MATHEMATICS

T. D. VENTSEL

ON SOME QUASILINEAR PARABOLIC SYSTEMS

(Presented by Academician I. G. Petrovskii on May 6, 1957)

1. In the present note, for systems of the form

\[ \frac{\partial^{2}u_i}{\partial x^{2}} = \frac{\partial u_i}{\partial t} + \sum_{j=1}^{N} b_{ij}(x,t,u_1,\ldots,u_N)\frac{\partial u_j}{\partial x} + \sum_{j=1}^{N} c_{ij}(x,t,u_1,\ldots,u_N)u_j + \]
\[ + f_i(x,t), \qquad i=1,\ldots,N, \tag{1} \]

the existence of a solution of the Cauchy problem and of the first boundary-value problem in the rectangle \(R_T\{x_1\le x\le x_2,\; 0\le t\le T\}\) is proved with boundary conditions

\[ u_i(x,0)=u_i^{(0)}(x), \qquad u_i(x_j,t)=u_i^{(j)}(t), \qquad i=1,\ldots,N,\quad j=1,2. \tag{2} \]

For the system

\[ \varepsilon \frac{\partial^{2}u}{\partial x^{2}} = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{\partial v}{\partial t}, \qquad 0= \frac{\partial v}{\partial t} + v\frac{\partial u}{\partial x} + u\frac{\partial v}{\partial x}, \tag{3} \]

which occurs in mechanics, the existence of a solution of the Cauchy problem and of the boundary-value problem in \(R_T\) is proved with conditions

\[ u(x,0)=u_0(x), \qquad u(x_1,t)=u(x_2,t)=0, \qquad v(x,0)=v_0(x). \tag{4} \]

1. Theorem 1. A solution of problem (1), (2) in \(R_T\), continuous in \(R_T\), whose derivatives occurring in the system are bounded in \(R_T\) and continuous inside \(R_T\), exists and is unique if the following assumptions are satisfied:

1) For \((x,t)\in R_T\) and for arbitrary \(u_k\),

\[ |b_{ij}(x,t,u_1,\ldots,u_N)|<B, \qquad \sum c_{ij}(x,t,u_1,\ldots,u_N)\lambda_i\lambda_j \ge c\sum \lambda_i^2, \]

where \(B,c\) are constants.

2) For \((x,t)\in R_T\) and

\[ \sum u_i^2 < M = \max\left\{ \max_{\substack{(x,t)\in R_T\\ j=0,1,2}} \sum (u_i^{(j)})^2 e^{2CT}, \frac{NF^2}{C}e^{2CT} \right\}, \tag{5} \]

where \(C>B^2N^2/2+2|c|+1\), \(F=\max |f_i(x,t)|\); the coefficients \(b_{ij}\), their first-order derivatives and second derivatives with respect to \(x\) and \(u_k\) are continuous; the derivatives

\[ \frac{\partial^{2}b_{ij}}{\partial x\,\partial t},\qquad \frac{\partial^{2}b_{ij}}{\partial u_k\,\partial t},\qquad \frac{\partial^{3}b_{ij}}{\partial x^{2}\partial t},\qquad \frac{\partial^{3}b_{ij}}{\partial x\,\partial t\,\partial u_k}, \]
\[ \frac{\partial^{3}b_{ij}}{\partial x\,\partial u_k\,\partial u_l},\qquad \frac{\partial^{3}b_{ij}}{\partial t\,\partial u_k\,\partial u_l},\qquad \frac{\partial^{3}b_{ij}}{\partial x^{2}\partial u_k},\qquad \frac{\partial^{3}b_{ij}}{\partial u_k\,\partial u_l\,\partial u_m} \tag{6} \]

are continuous inside \(R_T\). The functions \(c_{ij}\) and \(f_i\) possess the same smoothness.

3) The functions \(u_i^{(0)}(x)\) have second derivatives, bounded for \(x_1\le x\le x_2\) and continuous for \(x_1<x<x_2\); the functions \(u_i^{(1)}(t)\), \(u_i^{(2)}(t)\) have continuous first derivatives.

4) The compatibility conditions are satisfied

\[ u_i^{(0)}(x_j)=u_i^{(j)}(0),\qquad i=1,\ldots,N,\quad j=1,2. \tag{7} \]

Remark. The derivatives \(\partial u_i/\partial x\) of the solution obtained, for each fixed \(t\), have limits as \(x\to x_1\) and \(x\to x_2\).

