen/preprints/cpmthd/

Next: この文書について... Up: compensated compactness と保存則方程式について Previous: 8.4 未解決問題

Shigeharu TAKENO
2001年 12月 17日