Existence theorem of weak solutions of $u_t+f(u)u_x=0$

Question

Consider this PDE:

$\begin{cases}u_t+f(u)u_x=0\\ u(x,0)=\varphi(x)\end{cases}$

Has this PDE weak solutions whatever is $f$ or $\varphi$? I want to find an existence theorem and bibliography about that?

Can anyone help me?

Thanks in advance!

Answer 1

The existence and uniqueness of a generalized solution $u=u(t,x)$ to the Cauchy problem for such kind of equations and for the more general one $$ u_t + \sum_{i=1}^n \frac{\partial}{\partial x_i}\varphi_i(t,x,u)+\psi(t,x,u)=0 $$ have been proved more than fifty years ago by Kruzhkov in his paper [1]. Kruzhkov builds $u$ in the class of bounded measurable function by using the method of vanishing viscosity introduced earlier by H. Hopf: the differentiability conditions required for the functions $\varphi(t,x,\cdot)$, $i=1,\ldots, n$ and $\psi(t,x,\cdot)$ are mild.
Moreover, the first paragraph of the paper gives a nice historical introduction to this field of research, supported by the bibliography at its end.

Reference

[1] Stanislav Nikolaevich Kruzhkov, "First order quasilinear equations in several independent variables" (English, Russian original), Mathematics of the USSR, Sbornik, vol. 10 pp. 217-243 (1970), DOI:SM1970v010n02ABEH002156, MR0267257, Zbl 0215.16203.