Theorem 1 is proved by means of Rothe’s method \((^1)\) and estimates analogous to the estimates in \((^{2,3})\).

Using Theorem 1, one can prove the solvability of the Cauchy problem for the system (1).

Theorem 2. A solution \(u=\{u_1,\ldots,u_N\}\) of the system (1), satisfying the initial conditions \(u_i(x,0)=u_i^{(0)}(x)\), \(i=1,\ldots,N\), continuous in the strip \(S_T\{-\infty<x<\infty,\ 0\le t\le T\}\), whose derivatives entering the system are continuous inside \(S_T\), exists and is unique if the following assumptions are satisfied:

1) For \((x,t)\in S_T\), \(|f_i(x,t)|<F\).

2) For \((x,t)\in S_T\) and arbitrary \(u_k\), \(|b_{ij}(x,t,u)|<B\), \(\sum c_{ij}(x,t,u)\lambda_i\lambda_j\ge c\sum \lambda_i^2\); \(B,F,c\) are constants.

3) For \((x,t)\in S_T\) and

\[ \sum u_i^2\le M_0=\max\left\{\max\sum\bigl(u_i^{(0)}\bigr)^2 e^{2CT},\ \frac{NF^2}{C}e^{2CT}\right\}, \]

where \(C>N^2B^2/2+2|c|+1\), the coefficients \(b_{ij}, c_{ij}, f_i\), their first partial derivatives and second derivatives with respect to \(x\) and \(u_k\) are continuous and bounded, and the derivatives (6) and the same derivatives of \(c_{ij}\) and \(f_i\) are continuous inside \(S_T\).

4) The functions \(u_i^{(0)}(x)\) have two continuous and bounded derivatives.

The solution of the Cauchy problem is obtained as the limit of solutions of the first boundary-value problem in an expanding sequence of rectangles. In proving the existence of this limit, the estimates obtained for the solutions of the first boundary-value problem and their derivatives are used.

Applying Theorem 1, inequality (5), and estimates for the derivatives of solutions of the first boundary-value problem, one can prove the theorem on existence of a solution of problem (1), (2), for sufficiently small \(T\), for arbitrary systems of the form (1) with sufficiently smooth coefficients.

Theorem 3. Suppose the following assumptions are satisfied:

1) In the whole space \(\{u_1,\ldots,u_N\}\), except for a certain closed set \(S\) containing at least one point all of whose coordinates are finite, the coefficients \(b_{ij}, c_{ij}, f_i\) satisfy the smoothness conditions specified in item 2) of Theorem 1, and the inequality \(\sum c_{ij}\lambda_i\lambda_j\ge c\sum \lambda_i^2\) holds.

2) The functions (2) do not take values belonging to \(S\), and satisfy condition 3) of Theorem 1.

3) The compatibility conditions (7) are satisfied.

Then the solution of problem (1), (2), possessing all the properties indicated in Theorem 1 and satisfying the condition \(\rho(u,S)\ge \rho_1>0\), where \(\rho_1\) is a constant for which the inequality \(\rho_1<\rho(u_i^{(j)},S)=\rho_0\) is valid, exists and is unique for \(t\le \tau(\rho_1,\rho_0)\), where \(\tau\) is a nonincreasing function of the first argument and a nondecreasing function of the second argument.

Remark. Theorem 3 is also valid in the case when the set \(S\) contains no point all of whose coordinates are finite, under the corresponding definition of the distance from a point with finite coordinates to the set \(S\).

Theorem 4. Suppose all the conditions of Theorem 3 are satisfied and, in addition, suppose that for all solutions of (1), (2) the a priori estimate

\[ \rho(u,S)\ge a>0. \tag{8} \]

is valid.

Then the solution of problem (1), (2) exists in the rectangle \(R_T\) for any \(T\).

Proof. According to Theorem 3, the desired solution exists for \(t \leqslant \tau(a/2,a)\). Suppose the solution exists for \(t \leqslant \tau_1\). The solution can be continued, by virtue of Theorem 3, by solving the first boundary-value problem with the same boundary data and with initial data at \(t=\tau_1\). The height of the rectangle to which the solution can be continued, by virtue of estimate (8) and the estimates for the derivatives of the solutions \(u_i\) entering the system, does not depend on \(\tau_1\). Therefore the solution exists everywhere in \(R_T\).

