The Porous Medium Equation
The Porous Medium Equation
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DOI：
10.1093/acprof:oso/9780198569039.001.0001
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发表时间：
2006-10
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影响因子：
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通讯作者：
J. Vázquez
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文献类型：
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作者：
J. Vázquez
动力学我们已经得到了一个有趣的概念，将解看作是在无限维度量空间X（这里是函数空间L1（Ω））中移动的连续曲线。将解视为一般空间中的连续曲线是微分方程抽象理论的起点，这是我们将经常使用的方法。在所谓的抽象动力学中，通常会忘记符号中的变量x，而查看映射t7 → u（t）∈ X，其中u（t）是u（·，t）的缩写形式。备注。(1)注意，该定理允许定义极限解（特别是弱解）u在任何时间t > 0的值u（t）作为L1（Ω）的定义良好的元素。实际上，在许多情况下，如当Φ是超线性的且f是有界的时，它是L∞（Ω）的元素。(2)如果u 0和f是有界的，则初始正则性更好。在这种情况下，初始数据是在Lp意义下取的：对于每个p 0，在Lp（Ω）中的u（t）→ u（0）;如果u 0是连续的，则当t → 0时，在x上一致收敛，见7.5.1节。(3)不幸的是，对于非齐次数据g 6= 0的Dirichlet问题，没有等价的L1估计。我们用一个简单但非常有用的结论来结束这一小节。推论6.3设u是极限解，其中u_0 ∈ L_1（Ω），f ∈ L_1（Q）。如果t1 > 0，则<$0（x，t）= u（x，t + t1）是数据<$0（x）= u（x，t1）和强迫项f（x，t）= f（x，t + t1）的极限解。这个重要的结果对于近似是直接的。我们把细节留给读者。备注。让我们注意到，任何极限解的概念都取决于可接受的近似的类型和取极限的函数设置。我们提出的定义适用于L1设置。如果需要，这些解将被称为L1极限解。有关扩展，请参见第6.6节。6.2关于L1-范数的连续依赖性是一个强大的性质。它允许我们推广前一节弱解的存在性结果，
dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions as continuous curves in a general space is the starting point of the abstract theory of differential equations, a way that we will travel quite often. In the so-called Abstract Dynamics it is typical to forget the variable x in the notation and look at the map t 7→ u(t) ∈ X, where u(t) is the abbreviated form for u(·, t). Remarks. (1) Note that the theorem allows to define the value u(t) of a limit solution (in particular, of a weak solution) u at any time t > 0 as a well-defined element of L1(Ω). Actually, in many cases, as when Φ is superlinear and f is bounded, it is an element of L∞(Ω). (2) If u0 and f are bounded the initial regularity is better. In that case the initial data are taken in the Lp sense: ũ(t) → ũ(0) in Lp(Ω), for every p 0; if u0 is continuous, then the convergence takes place uniformly in x as t → 0, see Section 7.5.1. (3) Unfortunately, there are no equivalent L1 estimates for the Dirichlet Problem with nonhomogeneous data g 6= 0. We end this subsection with a simple but very useful consequence. Corollary 6.3 Let u be a limit solution with data u0 ∈ L1(Ω) and f ∈ L1(Q). If t1 > 0, then ũ(x, t) = u(x, t + t1) is the limit solution with data ũ0(x) = u(x, t1) and forcing term f(x, t) = f(x, t + t1). This important result is immediate for the approximations. We leave the details to the reader. Remark. Let us note that any concept of limit solution depends on the type of admissible approximations and on the functional setting in which limits are taken. The definition we propose applies in the L1 setting. If needed, these solutions will be called L1-limit solutions. For an extension see Section 6.6. 6.2 Theory of very weak solutions The continuous dependence with respect to the L1-norm is a powerful property. It has allowed us to extend the existence result for weak solutions of the preceding section and