First order quasilinear equations in several independent variables

S. N. Kruzhkov

Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 217-243

Résumé

In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations $$ u_t+\sum_{i=1}^n\frac d{dx_i}\varphi_i(t,x,u)+\psi(t,x,u)=0 $$ in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the “vanishing viscosity method”; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in $L_1$ of the solution of this problem which do not depend on small viscosity. Bibliography: 22 titles.

Bibliographie

[1] E. Hopf, “The partial differential equation $u_t+u u_x=\mu u_{xx}$”, Comm. Pure Appl. Math., 3:3 (1950), 201–230 | DOI | MR | Zbl

[2] O. A. Oleinik, “Razryvnye resheniya nelineinykh differentsialnykh uravnenii”, Uspekhi matem. nauk, 12:3 (1957), 3–73 | MR

[3] L. D. Landau, E. M. Livshits, Mekhanika sploshnykh sred, Gostekhizdat, Moskva, 1953

[4] O. A. Oleinik, “O zadache Koshi dlya nelineinykh uravnenii v klasse razryvnykh funktsii”, DAN SSSR, 95:3 (1954), 451–455 | MR

[5] A. N. Tikhonov, A. A. Samarskii, “O razryvnykh resheniyakh kvazilineinogo uravneniya pervogo poryadka”, DAN SSSR, 99:1 (1954), 27–30 | Zbl

[6] P. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation”, Comm. Pure Appl. Math., 7:1 (1964), 159–193 | DOI | MR

[7] S. N. Kruzhkov, “K metodam postroeniya obobschennykh reshenii zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Uspekhi matem. nauk, 20:6 (1965), 112–118 | MR

[8] O. A. Oleinik, “O edinstvennosti i ustoichivosti obobschennogo resheniya zadachi Koshi dlya kvazilineinogo uravneniya”, Uspekhi matem. nauk, 14:2 (1959), 165–170 | MR

[9] I. M. Gelfand, “Nekotorye zadachi teorii kvazilineinykh uravnenii”, Uspekhi matem. nauk, 14:2 (1969), 87–158

[10] A. S. Kalashnikov, “Postroenie obobschennykh reshenii kvazilineinykh uravnenii pervogo poryadka bez usloviya vypuklosti kak predelov reshenii parabolicheskikh uravnenii s malym parametrom”, DAN SSSR, 127:1 (1959), 27–30 | Zbl

[11] Wu Zhuo-qun, “On the existence and uniqueness of the generalised solutions of the Cauchy problem for quasi-linear equations of first order without convexity conditions”, Chinese Math., 4 (1964), 561–577 | MR

[12] E. Conway, J. Smoller, “Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables”, Comm. Pure Appl. Math., 19:1 (1966), 95–105 | MR | Zbl

[13] A. I. Volpert, “Prostranstva $BV$ i kvazilineinye uravneniya”, Matem. sb., 73(115) (1967), 255–302 | MR

[14] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Gostekhizdat, Moskva, 1957 | MR

[15] S. N. Kruzhkov, “Rezultaty o kharaktere nepreryvnosti reshenii parabolicheskikh uravnenii i nekotorye ikh primeneniya”, Matem. zametki, 6:1 (1969), 97–108 | Zbl

[16] N. N. Kuznetsov, “O slabom reshenii zadachi Koshi dlya mnogomernogo kvazilineinogo uravneniya”, Matem. zametki, 2:4 (1967), 401–410 | Zbl

[17] S. N. Kruzhkov, “Obobschennye resheniya zadachi Koshi v tselom dlya nelineinykh uravnenii pervogo poryadka”, DAN SSSR, 187:1 (1969), 29–32 | Zbl

[18] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, LGU, Leningrad, 1950

[19] A. Fridman, Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, Moskva, 1968

[20] O. A. Oleinik, S. N. Kruzhkov, “Kvazilineinye parabolicheskie uravneniya vtorogo poryadka so mnogimi nezavisimymi peremennymi”, Uspekhi matem. nauk, 16:5 (1961), 115–155 | MR | Zbl

[21] S. D. Eidelman, Parabolicheskie sistemy, Nauka, Moskva, 1964 | MR

[22] S. K. Godunov, “Problema obobschennogo resheniya v teorii kvazilineinykh uravnenii i v gazovoi dinamike”, Uspekhi matem. nauk, 17:3 (1962), 147–158 | MR | Zbl