1. Applying the theorems proved, one can prove the existence of a solution of the first boundary-value problem and of the Cauchy problem for the system

\[ \varepsilon \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} - \frac{\partial \varphi(v)}{\partial x}, \qquad \varepsilon \frac{\partial^2 v}{\partial x^2} = \frac{\partial v}{\partial t} - \frac{\partial u}{\partial x}, \tag{9} \]

where the function \(\varphi(v)\) is defined and four times continuously differentiable for \(v>v_0 \geqslant -\infty\); the conditions \(\varphi'(v)>0\), \(\varphi''(v)\leqslant 0\) are fulfilled, and, as \(v\to v_0\),

\[ F(v)=\int \sqrt{\varphi'(v)}\,dv \to -\infty . \]

Consider the first boundary-value problem for system (9); in doing so we shall assume that \(v(x,0)\), \(v(x_1,t)\), \(v(x_2,t)>v_0\).

According to Theorem 4, in order to prove the existence of a solution it is sufficient to establish that, if \(u,v\) is a solution of the stated boundary-value problem, then \(v\geqslant v_1>v_0\).

Since \(\varphi'(v)>0\), the first-order system obtained from system (9) for \(\varepsilon=0\) is hyperbolic. (For \(\varphi(v)=-1/2v^2\) this system describes the motion of “shallow water,” as well as one-dimensional isentropic motion of a gas for \(c_p/c_v=2\) in Lagrangian coordinates. System (3) for \(\varepsilon=0\) describes the same motions in Eulerian coordinates.)

The first-order system can be brought to the form

\[ \frac{\partial f^+}{\partial t} + \sqrt{\varphi'(v)}\, \frac{\partial f^+}{\partial x} =0, \qquad \frac{\partial f^-}{\partial t} - \sqrt{\varphi'(v)}\, \frac{\partial f^-}{\partial x} =0, \]

where \(f^+\) and \(f^-\) are new unknown functions (Riemann invariants):

\[ f^+=-F(v)+u,\qquad f^-=-F(v)-u. \]

If in system (9) one also passes from the functions \(u\) and \(v\) to the functions \(f^+\) and \(f^-\), it takes the form

\[ \varepsilon \frac{\partial^2 f^+}{\partial x^2} - \frac{\partial f^+}{\partial t} = \sqrt{\varphi'(v)}\, \frac{\partial f^+}{\partial x} + \varepsilon \frac{\partial^2 f^+}{\partial v^2} \left(\frac{\partial v}{\partial x}\right)^2; \]

\[ \varepsilon \frac{\partial^2 f^-}{\partial x^2} - \frac{\partial f^-}{\partial t} = -\sqrt{\varphi'(v)}\, \frac{\partial f^-}{\partial x} + \varepsilon \frac{\partial^2 f^-}{\partial v^2} \left(\frac{\partial v}{\partial x}\right)^2 . \tag{10} \]

Since \(\varphi''(v)\leqslant 0\), we have \(\partial^2 f^+/\partial v^2\geqslant 0\), \(\partial^2 f^-/\partial v^2\geqslant 0\), so that from equations (10) there follow the inequalities

\[ f^+ \leqslant M^+ = \max_{x=x_1,x_2,\; t=0} f^+, \qquad f^- \leqslant M^- = \max_{x=x_1,x_2,\; t=0} f^- . \tag{11} \]

We have

\[ F(v) = -\frac{f^+ + f^-}{2} \geqslant -\frac{M^+ + M^-}{2}, \tag{12} \]

and since \(F'(v)=\sqrt{\varphi'(v)}>0\), it follows from (12) that \(v\geqslant v_1\). The inequality \(v_1>v_0\) follows from the monotonicity of \(F(v)\) and from the fact that \(F(v_0)=-\infty\). The existence of a solution of the first boundary-value problem is proved.

Using solutions of the first boundary-value problem, one can construct a solution of the Cauchy problem for system (9).

