On some analogy between different approaches to first order PDE’s with non smooth coefficients.(English)Zbl 0865.35032

The paper deals with the initial value problem for scalar first-order partial differential equation $\partial_tu+\text{div}(b(t,x)u)+ c(t,x)u=0\quad\text{for}\quad t\geq 0,\quad x\in\mathbb{R}^N,$
$u(0,x)=\psi(x)\in L^q(\mathbb{R}^N)\quad\text{with}\quad q\in[1,+\infty)$ or its strong form version, where the drift $$b(t,x)$$ is not Lipschitz continuous. The authors approach the problem in three different ways (entropy method, viscosity solution approach and regularization technique), and reach the conclusion that to assure the uniqueness of solutions each way leads to the same kind of seemingly natural assumptions on $$b(t,x)$$, which include that the symmetric part of its Jacobian matrix belongs to some $$L^p$$ space.

MSC:

 35F10 Initial value problems for linear first-order PDEs