Mathematics

E. S. SABININA

ON THE CAUCHY PROBLEM FOR THE EQUATION OF NONSTATIONARY GAS FILTRATION WITH MANY SPATIAL VARIABLES

(Presented by Academician I. G. Petrovskii, 16 VII 1960)

We shall consider the Cauchy problem in the domain \(G\{-\infty < x < \infty,\ 0 \le t \le T\}\) for the equation

\[ \partial u/\partial t = \Delta \varphi(u), \tag{1} \]

where \(\Delta\) is the Laplace operator; \(\varphi'(u)>0\) for \(0<u<\infty\); \(\varphi'(0)\ge 0\); \(\varphi(0)=0\); \(\varphi(u)\in C^{2+\alpha}\), with the condition

\[ u\big|_{t=0}=u_0(x),\qquad x=(x_1,\ldots,x_N),\qquad 0\le u_0(x)\le M,\quad \lim_{x\to\infty}u_0(x)=u_\infty . \tag{2} \]

Equation (1) describes nonstationary gas filtration in a multidimensional porous medium. In the case of one independent spatial variable this equation was studied in \((^{1,2})\).

In our work a definition of a generalized solution of problem (1)—(2) is introduced and its uniqueness is proved. Under the assumption that \(u_0(x)\ge \delta>0\), we construct a classical solution of problem (1)—(2), using two different methods of constructing a generalized solution and Nash a priori estimates. For \(u_0(x)\ge 0\) the generalized solution of problem (1)—(2) is obtained as the limit of classical solutions with initial function \(u_0(x)+\varepsilon\) as \(\varepsilon\to 0\).

1. Definition of a generalized solution and its uniqueness. A function \(u(x,t)\) bounded in \(G\) will be called a generalized solution of the Cauchy problem (1)—(2) if the generalized derivatives \(\partial\varphi(u)/\partial x_i\) exist and are square-summable in \(G\),

\[ \iint_G [u(x,t)-u_\infty]^2\,dx\,dt<\infty, \]

and for any continuously differentiable function \(f(x,t)\), equal to zero outside a finite region of the half-space \(t\le T\), the equality

\[ \iint_G \left[ u\,\frac{\partial f}{\partial t} -\sum_{i=1}^{N}\frac{\partial\varphi(u)}{\partial x_i}\, \frac{\partial f}{\partial x_i} \right]\,dx\,dt + \int_{-\infty}^{\infty}u_0(x)f(x,0)\,dx =0 \tag{3} \]

holds.

The uniqueness of the generalized solution of problem (1)—(2) is proved analogously to how this was done in \((^1)\).

2. The first method of constructing a generalized solution for \(u_0(x)\ge \delta>0\). We construct a generalized solution of the Cauchy problem (1)—(2) under the assumption that the generalized derivatives \(\partial\varphi(u_0)/\partial x_i\) and the difference \(\varphi(u_0)-\varphi(u_\infty)\) are square-summable.

The change of variables \(\varphi(u)=v,\ u=\Phi(v)\) transforms equation (1) into the equation

\[ \Phi'(v)\,\partial v/\partial t=\Delta v \]

with the initial condition \(v|_{t=0}=v_0(x)\), where \(v_0(x)=\varphi(u_0(x))\), \(v_\infty=\varphi(u_\infty)\),

Let \(v_0^h(x)\) be a sequence of functions satisfying the conditions:

1) \(v_0^h(x)\in C^{2+\alpha}\) in the space \(x\);

2) \(v_0^h(x)\) converges in the mean as \(h\to 0\) to \(v_0(x)\), together with the first-order derivatives, and \(v_0^h>\alpha>0\);

3) the integral of the square of the derivatives of the functions \(v_0^h(x)\) and \(|v_0^h(x)|\) is bounded by a constant \(C\) independent of \(h\).

