O. A. OLEINIK and N. D. VVEDENSKAYA

SOLUTION OF THE CAUCHY PROBLEM AND THE BOUNDARY-VALUE PROBLEM FOR NONLINEAR EQUATIONS IN THE CLASS OF DISCONTINUOUS FUNCTIONS

(Presented by Academician I. G. Petrovskii, 18 X 1956)

Below we shall give a correct formulation of the Cauchy problem and of the boundary-value problem for the equation

\[ \partial u/\partial t+\partial \varphi(t,x,u(t,x))/\partial x+\psi(t,x,u(t,x))=0 \tag{1} \]

in a large domain with discontinuous initial and boundary conditions. We shall define the generalized solution of these problems analogously to the way this was done in \((^1)\). This definition, as will be shown, is equivalent to the definition of generalized solutions by means of introducing “vanishing viscosity,” i.e. as the limit, as the parameter \(\varepsilon\) tends to zero, of solutions of the corresponding problems for the parabolic equation

\[ \varepsilon\,\partial^{2}u/\partial x^{2}=\partial u/\partial t+\partial \varphi(t,x,u)/\partial x+\psi(t,x,u). \tag{2} \]

1. The Cauchy problem. Suppose that \(\varphi(t,x,u)\) and \(\psi(t,x,u)\) have continuous derivatives of the second order; \(\varphi''_{uu}\geqslant 0\); \(u_0(x)\) is a measurable function bounded for all \(x\).

A generalized solution of the Cauchy problem for equation (1) in the domain \(G\{0\leqslant t\leqslant T,\ -\infty<x<+\infty\}\) with the initial condition

\[ u(0,x)=u_0(x) \tag{3} \]

will mean a bounded measurable function \(u(t,x)\), if:

1) For every smooth finite function \(f(t,x)\), equal to zero for \(t\geqslant T\), the relation holds

\[ \iint_G [u\,\partial f/\partial t+\varphi(t,x,u)\,\partial f/\partial x-\psi(t,x,u)f]\,dx\,dt +\int_{-\infty}^{+\infty} f(0,x)u_0\,dx=0. \tag{4} \]

2) There exists a function \(K(t,x_1,x_2)\), continuous for \(0<t\leqslant T\) and \(-\infty<x_i<+\infty\) \((i=1,2)\), such that for all \((t,x_1),(t,x_2)\) in \(G\), for \(t>0\),

\[ [u(t,x_1)-u(t,x_2)]/(x_1-x_2)\leqslant K(t,x_1,x_2). \tag{5} \]

Suppose that \(\varphi'_u\) is bounded for all \((t,x)\subset G\) and bounded \(u\).

Theorem 1. The generalized solution of the Cauchy problem (1), (3) is unique.

Let \(u_1\) and \(u_2\) be two solutions of problem (1), (3).

We shall show that

\[ \iint_G F(u_1-u_2)\,dx\,dt=0 \]

for any smooth function \(F(t,x)\), equal to zero for \(0\leqslant t\leqslant \delta\) and outside some finite domain \(D\) (i.e. that \(u_1=u_2\) almost everywhere in \(G\)). Let \(u_1^h\) and \(u_2^h\) be the mean functions for \(u_1\) and \(u_2\) with averaging radius \(h\) and with a smooth averaging kernel, and let \(M=\max\{|u_1|,|u_2|\}\), \(A=\max|\varphi'_u|\) for \((t,x)\subset G\) and \(|u|<M\). Consider the solution, vanishing for \(t=T\), of the equation

\[ \partial f/\partial t+\Phi_h\,\partial f/\partial x-\Psi_\rho\,\omega_\rho f=F, \]

where

\[ \Phi_h=\int_0^1 \varphi'_u\bigl(t,x,u_1^h+\tau(u_2^h-u_1^h)\bigr)\,d\tau,\qquad \Psi_\rho=\int_0^1 \psi'_u\bigl(t,x,u_1^\rho+\tau(u_2^\rho-u_1^\rho)\bigr)\,d\tau; \]

