论文标题
在诺伊曼边界条件下的奇异扩散
Singular Diffusion with Neumann boundary conditions
论文作者
论文摘要
在本文中，我们为非线性初始实物值问题开发了一种存在理论，具有单数扩散$ \ partial_t u = \ text {div}（k（x）\ nabla g（u））$，$ | _ | _ {t = 0} = u_0 = u_0 $ neumann边界条件$ k（x）\ nabla \ nabla g（u nabla g（u）这里$ x \ in B \ subset \ mathbb {r}^d $，一个有局部lipchitz边界的有界开放式集合，并用$ν$作为单位外部正常。函数$ g $是Lipschitz的连续且无折叠，而$ k（x）$是对角矩阵。我们表明，任何两个弱的熵解决方案$ u $和$ v $满足$ \ vert {u（t）-v（t）} \ vert_ {l^1（b）} \ le \ le \ le \ vert {u | _ | _ {t = 0} $ t \ ge 0 $，和一个常数$ c = c（k，g，b）$。如果我们仅限于$ k $的条目$ k_i $仅取决于相应的组件，即$ k_i = k_i（x_i）$，我们表明存在一个熵解决方案，因此在这种情况下确定问题是在哈达玛德的体验中得到很好的解决方案。
In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion $\partial_t u = \text{div}(k(x)\nabla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x)\nabla G(u)\cdot ν= 0$. Here $x\in B\subset \mathbb{R}^d$, a bounded open set with locally Lipchitz boundary, and with $ν$ as the unit outer normal. The function $G$ is Lipschitz continuous and nondecreasing, while $k(x)$ is diagonal matrix. We show that any two weak entropy solutions $u$ and $v$ satisfy $\Vert{u(t)-v(t)}\Vert_{L^1(B)}\le \Vert{u|_{t=0}-v|_{t=0}}\Vert_{L^1(B)}e^{Ct}$, for almost every $t\ge 0$, and a constant $C=C(k,G,B)$. If we restrict to the case when the entries $k_i$ of $k$ depend only on the corresponding component, $k_i=k_i(x_i)$, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.