Denote by \(v^h(x,t)\) the solution of the equation

\[ \Phi'(v(x,t-h))\,\frac{\partial v}{\partial t}=\Delta v \tag{4} \]

for \(t>0\), with the condition

\[ v\big|_{t=0}=v_0^h(x),\qquad \text{where } h>0 \text{ and } v(x,t)\equiv v_0^h(x)\quad \text{for } t<0. \tag{5} \]

Let \(G_m\) be the domain \(\{-m\le x_i\le m,\ i=1,\ldots,N,\ 0\le t\le T\}\), and let \(S_m^\tau\) be the domain \(\{-m\le x_i\le m,\ i=1,\ldots,N,\ t=\tau\}\), \(m=1,2,\ldots\). For fixed \(h>0\), we obtain the solution of problem (4)—(5) as the limit, as \(m\to\infty\), of the solutions \(v_m^h(x,t)\) of equation (4) in the domain with the condition

\[ v_m^h\big|_{t=0}=v_0^h(x)\quad \text{for } x\in S_{m-2}^0;\qquad v_m^h\big|_{t=0}=v_\infty\quad \text{for } x\in S_m^0\setminus S_{m-1}^0; \]

\[ v_m^h\big|_{|x_i|=m}=v_\infty,\quad i=1,\ldots,N;\qquad |v_m^h|_{t=0}|<c_2;\quad v_m^h\big|_{t=0}\in C^{2+\alpha} \]

with a constant independent of \(m\), and

\[ \int_{-\infty}^{\infty}\left(\frac{\partial v_m^h}{\partial x_i}\right)^2\,dx<c_1. \]

From Theorems 3 and 4 of the work \((^4)\) it follows that the functions \(v_m^h\) exist and belong to the class \(C^{2+\alpha}\) in \(G_m\), with a constant \(k\) independent of \(m\), since in Theorem 3 the norm of the solution in \(C^{2+\alpha}\) does not depend essentially on the dimensions of the domain. Hence it follows that, for fixed \(h\), the set \(\{v_m^h\}\) is compact in the sense of \(C^{2+\alpha}\) in any finite subdomain of \(G\), i.e., there exists a subsequence converging, as \(h\to 0\), to a function \(v^h\in C^{2+\alpha}\). It is obvious that \(v^h\) is a solution of problem (4)—(5).

By virtue of the maximum principle, \(|v^h|\le c_2\). Multiplying equation (4), written for the function \(v_m^h(x,t)\), by \(\partial v_m^h/\partial t\) and integrating the resulting equality over the domain \(G\), it is easy to prove the uniform boundedness of \(v^h-v_\infty\) in the norm \(W_2'(G)\).

Integrating by parts, we obtain

\[ \iint_G \Phi'(v_m^h(x,t-h)) \left(\frac{\partial v_m^h}{\partial t}\right)^2 \,dx\,dt + \frac12\int_{S^T} \sum_{i=1}^N \left(\frac{\partial v_m^h}{\partial x_i}\right)^2 \,dx = \]

\[ = \frac12\int_{S^0} \sum_{i=1}^N \left(\frac{\partial v_m^h}{\partial x_i}\right)^2 \,dx \le Nc_1. \]

Moreover,

\[ v^h(x,t)-v^h(x,0)=\int_0^t \frac{\partial v^h}{\partial t}\,dt, \]

\[ \int_{-\infty}^{\infty}(v^h(x,t)-v_\infty)^2\,dx \le 2T^2\iint\left(\frac{\partial v^h}{\partial t}\right)^2\,dx\,dt + \int_{-\infty}^{\infty}(v_0^h-v_\infty)^2\,dx \le c_3. \]

By the embedding theorems, the \(v^h\) are compact in \(L_2\) in any finite domain. The limiting function \(v\) has generalized derivatives with respect to \(x\) and to \(t\),

which are the weak limit of the corresponding derivatives of the functions \(v^h\), and therefore passage to the limit as \(h\to 0\) is legitimate in the integral identity

\[ \iint_G \left(\Phi'(v^h(x,t-h))\frac{\partial f^h}{\partial t}f +\sum_{i=1}^{N}\frac{\partial v^h}{\partial x_i}\frac{\partial f}{\partial x_i}\right)\,dx\,dt=0. \]

Since \(v=\varphi(u)\) and \(\Phi(v)=u\), (3) is valid for \(u\). It is obvious that \(u(x,t)\) satisfies all the requirements of a generalized solution of problem (1)—(2).

