Continuity of the density of a gas flow in a porous medium
Continuity of the density of a gas flow in a porous medium
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多孔介质中气流密度的连续性
DOI：
10.1090/s0002-9947-1979-0534112-2
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发表时间：
1979
影响因子：
1.3
通讯作者：
A. Friedman
中科院分区：
文献类型：
--
作者：
L. Caffarelli;A. Friedman
多孔介质中的气体方程是一个简并的非线性抛物线方程。众所周知，存在唯一的广义解。本文证明了广义解是连续的。 0. 简介。多孔介质中气体的密度 u(x, t) 对于 x G R"、t > 0 满足方程 du/dt = Aum (m > 1) (0.1)，且初始条件 m(x, 0) = m0(x)。 (0.2) 这里 u0(x) > 0 且 u(x, t) > 0。方程 (0.1) 是一个非线性抛物线方程，在 u 处退化= 0. (0.1), (0.2) 的解的概念是在某种弱意义上进行的（在第 1 节中进行了精确定义）。 (0.3) 当 n = 1 时，该结果是已知的；参见 [9]、[10]、[1] 和 [5]。 3 我们建立了 t > 0 的初步估计，对于 / = 0，在 §5 中给出。主要结果是令 m0(x) 为 R" 中定义的函数，并满足： 0 < m0(x) < TV (TV < oo)，(1.1) f (u0(x))2 dx < oo，(1.2) u0(x) 在 R" 中连续，并且在 m0 > 0 的每个紧凑集中一致保持连续。编辑于 1978 年 8 月 7 日接收。AMS (MOS) 主题分类 (1970) '这项工作部分得到国家科学基金会拨款 74 06375 A01 和 MC 575-21416 AOL © 1979 美国数学会的支持。 0002-9947/79/0000-035 3/$04.75 99 许可或版权限制可能适用于再分发；参见 http://www.ams.org/journal-terms-of-use 100 L. A. CAFFARELLI 和 AVNER FRIEDMAN 我们考虑 R" X (0, oo), (1.4) 中的柯西问题 du/dt = Aum m(x, 0) = Mq(x) inR", (1.5) 其中 m 是固定数，m > 1。通过 (1.4), (1.5) 的解，我们指的是函数 u(x, t)，对于任何 T < oo, fT f \(u(x, t))2 + \Vxum(x, t)\2] dxdt<<x> (1.6) J0 JR»L和 /oX("í " V*"m ' Vx/) dX dt + hUo(x)Áx) ** = ° (1-7) 对于任何连续可微函数 / 在 R" X [0, T) 中具有紧凑支持 我们回想一下 [11]，在条件 (1.1)、(1.2) 下，可以给出解决方案的其他概念，它允许 x = oo 处的不同衰减条件。 (1.2) 本文的结果不受这些解的其他概念的影响，解 u(x, t) 可以作为方程 (1.4) 的解 m,(x, t) (tj|0) 的极限来获得，初始条件为 R" 中的 u(x, 0) = m0(x) + 7]； (1.8) 参见[11]。请注意，(1.4)、(1.8) 的解 uv 是在经典意义上获取的，如果 tj < 17'，则 uv < u^，并且 R" X (0, 00) 中的 tj < uv(x, t) < TV + 7}。 (1.9) 我们定义两点 (x1, f1), (x2, t2) 之间的抛物线距离： d((x\t>),(x2,t2))=\x>-x2\+\tl-t2\X/2。当我们谈到函数 v(x, t) 的连续性模时，我们总是指两点之间的距离为抛物线距离：<oe(/-) = qiogr|"e (0<6<2/h), (1.10) <b(r) = C2-cllos'l'/2 (1.11) 其中C > 0，c > 0。本文的主要结果由以下定理说明。定理 1.1 (i) (1.4)、(1.5) 的解 u 在 R" X [0, 00) 中连续； (ii) 对于任何 80 > 0，um 在 R" X [8Q, 00) 中具有连续模 ae(r)，对于任何 0 < e < 2/n，如果 n > 3，并且在 R" X [50, 00) 中具有连续模 <b(r) if « = 2. 许可或版权限制可能适用于再分发；参见 http://www.ams.org/journal-terms-of-use 气体流密度的连续性 101 t > 0 的连续性证明和 (ii) 的证明在第 4 节中给出，从证明中可以看出，这部分定理不需要条件 (1.3)。 §5 中给出了 §§2 和 3 中所需的一些估计。 2. 在本节和§3 中，我们获得了 (1.4)、(1.8) 的解 ^(x, t) 的各种辅助结果。为了简单起见，我们将用 u(x, t) 表示该解；我们还将获得 0 < tj < 1。 tj. 我们设置 M = TV + 1，因此，通过 (1.9)，0 < m(x, t) < M。 (2.1) 引理 2.1 以下不等式成立： '£■>-"-7, (2-2) at m — I dum m _ ,„ _,, t-r>-r Um。 (2.3) at m — I 该结果是由 Aronson 和Benilan [4]. 由于证明很短，我们在这里简单地给出函数 w = t(du/dt) 满足 dw/dt = mA(um~lw) + AmTM. 