Geodesic
First order quasilinear equations in several independent variables
Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 217-243

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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. 
@article{SM_1970_10_2_a5,
     author = {S. N. Kruzhkov},
     title = {First order quasilinear equations in several independent variables},
     journal = {Sbornik. Mathematics},
     pages = {217--243},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {1970},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_2_a5/}
}
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S. N. Kruzhkov. First order quasilinear equations in several independent variables. Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 217-243. http://geodesic.mathdoc.fr/item/SM_1970_10_2_a5/