1. A second method for constructing a generalized solution of problem (1)—(2) when \(u_0(x)\geqslant \delta>0\). In order to prove that the constructed generalized solution is classical, we shall construct it in another way and show that it satisfies the Hölder condition for \(t>t_0>0\). Analogously to how this was done in Sec. 2, one can show that the solutions \(u_m^h\) in the domain \(G_m\) of the equation

\[ \frac{\partial u}{\partial t} = \sum_{i=1}^{N}\frac{\partial}{\partial x_i} \left[\varphi'(u(x,t-h))\frac{\partial u}{\partial x_i}\right] \tag{6} \]

with the conditions

\[ u_m^h\big|_{t=0}=\Phi(v_m^h)\big|_{t=0},\qquad u_m^h\big|_{|x_i|=m}=\Phi(v_m^h)\big|_{|x_i|=m} \]

are compact, for fixed \(h\), in the sense of the norm \(C^{2+\alpha}\) in any finite domain. The limiting function \(u^h\) is a solution of equation (6) with the condition \(u^h|_{t=0}=u_0^h(x)\).

By the maximum principle the \(u^h\) are uniformly bounded with respect to \(h\). Nash’s theorem \((^5)\) is applicable to the functions \(u^h(x)\). According to this theorem, all the functions \(u^h\) satisfy the Hölder condition for \(t>t_0>0\) (\(t_0\) is an arbitrary number) with a constant independent of \(h\). Therefore there exists a subsequence \(u^h\) converging, as \(h\to 0\), to a function \(u\) which also satisfies the Hölder condition for \(t>t_0\). The function \(u(x,t)\) is a generalized solution of problem (1)—(2). Indeed, for \(u_m^h\) the equality

\[ \frac12\iint_{G_m} \left(\frac{\partial(u_m^h-u_\infty)}{\partial t}\right)^2 \,dx\,dt - \iint_{G_m}\sum_{i=1}^{N} \frac{\partial}{\partial x_i} \left(\varphi'(u_m^h(x,t-h))\right) (u_m^h-u_\infty)\,dx\,dt=0, \]

\[ \frac12\int_{S_m^T}(u_m^h-u_\infty)^2\,dx + \iint_{G_m}\varphi'(u_m^h(x,t-h)) \sum_{i=1}^{N} \left(\frac{\partial u_m^h}{\partial x_i}\right)^2 \,dx\,dt \leqslant \]

\[ \leqslant \frac12\int_{-\infty}^{\infty}(u_m^h-u_\infty)^2\,dx \leqslant c_4. \]

From the estimate obtained it follows that passage to the limit as \(h\to 0\) is possible in the integral identity (3), where the function \(u^h(x,t)\) has been substituted for \(u(x,t)\). By virtue of the uniqueness theorem, the function \(u(x,t)\) coincides with the generalized solution constructed in Sec. 2.

1. Existence of a classical solution of problem (1)—(2). We shall show that \(u(x,t)\) is a classical solution of problem (1)—(2), i.e., it is continuous for \(t\geqslant 0\), has continuous derivatives for \(t>0\) appearing in equation (1), and satisfies this equation and condition (2) in the ordinary sense.

Theorem 1. A bounded classical solution of the Cauchy problem (1)—(2) exists if \(u_0(x)\geqslant \delta>0\), is continuous, and the generalized derivatives of the functions \(\varphi(u_0(x))\) and the difference \(\varphi(u_0(x))-\varphi(u_\infty)\) are square integrable.

Proof. The equation \(\Phi'(\varphi(u))\,\partial z/\partial t=\Delta z\), where \(u\) is a generalized solution of problem (1)—(2) satisfying the Hölder condition for \(t\ge t_0>0\), has, as is known, a classical solution \(z(x,t)\) satisfying the initial condition
\[ z\big|_{t=0}=\varphi(u_0(x)) \]
(see, for example, \({}^{4}\)). It is easy to show, similarly to how this was done in Sec. 2, that the function \(z-\varphi(u_\infty)\) is square-summable over the domain \(G\) together with all its first derivatives. The difference \(v^h-z\) satisfies the equation
\[ \Phi'(v^h(x,t-h))\frac{\partial(v^h-z)}{\partial t} = \Delta(v^h-z) + \frac{\partial z}{\partial t}\,\Phi''(\varphi(u)-v^h). \tag{7} \]