Theorem 5. In the strip \(S_T\) there exists a unique solution of system (9), satisfying the conditions

\[ u(x,0)=u_0(x), \qquad v(x,0)=v_0(x) \]

and possessing all the properties indicated in Theorem 2, if the following assumptions are fulfilled:

1) The function \(\varphi(v)\) is defined and four times continuously differentiable for \(v>v_0\ge -\infty\), with \(\varphi'(v)>0\), \(\varphi''(v)\le 0\), \(F(v)\to -\infty\) as \(v\to v_0\).

2) \(v_0(x)\ge v_1>v_0\).

3) The functions \(u_0\) and \(v_0\) are twice continuously differentiable and bounded together with their first and second derivatives for \(-\infty<x<\infty\).

The indicated method for obtaining an a priori estimate for the solution of the first boundary-value problem is applicable to a whole series of systems of the form (1) with two spatial variables.

1. Consider problem (3), (4). The substitution

\[ w=\int_{x_1}^{x} v\,dx,\qquad z=\varepsilon u-w \]

brings system (3) into the form

\[ \varepsilon \frac{\partial^2 z}{\partial x^2} = \frac{\partial z}{\partial t} + \frac{w+z}{\varepsilon}\frac{\partial z}{\partial x}, \qquad 0= \frac{\partial w}{\partial t} + \frac{w+z}{\varepsilon}\frac{\partial w}{\partial x}. \tag{13} \]

The initial and boundary values for \(w\) and \(z\) are determined from conditions (4); for \(x=x_1,\ x=x_2\) the functions \(w\) and \(z\) are constant. Obviously,

\[ \max |z|\le \max |z_0|,\qquad \max |w|\le \max |w_0|. \]

It can be proved that the first boundary-value problem for system (13) has a solution for

\[ T<\frac{\varepsilon\ln 2}{4\max\bigl(|\partial w_0/\partial x|+|\partial z_0/\partial x|\bigr)}. \]

To prove the existence of a solution of the problem for arbitrary \(T\), it is necessary to establish a priori estimates for \(|\partial w/\partial x|=|w_x|\), \(|\partial z/\partial x|=|z_x|\). For \(x=x_1,\ x=x_2\) the function \(z_x\) is estimated in the same way as the analogous derivative in (2). Differentiating the second of equations (13) with respect to \(x\), we obtain

\[ 0= \frac{\partial w_x}{\partial t} + \frac{w+z}{\varepsilon}\frac{\partial w_x}{\partial x} + \frac{w_x+z_x}{\varepsilon}w_x, \]

whence it follows that at the point of maximum of \(|w_x|\) the estimate \(|w_x|\le \max |z_x|\) is valid. To estimate \(\max |z_x|\) in the case when this maximum is attained inside \(R_T\) or at \(t=T\), one makes the substitution

\[ z=\varphi(\bar z)=k\int_{0}^{\bar z} e^{-s^2/2}\,ds,\qquad k=\frac{2}{\sqrt{\pi}}\max |z|. \]

Then

\[ L\bar z_x = \frac{\partial^2\bar z_x}{\partial x^2} - \frac{\partial \bar z_x}{\partial t} - \frac{w+z}{\varepsilon}\frac{\partial \bar z_x}{\partial x} \]

\[ = - \left(\frac{\varphi''}{\varphi'}\right)'\bar z_x^{\,3} - 2\frac{\varphi''}{\varphi'}\frac{\partial\bar z_x}{\partial x}\bar z_x + \frac{w_x+\bar z_x\varphi'(\bar z)}{\varepsilon}\bar z_x . \]

At the point of maximum of \(|\bar z_x|\),

\[ \bar z_x L\bar z_x \ge \varepsilon \bar z_x^{\,4}-2\bar z_x^{\,3}k/\varepsilon, \]

whence \(|\bar z_x|<2k/\varepsilon^2\). From the boundedness of \(|\bar z_x|\) follows the boundedness of \(|z_x|\). The solution of the Cauchy problem for system (3) is obtained as the limit of the solutions of problems (3), (4) as \(x_1\to-\infty,\ x_2\to\infty\).

Received
6 V 1957

REFERENCES

1. E. Rothe, Math. Ann., 102, 650 (1930).
2. O. A. Oleinik, T. D. Venttsel, Matem. sborn., 41 (83), 105 (1957).
3. T. D. Venttsel, Matem. sborn., 41 (83), 4, 499 (1957).