函数 z = — u/(m — 1) 满足相同的方程，并且 z(x, 0) < 0 = w(x, 0)（如果 m0(x) 是光滑的，则 w 在 t = 0 处连续）。如果 u0(x) 是平滑的，则可得出 (2.2)；对于一般的 m0，可立即从 (2.2) 得出不等式。对于任何 x° G R"，t° > 250，我们引入集合 Cr,h{x°, t°) = {(x, t); \x-x°\<r,t°-h<t< t°}，Br(x°)={x;\x-x°\<r}，其中 h < 80。用 \Br(x°)\ 表示 Br(x°) 的体积。设 C0 = m/ (ml)80 (2.4) 并令 h0 = h0(S0) 为任意正数，满足：e2C°*°<f, h0<80。 (2.5) 引理 2.2。对于任何 x° G Rn, t° > 250, X>0, r > 0, 0 <h < h0(80)，以下为真： if -i-— f um(x, t°h)dx>X (2.6) \Br(x°)\ Jb,^ 并且如果 h > Mynr2/X， (2.7) 许可或版权限制可能适用于再分发；参见 http://www.ams.org/journal-terms-of-use 102 L. A. CAFFARELLI 和 AVNER FRIEDMAN 然后 um(x°, t°) > i X; (2.8) 这里 yn 是一个正数，仅取决于维度 n。证明。根据 (2.3), (2.4)，dum/dt > (m/(m \)t)um > C0um 如果 t° h0 <t<t°（因为 t > 280 h0 > 80）。因此 Mm(x°, t°) > e-co<'-'°+A>M'"(x0, t) > e-c^">um(x0, t)。 (2.9) 再次，根据(2.3)、(2.4)，函数 1 w(t) =-— f um(x, t) dx |Rr(x°)|V°) 满足 <p'(t) > (m/(m l)t)tp(t) > -C0tp(0 使得 <p(t) > e-c»('-'°+*V('o h)> e-c^(t0 h)> e~c^X, (2.10) 其中 (2.6) 用于最后一个不等式，假设 n > 3 并令 C7(P) = p2'" r2'" ((n 2)/2)r-"(r2 p2), p = |x x°|。 (2.11) 请注意，G(r) = 0，G'(r) = 0 因为如果 p，则 G'(p) < 0。 < r, G(p) 在 Br(x°) 中为正。根据格林公式，um(x°, t)=\y( GAum dx + —i— f um(x, t) dx (2.12) JBr{x°) \Br(x°)\ JBr(X») 其中 y„ 是仅取决于 n 的正常数。现在假设断言 (2.8) 不成立。则 (2.9) 给出um(x°, t) < i Xec^o 将这个和 (2.10) 代入 (2.12) 并使用 (2.5) 的第一个不等式，我们得到 Ï A < i y„ f GAum dx.JBr{x°) 将此不等式关于 t, t° — h < t < t° 积分，我们得到 Xh<y„[' [ G(p)Aum(x, t) dx dt Jt°-h JBr(x°)
The equation of gas in a porous medium is a degenerate nonlinear parabolic equation. It is known that a unique generalized solution exists. In this paper it is proved that the generalized solution is continuous. 0. Introduction. The density u(x, t) of gas in a porous medium satisfies the equation du/dt = Aum (m > 1) (0.1) for x G R", t > 0, and an initial condition m(x, 0) = m0(x). (0.2) Here u0(x) > 0 and u(x, t) > 0. The equation (0.1) is a nonlinear parabolic equation, degenerating at the points where u = 0. The concept of a solution of (0.1), (0.2) is taken in some weak sense (to be defined precisely in §1). The purpose of this paper is to prove that m(x, t) is continuous. (0.3) This result is known for n = 1; see [9], [10], [1] and [5]. In § 1 we state this result more precisely, giving also a uniform modulus of continuity. In §§2 and 3 we establish preliminary estimates. The proof of (0.3) for t > 0 is given in §4 and, for / = 0, in §5. 1. The main results. Let m0(x) be a function defined in R" and satisfying: 0 < m0(x) < TV (TV < oo), (1.1) f (u0(x))2 dx < oo, (1.2) u0(x) is continuous in R", and uniformly Holder continuous in every compact set where m0 > 0. Received by the editors August 7, 1978. AMS (MOS) subject classifications (1970). Primary 35K55. 