L. I. Kamynin

ON A PROBLEM IN HYDRAULIC ENGINEERING

(Presented by Academician S. L. Sobolev on 25 VII 1961)

The note considers the solution of a mixed problem for a parabolic equation with coefficients that have a discontinuity along a moving and initially unknown line. Problems of this kind arise in the practice of hydraulic engineering, for example (see (¹)) in the injection of binding solutions into rock formations to increase the strength and water impermeability of the foundations of hydraulic structures. If injection is carried out simultaneously from a number of wells located on one straight line at equal and sufficiently small distances from one another, then, in order to determine the pressure \(u_1(x,t)\) of the injected liquid and the pressure \(u_2(x,t)\) of the displaced liquid, moving in a porous medium without mixing (where \(x=h(t)\) is the line of their interface), one may formulate the following mathematical problem, which in what follows we shall call Verigin’s problem.

It is required to find three functions \(u_1(x,t)\), \(u_2(x,t)\), and \(h(t)\) satisfying the system of equations of parabolic type

\[ \frac{\partial u_1}{\partial t} = \frac{\partial}{\partial x} \left( a_1(x,t)\frac{\partial u_1}{\partial x} \right), \qquad 0<x<h(t); \]

\[ \frac{\partial u_2}{\partial t} = \frac{\partial}{\partial x} \left( a_2(x,t)\frac{\partial u_2}{\partial x} \right), \qquad h(t)<x<l;\; 0<t<T; \tag{1} \]

the initial conditions

\[ u_1(x,0)=\psi_1(x), \qquad 0\le x\le h(0)=c; \]

\[ u_2(x,0)=\psi_2(x), \qquad c\le x\le l; \tag{2} \]

the boundary conditions

\[ \frac{\partial u_1(0,t)}{\partial x}=\varphi_1(t); \]

\[ \frac{\partial u_2(l,t)}{\partial x}=\varphi_2(t), \qquad 0\le t\le T; \tag{3} \]

the conjugation conditions on the unknown line of discontinuity of the coefficients \(x=h(t)\)

\[ u_1(h(t),t)=u_2(h(t),t); \]

\[ a_1(h(t),t)\frac{\partial u_1}{\partial x}(h(t),t) = a_2(h(t),t)\frac{\partial u_2}{\partial x}(h(t),t), \qquad 0\le t\le T, \tag{4} \]

where, moreover, for the line of discontinuity the differential equation must hold

\[ \frac{dh(t)}{dt} = -\beta(h(t),t)\frac{\partial u_1}{\partial x}(h(t),t), \tag{5} \]

where

\[ \beta(x,t) = B\, \frac{\alpha_1(x,t)a_2(x,t)-a_2(x,t)a_1(x,t)} {a_2(x,t)} \]

\[ (B>0 \text{ is a constant}). \]

In addition, the compatibility conditions are satisfied

\[ \begin{gathered} \psi'_1(0)=\varphi_1(0);\\ \psi'_2(l)=\varphi_2(l);\\ \psi_1(c)=\psi_2(c);\\ \alpha_1(c,0)\psi'_1(c)=\alpha_2(c,0)\psi'_2(c). \end{gathered} \tag{6} \]

With regard to the coefficients of equation (1), the initial functions (2), and the boundary functions (3)—(5), we shall henceforth assume that the following conditions are fulfilled:

I. The functions \(a_i(x,t)\) are continuous in the closed domain
\[ \overline{G}_T=\{(x,t);\;0\leq x\leq l,\;0\leq t\leq T\} \]
together with the derivatives \(\partial a_i(x,t)/\partial x\), \(\partial a_i(x,t)/\partial t\), and \(\partial^2 a_i(x,t)/\partial x^2\) \((i=1,2)\), and \(a_i(x,t)\), \(\partial a_i(x,t)/\partial x\), and \(\partial a_i(x,t)/\partial t\) satisfy a Hölder condition in \(x\) and \(t\) with nonzero exponent; moreover,
\[ 0<a_i\leq a_i(x,t)\leq A_i, \]
where \(a_i,A_i\) \((i=1,2)\) are constants.

II. The functions \(\varphi_i(t)\) \((i=1,2)\) satisfy a Hölder condition in \(t\) with nonzero exponent; the functions \(\psi_i(x)\) \((i=1,2)\) are continuous together with \(\psi'_i(x)\), and \(\psi'_i(x)\) satisfy a Hölder condition in \(x\) with nonzero exponent; moreover, \(\varphi_i(t)<0\) and \(\psi'_i(x)<0\) \((i=1,2)\).

III. In the domain \(\overline{G}_T\) the functions \(a_i(x,t)\) \((i=1,2)\) satisfy a Hölder condition in \(x\) and \(t\) with nonzero exponent, and
\[ \sqrt{a_1(x,t)}\,a_2(x,t)+\sqrt{a_2(x,t)}\,a_1(x,t)\neq 0, \]
\(\alpha_i(x,t)>0\) \((i=1,2)\), \(A\geq \beta(x,t)\geq a>0\), where \(A\) and \(a\) are constants.

IV. The condition \(0<c<l\) is fulfilled.

When conditions I—IV are fulfilled, the existence of a solution of the Verigin problem (1)—(6) is proved. We note that, in the case of constant \(a_i(x,t)\) and \(\alpha_i(x,t)\), the Verigin problem (1)—(6) was considered by N. N. Verigin \((^1)\) and L. I. Rubinshtein \((^2)\). We outline the scheme of the proof.

