Kinetic Formulation for a Parabolic Conservation Law. Application to Homogenization
Anne-Laure Dalibard · SIAM Journal on Mathematical Analysis · 2007
We derive a kinetic formulation for the parabolic scalar conservation law $\partial_t u + \mathrm{div}_y A(y,u) - \Delta_y u=0$. This allows us to define a weaker notion of solutions in $L^1$, which is enough to recover the $L^1$ contraction principle. We also apply this kinetic formulation to a homogenization problem studied in a previous paper; namely, we prove that the kinetic solution $u^{\varepsilon}$ of $\partial_t u^{\varepsilon} + \mathrm{div}_x A\left({x}/{\varepsilon}, u^{\varepsilon} \right)- \varepsilon\Delta_x u^{\varepsilon}=0$ behaves in $L^1_{\text{loc}}$ as $v\left( {x}/{\varepsilon}, \bar{u}(t,x)\right)$, where v is the solution of a cell problem and $\bar{u}$ the solution of the homogenized problem.