Abstract
Full Text
S. L. KAMENOMOSTSKAYA
ON A PROBLEM IN FILTRATION THEORY
(Presented by Academician I. G. Petrovskii, 9 III 1957)
In filtration theory, under an elastic-plastic regime \((^{1,2})\), the following problem arises. Find a function \(p\) that satisfies the equation \(\partial p/\partial t = a_1^2 \partial^2 p/\partial x^2\) for \(\partial p/\partial t \geqslant 0\), the equation \(\partial p/\partial t = a_2^2 \partial^2 p/\partial x^2\) for \(\partial p/\partial t < 0\), and the conditions:
\[ p(x,0)=p_0(x)\quad (0<x<\infty),\qquad \left.\frac{\partial p}{\partial x}\right|_{x=0}=0\quad (t>0). \tag{1} \]
Below a solution of this problem is given.
Instead of the two indicated equations for \(p\), we shall consider the single equation equivalent to them
\[ \alpha \frac{\partial p}{\partial t} +\beta \left|\frac{\partial p}{\partial t}\right| = \frac{\partial^2 p}{\partial x^2}, \tag{2} \]
where \(\alpha+\beta=1/a_1^2,\ \alpha-\beta=1/a_2^2\). We note that \(\alpha>0\) and \(|\beta|<\alpha\). In addition, we replace the conditions (1) for \(p\) by one initial condition
\[ p(x,0)=\varphi(x)\quad (-\infty<x<\infty), \tag{3} \]
where \(\varphi(x)\) is an even function of \(x\), equal to \(p_0(x)\) for \(x>0\). We shall study the problem (2), (3) in the strip \(R_T(-\infty<x<\infty,\ 0\leqslant t\leqslant T)\) of arbitrary fixed width \(T\).
Theorem 1. Let \(\varphi(x)\) be bounded on the whole line and have three generalized (in the sense of Sobolev \((^3)\)) derivatives belonging to \(L_2(-\infty,\infty)\). Then there exists a continuous solution \(p(x,t)\), bounded in \(R_T\), of equation (2) under condition (3), having continuous derivatives \(\partial p/\partial t,\ \partial p/\partial x,\ \partial^2p/\partial x^2\) from \(L_2(R_T)\) and the generalized derivative \(\partial^2p/\partial x\,\partial t \in L_2(R_T)\).
We give a brief proof. Consider the function \(v(x,t)=\partial p/\partial t\). For \(v(x,t)\) we have the equation
\[ \frac{\partial(\alpha v+\beta |v|)}{\partial t} = \frac{\partial^2 v}{\partial x^2} \tag{4} \]
and the initial condition
\[ v(x,0)=\psi(x), \tag{5} \]
where
\[ \psi(x)=\frac{\alpha\varphi''-\beta|\varphi''|}{\alpha^2-\beta^2}. \]
It is clear that \(\psi(x)\in L_2(-\infty,\infty)\) and that there exists a generalized derivative \(\psi'(x)\in L_2(-\infty,\infty)\). Introduce a quantity \(K\) such that
\[ \int_{-\infty}^{\infty}[\psi(x)]^2\,dx<K \]
and
\[ \int_{-\infty}^{\infty}[\psi'(x)]^2\,dx<K. \]
We shall find in the rectangle \(Q_l(|x|\le l,\; 0\le t\le T)\) a continuous function \(v(x,t)\) that has the generalized derivatives entering the equation, satisfies almost everywhere in \(Q_l\) equation (4) and the conditions
\[ v(l,t)=v(-l,t)=0,\qquad v(x,0)=\psi_1(x). \tag{6} \]
Here \(\psi_1(x)\) is continuous, has a first generalized derivative, \(\psi_1(x)=\psi(x)\) for \(|x|\le l-1\), and \(\psi_1(x)=0\) for \(|x|\ge l\). It is clear that \(\psi_1(x)\) can be chosen so that
\[ \int_{-l}^{l}[\psi_1(x)]^2\,dx<K_1 \quad\text{and}\quad \int_{-l}^{l}[\psi'_1(x)]^2\,dx<K_1, \]
where \(K_1\) does not depend on \(l\).
To construct the solution, we replace equation (4) in \(Q_l\) by the difference-differential equation, according to the Rothe method \((^4)\),
\[ L_h v_h=\alpha\,\frac{v_h^n-v_h^{n-1}}{h} +\beta\,\frac{|v_h^n|-|v_h^{n-1}|}{h} -\frac{d^2v_h^n}{dx^2}=0. \tag{7} \]
Put \(v_h^0=\psi_1(x)\). Then, successively on each line \(t_n=nh\), we determine \(v_h^n(x)\) as a solution of the ordinary differential equation (7) under the boundary conditions \(v_h^n(-l)=v_h^n(l)=0\). The existence of such a solution is proved, for example, in \((^5)\).
Let \(h\) run through the sequence \(h_i=\dfrac{T}{2^i}\). We shall find some estimates independent of \(h\).
Consider the expression
\[ h\sum_{n=1}^{N}\int_{-l}^{l} L_hv_h\,\frac{v_h^n-v_h^{n-1}}{h}\,dx = h(\alpha-|\beta|)\sum_{n=1}^{N}\int_{-l}^{l} \left(\frac{v_h^n-v_h^{n-1}}{h}\right)^2\,dx + \]
\[ +|\beta|\sum_{n=1}^{N}\int_{-l}^{l} \frac{\bigl[v_h^n-v_h^{n-1}+\operatorname{sign}\beta\,(|v_h^n|-|v_h^{n-1}|)\bigr](v_h^n-v_h^{n-1})}{h}\,dx - \]
\[ -\sum_{n=1}^{N}\int_{-l}^{l} \frac{d^2v_h^n}{dx^2}\,(v_h^n-v_h^{n-1})\,dx =0; \tag{8} \]
here \(N\le T/h\).
