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) no. 5, pp. 407-425.
@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},
     pages = {407--425},
     publisher = {Gauthier-Villars},
     volume = {7},
     number = {5},
     year = {1990},
     mrnumber = {1138530},
     zbl = {0726.35009},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1990__7_5_407_0/}
}
TY  - JOUR
AU  - Weinberger, H. F.
TI  - Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 1990
SP  - 407
EP  - 425
VL  - 7
IS  - 5
PB  - Gauthier-Villars
UR  - http://www.numdam.org/item/AIHPC_1990__7_5_407_0/
LA  - en
ID  - AIHPC_1990__7_5_407_0
ER  - 
%0 Journal Article
%A Weinberger, H. F.
%T Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity
%J Annales de l'I.H.P. Analyse non linéaire
%D 1990
%P 407-425
%V 7
%N 5
%I Gauthier-Villars
%U http://www.numdam.org/item/AIHPC_1990__7_5_407_0/
%G en
%F 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) no. 5, pp. 407-425. http://www.numdam.org/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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl