Initial layer for the homogenization of a conservation law with vanishing viscosity (Q2642266)

In a previous paper [J. Math. Pures Appl. (9) 86, No. 2, 133--154 (2006; Zbl 1121.35014)] the author studied the asymptotic behaviour of the solutions, as \(\varepsilon\) goes to zero, of the problems (\(\varepsilon > 0\)) \[ \begin{aligned} \frac{\partial u}{\partial t}+ \operatorname{div} A \Big(\frac{x}{\varepsilon},u\Big)- \varepsilon \Delta u = f &\quad\text{in } {\mathbb R}^{N} \times [0,+\infty), \\ u(0,x) = u_0 \Big( x,\frac{x}{\varepsilon}\Big) &\quad\text{in } {\mathbb R}^{N}, \end{aligned} \] under the assumption that \(u_0(x,y) = v(y, \bar{u}_0(x))\) for some \(\bar{u}_0\) and \(v(y,p):= p + \widetilde{u}(y,p)\), where \(\widetilde{u}\) is the unique \(H^1\) and \(Y\)-periodic solution with \(\int_Y \widetilde{u} = 0\) (\(Y\) the unit cell) of \[ -\Delta_y \widetilde{u} + {\text{div}}_y\, A(y , p + \widetilde{u}) =0. \] Given \(\bar{u}\) the solution of the scalar convervation law \[ \frac{\partial \bar{u}}{\partial t} + \sum_{1=1}^N \frac{\partial \bar{A}_i}{\partial x_i} \Big(\frac{x}{\varepsilon}, \bar{u} \Big) = 0, \qquad \bar{u}(0,x) = \bar{u}_0(x) \] for suitable \(\bar{A}_i\), defined by a cell problem, the author proves that, as \(\varepsilon\) goes to \(0\), \[ u^{\varepsilon}(t,x) - v \big(x/\varepsilon, \bar{u}(t,x) \big) \to 0 \quad \text{in } L^2_{\text{loc}} \big( [0,+\infty) \times {\mathbb R}^N \big). \] Under suitable assumptions on the coefficients \(A_i\) and requiring on \(u_0\) the assumption \( v(y, \beta_1) \leqslant u_0(x,y) \leqslant v(y,\beta_2)\) for some \(\beta_1, \beta_2 \in {\mathbb R}\) she proves that, as \(\varepsilon\) goes to \(0\), \[ u^{\varepsilon}(t,x) - v \big(x/\varepsilon, \bar{u}(t,x) \big) \to 0 \quad \text{in } L^1((0,T) \times B_R) \] for every \(T, R > 0\).