\(\omega_\rho\) is a smooth finite function equal to zero for \(t\le \rho\) and to one for \(|x|\le d+AT\) and \(t\ge 2\rho\) (\(d\) is the greatest distance of the points of \(D\) from the origin); \(|\omega_\rho|\le 1\). It can be shown, using (5), that for fixed \(\rho\), \(\partial f/\partial x\) and \(\partial f/\partial t\) are bounded for \(t\ge \alpha>0\) uniformly in \(h\), while the variation of \(f\) as a function of \(x\) is uniformly bounded for all \(t\) and \(h\) \((^1)\). Using (4), we obtain

\[ \iint_G F(u_1-u_2)\,dx\,dt = \iint_G\bigl[(\Phi_h-\Phi)(u_1-u_2)\,\partial f/\partial x -(\Psi_\rho\omega_\rho-\Psi)(u_1-u_2)f\bigr]\,dx\,dt, \tag{6} \]

whence it easily follows that the right-hand side of (6) is arbitrarily small for small \(\rho\) and \(h\)

\[ \left( \Phi=\frac{\varphi(t,x,u_1)-\varphi(t,x,u_2)}{u_1-u_2}, \qquad \Psi=\frac{\psi(t,x,u_1)-\psi(t,x,u_2)}{u_1-u_2} \right). \]

Remark. In an analogous way one can prove the uniqueness of the generalized solution of problem (1), (3) under the assumption that \(K(t,x_1,x_2)\) becomes infinite for some set of values of \(t\), the closure of which has measure zero.

Assume further that the functions \(\varphi,\ \varphi'_x,\ \varphi''_{xu},\ \varphi''_{xx},\ \psi,\ \psi'_u\), and \(\psi'_x\) are bounded when \((t,x)\subset G\) and \(u\) is bounded, and that \(\varphi''_{uu}\ge c>0\) for bounded \(u\) and \(0\le t\le \beta\). Let there exist a constant \(M_1\) and a function \(C(v)\), \(C'(v)\ge 0\), such that

\[ \max |\varphi'_x+\psi|\le C(v), \quad\text{when }(t,x)\subset G\text{ and }|u|\le v, \qquad \int_m^{M_1}\frac{dv}{C(v)}\ge T, \]

where \(m=\sup |u_0(x)|\).

Theorem 2. A generalized solution of the Cauchy problem (1), (3) exists.

The proof of this theorem is obtained by the method of finite differences, as was done in \((^2)\), with the aid of Lax’s finite-difference scheme \((^3)\). Using the convergence of the solutions of the difference equations constructed by this scheme to the generalized solution of problem (1), (3), one can establish the following theorem.

Theorem 3. If

\[ \int_a^b |u_0(x)-\widetilde u_0(x)|\,dx\le \delta, \]

then for the generalized solutions \(u(t,x),\ \widetilde u(t,x)\) of problem (1), (3), corresponding to the initial conditions \(u_0,\ \widetilde u_0\), the inequality

\[ \int_{a+At}^{b-At}|u(t,x)-\widetilde u(t,x)|\,dx<N\delta \]

holds, where the constant \(N\) depends only on \(\sup\{|u|,|\widetilde u|\}\) and \(t\), \(a<b\).

We now consider the Cauchy problem for equation (2) with condition (3). By a generalized solution of problem (2), (3) in \(G\) we shall mean a bounded measurable function \(u_\varepsilon(t,x)\) satisfying, for every smooth finite function \(f\) equal to zero for \(t\ge T\), the relation

\[ -\iint_G \varepsilon\frac{\partial^2 f}{\partial x^2}u_\varepsilon\,dx\,dt = \iint_G\left[ \frac{\partial f}{\partial t}u_\varepsilon +\frac{\partial f}{\partial x}\varphi(t,x,u_\varepsilon) -f\psi(t,x,u_\varepsilon) \right]\,dx\,dt + \int_{-\infty}^{+\infty}fu_0\,dx. \]

The existence of a generalized solution of problem (2), (3) for any bounded measurable \(u_0\) is proved with the aid of Lax’s finite-difference scheme under the mesh relation \(\Delta x^2/\Delta t=2\varepsilon\). The solution \(u_\varepsilon\) obtained satis-

satisfies (5) with the function \(K(t,x_1,x_2)=K/t\), where \(K=\mathrm{const}\). It turns out that, for \(t>0\), \(u_\varepsilon\) has the continuous derivatives occurring in (2), and satisfies this equation. It can be shown that the generalized solution of problem (2), (3) is unique.

