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!



Existence theorem of weak solutions of $u_t+f(u)u_x=0$
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](https://doi.org/10.1070/SM1970v010n02ABEH002156), [MR0267257](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0267257), [Zbl 0215.16203](https://zbmath.org/0215.16203).