'This work is partially supported by National Science Foundation Grants 74 06375 A01 and MC 575-21416 AOL © 1979 American Mathematical Society 0002-9947/79/0000-035 3/$04.75 99 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 100 L. A. CAFFARELLI AND AVNER FRIEDMAN We consider the Cauchy problem du/dt = Aum in R" X (0, oo), (1.4) m(x, 0) = Mq(x) inR", (1.5) where m is a fixed number, m > 1. By a solution of (1.4), (1.5) we mean a function u(x, t) such that, for any T < oo, fT f \(u(x, t))2 + \Vxum(x, t)\2] dxdt<<x> (1.6) J0 JR»L and /oX("í " V*"m ' Vx/) dX dt + hUo(x)Áx) ** = ° (1-7) for any continuously differentiable function / with compact support in R" X [0, T). We recall [11] that under the conditions (1.1), (1.2), there exists a unique solution. Other concepts of a solution can be given which allow for a different decay condition at x = oo than in (1.2). The results of this paper are not affected by working with these other concepts of a solution. The solution u(x, t) can be obtained as a limit of solutions m,(x, t) (tj|0) of the equation (1.4) with the initial condition u(x, 0) = m0(x) + 7] in R"; (1.8) see [11]. Notice that the solution uv of (1.4), (1.8) is taken in the classical sense, uv < u^ if tj < 17', and tj < uv(x, t) < TV + 7} in R" X (0, 00). (1.9) We define the parabolic distance between two points (x1, f1), (x2, t2) by d((x\t>),(x2,t2))=\x>-x2\+\tl-t2\X/2. When we shall speak of a modulus of continuity of a function v(x, t), we shall always mean the distance between two points to be the parabolic distance. We now introduce two moduli of continuity: <oe(/-) = qiogr|"e (0<6<2/h), (1.10) <b(r) = C2-cllos'l'/2 (1.11) where C > 0, c > 0. The main result of the paper is stated in the following theorem. Theorem 1.1 (i) The solution u of (1.4), (1.5) is continuous in R" X [0, 00); (ii) For any 80 > 0, um has a modulus of continuity ae(r) in R" X [8Q, 00) for any 0 < e < 2/n, if n > 3, and a modulus of continuity <b(r) in R" X [50, 00) if « = 2. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CONTINUITY OF THE DENSITY OF A GAS FLOW 101 The proof of continuity for t > 0 and the proof of (ii) are given in §4. It will become obvious from the proof that the condition (1.3) is not required for this part of the theorem. The proof of continuity for t = 0 is given in §5. §§2 and 3 develop some estimates needed in §4. 2. Preliminary lemmas. In this section and in §3 we obtain various auxiliary results for the solution ^(x, t) of (1.4), (1.8). For simplicity we shall denote this solution by u(x, t); we also take 0 < tj < 1. All the estimates which we shall obtain, and all the constants will be independent of tj. We set M = TV + 1, so that, by (1.9), 0 < m(x, t) < M. (2.1) Lemma 2.1. The following inequalities hold: '£■>-"-7, (2-2) at m — I dum m _ ,„ _,, t-r>-r Um. (2.3) at m — I This result is due to Aronson and Benilan [4]. Since the proof is short, we briefly give it here. The function w = t(du/dt) satisfies dw/dt = mA(um~lw) + AmTM. The function z = — u/(m — 1) satisfies the same equation, and z(x, 0) < 0 = w(x, 0) (w is continuous at t = 0 if m0(x) is smooth). By comparison, then, (2.2) follows if u0(x) is smooth; for general m0, one uses approximation. The inequality (2.3) follows immediately from (2.2). Let ô0 be a fixed positive number. For any x° G R", t° > 250, we introduce the sets Cr,h{x°, t°) = {(x, t); \x-x°\<r,t°-h<t< t°}, Br(x°)={x;\x-x°\<r}, where h < 80. Denote the volume of Br(x°) by \Br(x°)\. Set C0 = m/ (m l)80 (2.4) and let h0 = h0(S0) be any positive number satisfying: e2C°*°<f, h0<80. (2.5) Lemma 2.2. For any x° G Rn, t° > 250, X>0, r > 0, 0 <h < h0(80), the following is true: if -i-— f um(x, t°h)dx>X (2.6) \Br(x°)\ Jb,^ and if h > Mynr2/X, (2.7) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 102 L. A. CAFFARELLI AND AVNER FRIEDMAN then um(x°, t°) > i X; (2.8) here yn is a positive number depending only on the dimension n. Proof. By (2.3), (2.4), dum/dt > (m/(m \)t)um > C0um if t° h0 <t<t° (since t > 280 h0 > 80). Hence Mm(x°, t°) > e-co<'-'°+A>M'"(x0, t) > e-c^">um(x0, t). (2.9) Again, by (2.3), (2.4), the function 1 w(t) =-— f um(x, t) dx |Rr(x°)|V°) satisfies <p'(t) > (m/(m l)t)tp(t) > -C0tp(0 so that <p(t) > e-c»('-'°+*V('o h)> e-c^(t0 h)> e~c^X, (2.10) where (2.6) was used in the last inequality. Suppose n > 3 and let C7(P) = p2'" r2'" ((n 2)/2)r-"(r2 p2), p = |x x°|. (2.11) Notice that G(r) = 0, G'(r) = 0 Since G'(p) < 0 if p < r, G(p) is positive in Br(x°). By Green's formula, um(x°, t)=\y( GAum dx + —i— f um(x, t) dx (2.12) JBr{x°) \Br(x°)\ JBr(X») where y„ is a positive constant depending only on n. Suppose now that the assertion (2.8) is not true. Then (2.9) gives um(x°, t) < i Xec^o. Substituting this and (2.10) into (2.12) and using the first inequality of (2.5), we obtain Ï A < i y„ f GAum dx. JBr{x°) Integrating this inequality with respect to t, t° — h < t < t°, we get Xh<y„[' [ G(p)Aum(x, t) dx dt Jt°-h JBr(x°)