Linear transport equations with discontinuous coefficients. (English) Zbl 0992.35104

From the introduction: We consider the one-dimensional homogeneous linear transport equation \[ u_t(t,x)+ a(t,x)u_x=0, \quad (t,x)\in \Omega:= [0,T]\times \mathbb{R},\tag{1} \] with initial condition \[ u(0,x)= u^0(x),\tag{2} \] where the velocity \(a\) is bounded and satisfies the one-sided Lipschitz condition \[ \bigl(a(t,x) -a(t,y)\bigr) (x-y)\geq-m (t)(x-y)^2,\quad m\in L^1 [0,T]. \] Here the coefficient \(a\) is possibly discontinuous and the theory of R. DiPerna and P. Lions cannot be applied.
Our main result is to introduce an entropy condition which selects a unique solution from the set of continuous weak solutions to (1), (2). This entropy solution has certain minimality properties and can also be described by using the Filippov flow for the characteristic equation. When \(u_0\in \text{Lip}1\), our entropy condition is equivalent to the ones introduced by Bouchut and James in the case of Lipschitz initial conditions.


35R05 PDEs with low regular coefficients and/or low regular data
35L65 Hyperbolic conservation laws
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