Theorem 4. As \(\varepsilon\to 0\), the generalized solutions \(u_\varepsilon\) of problem (2), (3) converge in \(L_1(G')\) to the generalized solution \(u\) of problem (1), (3), i.e., for any finite domain \(G'\subset G\)
\[ \iint_{G'} |u_\varepsilon-u|\,dx\,dt \to 0 \quad\text{as }\varepsilon\to 0 . \]

The proof of this theorem follows easily from the fact that the solutions of the difference equations constructed according to the Lax scheme, as \(\Delta t\to 0\) and \(\Delta x^2/\Delta t=2\varepsilon\), converge to the solution of problem (2), (3), while as \(\Delta t\to 0\) and \(\Delta x^2/\Delta t\to 0\) they converge to the solution of problem (1), (3). This theorem for smooth \(u_0\) and \(\psi=0\) was first proved in \({}^{(4)}\), and then in \({}^{(2)}\). Weak convergence of smooth \(u_\varepsilon\) to \(u\), for sufficiently smooth \(u_0\) and \(\psi\equiv 0\), was obtained by another method in \({}^{(5)}\).

Boundary-value problem. Consider in the rectangle \(R\{0\le t\le T,\ 0\le x\le 1\}\) the boundary-value problem for equation (1) with the conditions:
\[ u(0,x)=u_0(x),\qquad u(t,0)=u_1(t),\qquad u(t,1)=u_2(t), \tag{7} \]
where \(u_0,u_1,u_2\) are bounded measurable functions and
\(\varphi'_u(t,0,u_1(t))>0,\ \varphi'_u(t,1,u_2(t))<0\) for \(0\le t\le T\).

A bounded measurable function \(u(t,x)\) will be called a generalized solution of the boundary-value problem (1), (7) in \(R\) if:

a) For every continuously differentiable function \(f\) in \(R\), equal to zero for \(t=T\), the equality
\[ \iint_R \left[ u\frac{\partial f}{\partial t} +\varphi(t,x,u)\frac{\partial f}{\partial x} -\psi(t,x,u)f \right]\,dx\,dt + \int_0^1 u_0 f(0,x)\,dx +\int_0^T u_2 f(t,1)\,dt -\int_0^T u_1 f(t,0)\,dt =0. \]

b) There exists a function \(K(t,x_1,x_2)\), continuous for \(0<t\le T,\ 0<x_1<1,\ 0<x_2<1\), such that for any points \((t,x_1)\) and \((t,x_2)\) in \(R\), (5) is satisfied.

Analogously to the way Theorem 1 was proved, the following theorem is proved.

Theorem 5. The generalized solution of the boundary-value problem (1), (7) is unique in the following classes of functions:

I) in the class of functions satisfying condition b) with functions \(K(t,x_1,x_2)\) continuous for \(0<t\le T\) and \(0\le x_i\le 1\);

II) in the class of functions \(u(t,x)\) for each of which there exists \(\alpha(t)\) \((\alpha(t)>0\) for \(t>0)\) such that \(\varphi'_u(t,x,u(t,x))>0\) for \(0<t\le T,\ 0\le x\le \alpha(t)\), and \(\varphi'_u(t,x,u(t,x))<0\) for \(0<t\le T,\ 1-\alpha(t)\le x\le 1\).

Theorem 6. Let the functions \(u_0,u_1,u_2\) be such that
\[ \varphi(t,0,\bar u_1(t))-\varphi(t,0,w_1(t))>0,\qquad \varphi(t,1,\bar u_2(t))-\varphi(t,1,w_2(t))>0 \]
for \(0\le t\le T\), where
\[ \bar u_1(t)=\inf_{\tau\le t} u_1(\tau),\qquad \bar u_2(t)=\sup_{\tau\le t} u_2(\tau); \]
\[ w_1(t)=\inf_{\tau\le t}\{u_0,u_1(\tau),u_2(\tau)\}-t\,C(v(t)); \]
\[ w_2(t)=\sup_{\tau\le t}\{u_0,u_1(\tau),u_2(\tau)\}+t\,C(v(t)); \qquad C(v)\ge \max_{R,\ |u|<v}\{|\varphi'_u|+|\psi|\}; \]
\(v(t)\) is the solution of the equation \(dv/dt=C(v)\) with the condition
\[ v(0)=\sup\{|u_0|,|u_1|,|u_2|\}. \]
Then there exists a generalized solution of problem (1), (7), satisfying II). If \(u_1\) and \(u_2\) satisfy the Lipschitz condition, then I) also holds for the solution.

