Singular limit of solutions of $u_ t=\Delta u^ m-A\cdot \nabla (u^ q/q)$ as $q\to \infty$

Abstract:

We will show that the solutions of ${u_t} = \Delta {u^m} - A\nabla ({u^q}/q)$ in ${R^n} \times (0,T),T > 0,m > 1,u(x,0) = f(x) \in {L^1}({R^n}) \cap {L^\infty }({R^n})$ converge weakly in ${({L^\infty }(G))^ * }$ for any compact subset $G$ of ${R^n} \times (0,T)$ as $q \to \infty$ to the solution of the porous medium equation ${\upsilon _t} = \Delta {\upsilon ^m}$ in ${R^n} \times (0,T)$ with $\upsilon (x,0) = g(x)$ where $g \in {L^1}({R^n}),0 \leqslant g \leqslant 1$, satisfies $g(x) + {(g(x))_{{x_1}}} = f(x)\quad {\text {in}}\quad \mathcal {D}’\left ( {{R^n}} \right )$ for some function $\tilde {g}(x) \in {L^1}({R^n}),\quad \tilde {g}(x) \geqslant 0$ such that $g(x) = f(x),\quad \tilde {g}(x) = 0$ whenever $g(x) < 1$ a.e. $x \in {R^n}$. The convergence is uniform on compact subsets of ${R^n} \times (0,T)\quad {\text {if}}\quad f \in {C_0}({R^n})$.