We multiply equation (9) by \(\alpha_m\,\partial(v^h-z)/\partial t\) and integrate over the domain \(G\). The function \(\alpha_m\) is infinitely differentiable, is equal to zero outside the cube \(|x_j|=m\), is equal to one inside the cube \(|x_j|=m-1\), everywhere \(0\le \alpha_m(x)\le 1\), and \(\alpha_m'(x)\) and \(\alpha_m'^2(x)/\alpha_m(x)\) are bounded uniformly in \(m\). Integrating (7) by parts, after an elementary transformation we obtain
\[ \begin{aligned} &\left(1-\frac{N\varepsilon_1}{2}-\frac{\varepsilon_2}{2}\right) \iint \Phi'(v^h(x,t-h)) \left(\frac{\partial(v^h-z)}{\partial t}\right)^2\,dx\,dt \\ &\quad +\frac12\int_{S T}\sum_{i=1}^{N} \left(\frac{\partial(v^h-z)}{\partial x_i}\right)^2 \alpha_m\,dx \le \frac12\int_{S^0}\sum_{i=1}^{N} \left(\frac{\partial(v^h-z)}{\partial x_i}\right)^2 \alpha_m\,dx \\ &\quad +\frac{1}{2\varepsilon_1} \iint\sum_{i=1}^{N} \left(\frac{\partial(v^h-z)}{\partial x_i}\right)^2 \frac{1}{\Phi'} \frac{(\partial\alpha_m/\partial x_i)^2}{\alpha_m}\,dx\,dt + \frac{1}{2\varepsilon_1} \iint \left(\frac{\partial z}{\partial t}\right)^2 \frac{\Phi''^2}{\Phi'} (\varphi(u)-v^h)^2\alpha_m\,dx\,dt . \tag{8} \end{aligned} \]

We choose the numbers \(\varepsilon_1\) and \(\varepsilon_2\) so that the coefficient of
\[ \iint \Phi'\left(\frac{\partial(v^h-z)}{\partial t}\right)^2\alpha_m\,dx\,dt \]
is positive. Since \(\partial(v^h-z)/\partial x_i\) is square-summable in \(G\) and the integral
\[ \iint\sum_{i=1}^{N} \left(\frac{\partial(v^h-z)}{\partial x_i}\right)^2 \frac{1}{\Phi'} \frac{(\partial\alpha_m/\partial x_i)^2}{\alpha_m}\,dx\,dt \]
differs from zero only for \(x\in S_m\setminus S_{m-1}\), this integral tends to zero as \(m\to\infty\). Passing to the limit as \(m\to\infty\) in (8), we obtain
\[ \begin{aligned} &\left(1-\frac{N\varepsilon_1}{2}-\frac{\varepsilon_2}{2}\right) \iint \Phi'(v^h) \left(\frac{\partial(v^h-z)}{\partial t}\right)^2\,dx\,dt + \frac12\int_{S^T}\sum_{i=1}^{N} \left(\frac{\partial(v^h-z)}{\partial x_i}\right)^2\,dx \\ &\le \frac12\int_{S^0}\sum_{i=1}^{N} \left(\frac{\partial(v^h-z)}{\partial x_i}\right)^2\,dx + \frac{1}{2\varepsilon_2} \iint \left(\frac{\partial z}{\partial t}\right)^2 \frac{\Phi''^2}{\Phi'} (\varphi(u)-v^h)^2\,dx\,dt . \end{aligned} \]

The integrals on the right-hand side tend to zero as \(h\to0\). The first of them tends to zero by virtue of the conditions imposed on \(v^h\big|_{t=0}\). The second tends to zero as \(h\to0\), since \(\partial z/\partial t\) is square-summable in \(G\) and \(v^h\) tends to \(\varphi(u)\) in the mean as \(h\to0\) in any finite domain.

From the estimates obtained it follows that \(v-z=\mathrm{const}\). Since \(v=z\) when \(t=0\), we have \(v\equiv z\), i.e. \(u=\Phi(v)\) is a classical solution of problem (1)—(2).

In the case where \(u_0(x)\ge0\), the solution of problem (1)—(2) can be obtained as the limit of solutions with positive initial data.

The author is deeply grateful to his scientific adviser O. A. Oleinik for substantial and repeated assistance.

Moscow State University
named after M. V. Lomonosov

Received
16 VII 1960

References

1. O. A. Oleinik, DAN, 113, No. 6 (1957).
2. A. S. Kalashnikov, DAN, 115, No. 5 (1957).
3. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1959.
4. A. Friedman, J. Math. and Mech., 7, No. 5 (1958).
5. J. Nash, Am. J. Math., 80, No. 4 (1958).