This theorem is proved by the method of finite differences. For \(\psi \equiv 0\) it can be proved analogously to what was done in \((^1)\) for the Cauchy problem. To this end, suppose that any point of \(R\) can be connected in a unique way with a point of \(\Gamma\) by the projection of a characteristic of equation (1) (\(\Gamma\) are the sides of \(R\): \(x=0,\ x=1,\ t=0\)). Let \(U(t,t_1,x_1,\eta,\xi)\) be the function determined from the equation \(dx/dt=\varphi'_u(t,x,U)\), where \(x=x(t;t_1,x_1,\eta,\xi)\) is the projection of the characteristic of equation (1) passing through the points \((t_1,x_1)\) and \((\eta,\xi)\). Let \((t,x)\subset R,\ P\subset\Gamma\), and

\[ J(t,x,P)=\int_0^s [u_0(\xi)-U(0,t,x,0,\xi)]\,d\xi, \qquad \text{when } P=(0,s),\ 0\le s\le 1; \]

\[ J(t,x,P)=-\int_0^\sigma [\varphi(\eta,0,u_1(\eta))-\varphi(\eta,0,U(\eta,t,x,\eta,0))]\,d\eta, \]

\[ \qquad \text{when } P=(\sigma,0),\ 0<\sigma<t; \]

\[ J(t,x,P)=\int_0^1 [u_0(\xi)-U(0,t,x,0,\xi)]\,d\xi -\int_0^\sigma [\varphi(\eta,1,u_2(\eta))-\varphi(\eta,1,U(\eta,t,x,\eta,1))]\,d\eta, \]

\[ \qquad \text{when } P=(\sigma,1),\ 0<\sigma<t. \]

The function \(J\) has properties analogous to those established for \(J(t,x,s)\) in (6). Let the point \(P=(\eta(t,x),\xi(t,x))\) be the point of \(\Gamma\) at which \(J\), as a function of \(P\), assumes its least value. For \(\psi\equiv 0\), the solution of problem (1), (7) under the assumptions of Theorem 6 is given by the formula

\[ u(t,x)=U(t,t,x,\eta(t,x),\xi(t,x)). \]

This solution has the properties established for the solution of the Cauchy problem in (6).

Under some additional assumptions on \(\varphi,\psi,u_0,u_1,u_2\), in (7) the existence of a solution of the first boundary-value problem (2), (7) has been proved. One can construct generalized solutions of problem (2), (7), analogously to how this was done above for problem (2), (3), without the assumption of continuity of \(u_0,u_1,u_2\).

Theorem 7. The solutions of problem (2), (7), as \(\varepsilon\) tends to zero, converge to the generalized solution of problem (1), (7) in the sense of \(L_1(R)\), if \(u_0,u_1,u_2\) satisfy the conditions of Theorem 6.

For \(\psi\equiv 0\), the question of convergence of the solutions \(u_\varepsilon\) of problem (2), (7) as \(\varepsilon\to 0\) has been studied in detail in \((^4)\).

Mathematical Institute
named after V. A. Steklov
Academy of Sciences of the USSR

Received
18 X 1956

REFERENCES

\(^1\) O. A. Oleinik, DAN, 109, No. 6 (1956).
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\(^4\) O. A. Oleinik, Uspekhi Mat. Nauk, 10, issue 3 (65), 229 (1955).
\(^5\) O. A. Ladyzhenskaya, Reports at the All-Union Mathematical Congress, 1956.
\(^6\) O. A. Oleinik, Tr. Moscow Math. Soc., 5, 433 (1956).
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