After integration by parts we obtain
\[ \frac12\int_{-l}^{l}\left(\frac{dv_h^N}{dx}\right)^2 dx +h(\alpha-|\beta|)\sum_{n=1}^{N}\int_{-l}^{l} \left(\frac{v_h^n-v_h^{n-1}}{h}\right)^2 dx < \frac12\int_{-l}^{l}[\psi'_1(x)]^2dx =\frac12K_1. \]
Since \(N\) is arbitrary, the last inequality means that
\[ h\sum_{n=1}^{T/h}\int_{-l}^{l}\left(\frac{dv_h^n}{dx}\right)^2 dx + h\sum_{n=1}^{T/h}\int_{-l}^{l} \left(\frac{v_h^n-v_h^{n-1}}{h}\right)^2 dx <C_0, \tag{9} \]
where \(C_0\) is a constant independent of \(h\) and \(l\).
Denote by \(\bar v_h(x,t)\) the function in \(Q_l\) which for \(t=nh\) is equal to \(v_h^n(x)\) and is linear in \(t\) for \((n-1)h<t<nh\). From (9) it is not difficult to obtain
\[ \left\|\frac{\partial^2\bar v_h}{\partial x^2}\right\|_{L_2(Q_l)} + \|\bar v_h\|_{W_2^{(1)}(Q_l)} <C_1, \tag{10} \]
where \(C_1\) does not depend on \(h\) and \(l\).
It follows from (10) that there exists a subsequence \(h_j\) such that \(\{\bar v_{h_j}\}\) converges strongly in \(L_2(Q_l)\) and weakly in \(W_2^{(1)}(Q_l)\) to \(v(x,t)\in W_2^{(1)}(Q_l)\), while \(\{\partial^2\bar v_{h_j}/\partial x^2\}\) converges weakly to \(\partial^2v/\partial x^2\in L_2(Q_l)\) \((^3)\).
It can be proved that \(v(x,t)\) almost everywhere satisfies equation (4), is continuous, and satisfies the initial and boundary conditions.
The solution of the Cauchy problem for equation (4) will be found as the limit of the solutions \(v_l(x,t)\) of the first boundary-value problem for an infinitely expanding sequence of rectangles \(Q_l\) \((l=1,2,\ldots)\).
Set
\[
p(x,t)=\int_0^t v(x,\tau)\,d\tau+\varphi(x).
\]
It can be proved that \(p(x,t)\) is continuous, bounded in \(R_T\), satisfies the equation, and has the derivatives indicated in the theorem.
Lemma. If \(p_1(x,t)\) and \(p_2(x,t)\) are two solutions of equation (2), with
\(p_1(x,0)=\varphi_1(x)\), \(p_2(x,0)=\varphi_2(x)\), and if there exist continuous derivatives
\(\partial p_i/\partial t\), \(\partial p_i/\partial x\), \(\partial^2 p_i/\partial x^2 \in L_2(R_T)\) and generalized derivatives
\(\partial^2 p_i/\partial x\partial t \in L_2(R_T)\) \((i=1,2)\), then the inequality
\[
\left\|\frac{\partial(p_1-p_2)}{\partial x}\right\|^2_{L_2(R_T)}
+
\left\|\frac{\partial(p_1-p_2)}{\partial t}\right\|^2_{L_2(R_T)}
<
C\int_{-\infty}^{\infty}(\varphi_1'-\varphi_2')^2\,dx,
\tag{11}
\]
holds, where \(C=\mathrm{const}\) is independent of \(\varphi_1\) and \(\varphi_2\).
Theorem 2. If \(\varphi(x)\) is bounded on the whole line and has a generalized derivative
\(\varphi'(x)\in L_2(-\infty,\infty)\), then there exists a unique continuous solution \(p(x,t)\), bounded in \(R_T\), of equation (2) under condition (3), having, for \(t>0\), continuous derivatives
\(\partial p/\partial t\), \(\partial p/\partial x\), and \(\partial^2 p/\partial x^2\).
For the proof one must consider a sequence of functions \(\varphi_k(x)\) with the following properties:
a) \(\varphi_k(x)\) has three generalized derivatives in \(L_2(-\infty,\infty)\);
b) \(|\varphi_k(x)|<M\), where \(M\) does not depend on \(k\);
c) \(\varphi_k'(x)\to\varphi'(x)\) in \(L_2(-\infty,\infty)\) as \(k\to\infty\), and for every \(x_0\),
\(\varphi_k(x_0)\to\varphi(x_0)\).
Let \(p_k(x,t)\) be the solution of equation (2), constructed in Theorem 1, under the condition \(p_k(x,0)=\varphi_k(x)\). Applying estimate (11) to any two functions of the sequence \(\{p_k(x,t)\}\), it is not difficult to obtain that \(\{p_k(x,t)\}\) converges strongly in \(W^{(1)}_2(\Omega)\) to \(p(x,t)\in W^{(1)}_2(\Omega)\) (here \(\Omega\) is an arbitrary bounded domain in \(R_T\)).
It is clear that \(p(x,t)\) satisfies equation (2). It can be proved that the function \(p(x,t)\) is continuous, bounded in \(R_T\), and has, for \(t>0\), continuous derivatives \(\partial p/\partial t\), \(\partial p/\partial x\), and \(\partial^2 p/\partial x^2\); for \(t=0\), \(p(x,0)=\varphi(x)\).
The uniqueness of the solution with the indicated properties can be proved analogously to the way this is proved for the heat-conduction equation (7).
Institute of Chemical Physics
Academy of Sciences of the USSR
Received
28 II 1957
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