Assuming that \(h(t)\) is a monotonically nondecreasing function, by integrating equations (1) over the domains
\[ \overline{G}^{(1)}_T=\{(x,t),\;0\leq x\leq h(t),\;0\leq t\leq T\} \]
and
\[ \overline{G}^{(2)}_T=\{(x,t),\;h(t)\leq x\leq l;\;0\leq t\leq T\}, \]
respectively, and by using the initial and boundary conditions (2)—(4), one can obtain an integral equation for determining \(h(t)\) (cf., in this connection, the method of the authors who dealt with the solution of the Stefan problem \((^{6-9})\)):

\[ \begin{aligned} h(t)=Sh(t)\equiv c+B\Bigg\{& \int_{0}^{h(t)} u_1(x,t;h)\,dx+ \int_{h(t)}^{l} u_2(x,t;h)\,dx\\ &+\int_{0}^{t}\left[a_1(0,\tau)\varphi_1(\tau)-a_2(l,\tau)\varphi_2(\tau)\right]\,d\tau -\int_{0}^{c}\psi_1(x)\,dx-\int_{c}^{l}\psi_2(x)\,dx \Bigg\}. \end{aligned} \tag{7} \]

Consider the set \(\overline{M}\) obtained as the closure, in the metric of \(C\), of the set of continuously differentiable functions \(h(t)\) \((0\leq t\leq T)\) for which \(h(0)=c\) and \(0\leq dh(t)/dt\leq L\). On the set \(\overline{M}\) we consider the operator \(Sh\), \(g=Sh\), defined by means of the right-hand side of (7), where \(u_i(x,t;h)\) \((i=1,2)\) is the solution of the auxiliary problem (1)—(4), (6) with the given function \(h(t)\) from \(\overline{M}\). As was proved by the author in \((^3)\), the problem (1)—(4), (6) has a solution \(u_i(x,t;h)\) \((i=1,2)\), continuous together with
\[ \frac{\partial u_i}{\partial x}(x,t;h) \]
on the closure of \(\overline{G}^{(i)}_T\), provided conditions I—IV are fulfilled, if only \(h(t)\) satisfies a Hölder condition in \(t\) with exponent \(>1/2\), which holds in our case, since \(h(t)\in\overline{M}\) satisfies a Hölder condition

with exponent 1. The following lemmas are needed for the proof of the existence theorem.

Lemma 1. If \(u_i(x,t,h)\) is a solution of the auxiliary problem (1)—(4), (6), then, under conditions I—IV, there exists a constant \(D\), independent of the choice of \(h(t)\) from \(M\), such that in \(\bar G_T^{(i)}\)

\[ -D \leqslant \frac{\partial u_i}{\partial x}(x,t;h) \leqslant 0. \]

Lemma 2. If \(h(t)\) is a monotonically nonincreasing function satisfying a Hölder condition in \(t\) with exponent 1, then \(g(t)=Sh(t)\) will be a differentiable monotonically nonincreasing function for which

\[ g(0)=c,\qquad 0 \leqslant \frac{dg(t)}{dt} \leqslant D\max_{\bar G_T}\beta(x,t)=L. \]

Using the theorem on the continuous dependence of the solution of problem (1)—(4), (6) on the boundary, contained in our paper \((^4)\), one proves

Lemma 3. The operator \(Sh\) is continuous in the norm

\[ \|g\|=\max_{0\leqslant t\leqslant T}|g(t)|+ \sup_{0\leqslant t_1,t_2\leqslant T} \frac{|g(t_1)-g(t_2)|}{|t_1-t_2|} \]

on the set \(\overline M\).

Using the results of our paper \((^3)\) and of the paper of Gevrey \((^5)\), one proves

Lemma 4. There exists at least one fixed point of the mapping \(g=Sh\) of the set \(\overline M\) into itself.

Finally, with the aid of Lemma 4 and the integral equation (7), the main theorem is proved:

Theorem. Under conditions I—IV, the Verigin problem (1)—(6) has at least one solution \(\{u_1(x,t;h);\ u_2(x,t;h);\ h(t)\}\), and both

\[ u_i(x,t;h),\quad \text{as well as}\quad \frac{\partial u_i}{\partial x}(x,t;h) \]

satisfy a Hölder condition in \(x\) and \(t\) in \(\bar G_T^{(i)}\) with a nonzero exponent.

Moscow State University
named after M. V. Lomonosov

Received
8 VII 1961

CITED LITERATURE

\(^1\) N. N. Verigin, Izv. AN SSSR, OTN, No. 5, 674 (1952).
\(^2\) L. I. Rubinshtein, DAN, 113, No. 1, 50 (1957).
\(^3\) L. I. Kamynin, DAN, 139, No. 5 (1961).
\(^4\) L. I. Kamynin, DAN, 140, No. 6 (1961).
\(^5\) M. Gevrey, J. Math. pures et appl., 9, 305 (1913).
\(^6\) I. I. Kolodner, Comm. Pure and Appl. Math., 9, 1 (1956).
\(^7\) W. L. Miranker, Quart. Appl. Math., 16, 121 (1958).
\(^8\) G. W. Evans, Quart. Appl. Math., 9, 185 (1951).
\(^9\) W. T. Kyner, J. Math. and Mech., 8, No. 4, 483 (1959).