Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity
Weinberger, H. F.
Annales de l'I.H.P. Analyse non linéaire, Tome 7 (1990), p. 407-425 / Harvested from Numdam
Publié le : 1990-01-01
@article{AIHPC_1990__7_5_407_0,
     author = {Weinberger, H. F.},
     title = {Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {7},
     year = {1990},
     pages = {407-425},
     mrnumber = {1138530},
     zbl = {0726.35009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_1990__7_5_407_0}
}
Weinberger, H. F. Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity. Annales de l'I.H.P. Analyse non linéaire, Tome 7 (1990) pp. 407-425. http://gdmltest.u-ga.fr/item/AIHPC_1990__7_5_407_0/

[1] H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev., 43, 1915, pp. 163-170.

[2] P. Bauman and D. Phillips, Large-time behavior of solutions to certain quasilinear parabolic equations in several space dimensions, Am. Math. Soc., Proc., Vol. 96, 1986, pp. 237-240. | MR 818451 | Zbl 0611.35059

[3] P. Bauman and D. Phillips, Large-time behavior of solutions to a scalar conservation law in several space dimensions, Am. Math. Soc. Trans., Vol. 298, 1986, pp. 401-419. | MR 857450 | Zbl 0602.35074

[4] S.E. Buckley and M.C. Leverett, Mechanism of fluid displacement in sands, A.I.M.E., Vol. 146, 1942, pp. 107-116.

[5] J.M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Acad. Sci. Amsterdam, Vol. 43, 1940, pp. 2-12. | MR 1147 | Zbl 0061.45710

[6] J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., Ed. R.v. Mises and T.v. Karman, Vol. 1, 1948, pp. 171-199. | MR 27195

[7] J.D. Cole, On a quasi-linear prabolic equation occurring in aerodynamics, Quarterly Appl. Math., Vol. 9, 1951, pp. 225-236. | MR 42889 | Zbl 0043.09902

[8] A. Harten, J.M. Hyman, and P.D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math., Vol. 29, 1976, pp. 292-322. | MR 413526 | Zbl 0351.76070

[9] E. Hopf, The partial differential equation ut + uux = μ uxx, Comm. Pure Appl. Math., Vol. 3, 1950, pp. 201-230. | MR 47234 | Zbl 0039.10403

[10] A.M. Il'InandO.A. Oleinik,Behavior of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time, Dokl. Akad. Nauk S.S.S.R., Vol. 120, 1958, pp. 25-28; Am. Math. Soc. Trans., Vol. 42, 1964, pp. 19-23. | MR 101396 | Zbl 0082.08901

[11] A.M. Il'InandO.A. Oleinik,Asymptotic behavior of solutions of the Cauchy probem for some quasilinear equations for large values of time, Mat. Sbornik, Vol. 51 #2 (93), 1960, pp. 191-216. | MR 120469 | Zbl 0096.06601

[12] A.S. Kalashnikov, Construction of generalized solutions of quasilinear equations of first order without convexity conditions as limits of solutions of parabolic equations with small parameter, Dokl. Akad. Nauk S.S.S.R., Vol. 127, 1959, pp. 27-30. | MR 108651 | Zbl 0100.09203

[13] P.D. Lax, The initial value problem for nonlinear hyperbolic equations in two independent variables, Ann. Math. Studies 33, Princeton U. Press 1954, pp. 211-229. | MR 68093 | Zbl 0057.32502

[14] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., Vol. 10, 1957, pp. 537-566. | MR 93653 | Zbl 0081.08803

[15] T.-P. Liu, Invariants and asymptotic behavior of solutions of a conservation law, Am. Math. Soc. Proceedings, Vol. 71, 1978, pp. 227-231. | MR 500495 | Zbl 0392.35041

[16] O.A. Oleinik ,On Cauchy's problem for nonlinear equations in a class of discontinuous functions, Dokl. Akad. Nauk S.S.S.R., Vol. 95, 1954, pp. 451-455. | MR 64258 | Zbl 0058.32101

[17] O.A. Oleinik, Discontinuous solutions of differential equations, Uspekhi Mat. Nauk, 12 #3 (75), 1957, pp. 3-73. | MR 94541 | Zbl 0080.07701

[18] O.A. Oleinik, Construction of a generalized solution of the Cauchy problem for a quasilinear equation of first order by the introduction of "vanishing viscosity", Uspekhi Mat. Nauk, Vol. 14 #2 (86), 1959, pp. 159-164; Am. Math. Soc. Trans., Vol. 33, 1963, pp. 277-283. | MR 117426 | Zbl 0131.09101

[19] O.A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation, Uspekhi Mat. Nauk, Vol. 14 #2 (86), 1959, pp. 165-170; Am. Math. Soc. Trans., (2), 33, 1963, pp. 285-290. | MR 117408 | Zbl 0132.33303

[20] O.A. Oleinik and T.D. Ventsel', The first boundary value problem and the Cauchy problem for quasilinear equations of parabolic type, Matem. Sbornik, Vol. 41, 1957, pp. 105-128. | MR 86247

[21] D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, 1977.

[22] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations Prentice-Hall, Englewood Cliffs, N. J. 1967, Springer, New York, 1986. | MR 219861 | Zbl 0153.13602

[23] B. Keyfitz Quinn, Solutions with shocks: An example of an L1-contractive semi-group, Comm. Pure Appl. Math., Vol. 24, 1971, pp. 125-132. | MR 271545 | Zbl 0206.10401