Abstract Generated abstract
The paper addresses nonuniqueness for nonlinear parabolic systems with discontinuous initial data, following a correction to an earlier two-equation example. It constructs a three-equation system in one spatial variable by comparing self-similar solutions with diffusion coefficients depending on the similarity variable and then extending these coefficients smoothly as functions of the dependent variables. The construction yields coefficients that remain positive and bounded, while the same initial data, all components equal to the sign function, admit two distinct solutions. Thus the note provides an explicit mechanism showing that uniqueness can fail for parabolic systems of this type.
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MATHEMATICS
S. K. GODUNOV
ON NONUNIQUENESS FOR PARABOLIC SYSTEMS
(Presented by Academician I. G. Petrovskii, February 26, 1962)
In the work \((^1)\) I attempted to show that a nonlinear parabolic system of the form
\[ \frac{\partial u_i}{\partial t} = \frac{\partial}{\partial x} \left[ \sum_k a_{ik}\frac{\partial u_k}{\partial x} \right]. \tag{1} \]
need not have a unique solution in the case of discontinuous initial data. S. N. Kruzhkov drew my attention to an error that I had made in my calculations. He also proved that for the system considered in \((^1)\),
\[ \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \left[ a(u,v)\frac{\partial u}{\partial x} \right], \]
\[ \frac{\partial v}{\partial t} = \frac{\partial^2 v}{\partial x^2}, \]
the solution is unique also for discontinuous initial data. Nevertheless, the assertion that a nonunique solution is possible for systems of type (1) is correct. We shall now show this for a system of three equations, based on the same idea as in \((^1)\).
First let us note that the solution of the equation
\[ \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \left[ K(\xi)\frac{\partial u}{\partial x} \right] \qquad \left( \xi=\frac{x}{\sqrt{t}} \right) \]
for
\[ K(\xi)= \begin{cases} 1, & \text{for } -\infty<\xi<-2,\\ f(\xi), & \text{for } -2\leq \xi\leq -1,\\ 1, & \text{for } -1<\xi<+\infty \end{cases} \]
with the initial condition \(u|_{t=0}=\operatorname{sign} x\) is written by the formulas
\[ u(\xi) = -1+2\int_{-\infty}^{\xi}\frac{z(\eta)}{2K(\eta)}\,d\eta; \]
\[ z(\xi)= \begin{cases} A_1 \exp\left[-\dfrac{\xi^2}{4}\right], & \text{for } \xi\leq -2,\\[1.2em] A_2 \exp\left[-\displaystyle\int_{-2}^{\xi}\frac{\eta\,d\eta}{2f(\eta)}\right], & \text{for } -2<\xi\leq -1,\\[1.2em] A_3 \exp\left[-\dfrac{\xi^2}{4}\right], & \text{for } -1<\xi. \end{cases} \]
Of the expressions for the constants \(A_1, A_2, A_3\) we shall give only one:
\[ A_3=\left\{\int_{-\infty}^{+\infty} K^{-1}(\xi)\exp\left[-\int_0^\xi \frac{\eta\,d\eta}{2K(\eta)}\right]d\xi\right\}^{-1}. \]
The solution of the equation
\[ \frac{\partial w}{\partial t}=\frac{\partial^2 w}{\partial x^2} \qquad \left(w\big|_{t=0}=\operatorname{sign}x\right) \]
is defined as follows:
\[ w(\xi)=-1+2\int_0^\xi \bar z(\eta)\,d\eta, \]
where
\[ \bar z(\xi)=\left[\int_{-\infty}^{+\infty}\exp\left[-\frac{\xi^2}{4}\right]d\xi\right]^{-1} \exp\left[-\frac{\xi^2}{4}\right]. \]
It is clear from these formulas that in the plane \(u,w\) the curve defined by the parametric equations
\[ u=u(\xi),\qquad w=w(\xi), \]
connects the points \((-1,-1)\), \((+1,+1)\) and contains two rectilinear segments. One of these segments corresponds to the interval \(-1<\xi<+\infty\), and on it
\[ \frac{du}{dw}:\frac{z(\xi)}{\bar z(\xi)} = A_3\int_{-\infty}^{+\infty}\exp\left[-\frac{\xi^2}{4}\right]d\xi = \operatorname{const}=B. \]
We shall now show that \(B<1\). Since
\[ B= \left\{ \int_{-\infty}^{0}\exp\left[-\frac{\xi^2}{4}\right]d\xi + \int_0^{+\infty}\exp\left[-\frac{\xi^2}{4}\right]d\xi \right\} \times \]
\[ \times \left\{ \int_{-\infty}^{0}\frac{1}{K(\xi)} \exp\left[-\int_0^\xi \frac{\eta\,d\eta}{2K(\eta)}\right]d\xi + \int_0^\infty \exp\left[-\frac{\xi^2}{4}\right]d\xi \right\}^{-1}, \]
it suffices to verify that
\[ \int_{-\infty}^{0}\exp\left[-\frac{\xi^2}{4}\right]d\xi < \int_{-\infty}^{0}\frac{1}{K(\xi)} \exp\left[-\int_0^\xi \frac{\eta\,d\eta}{2K(\eta)}\right]d\xi. \tag{2} \]
Denote
\[ s(\xi)=-\left[2\int_0^\xi \frac{\eta\,d\eta}{K(\eta)}\right]^{1/2}. \]
The inverse function satisfies the equation
\[ \xi(s)=-\left[2\int_0^s K|\xi(\gamma)|\,\gamma\,d\gamma\right]^{1/2}. \]
Since \(K(\xi)\le 1\) and \(K(\xi)\not\equiv 1\), we have \(\xi(s)/s\le 1\) and \(\xi(s)/s\not\equiv 1\). The integral appearing on the right-hand side of inequality (3) can be rewritten as
\[ \int_{-\infty}^{0}\frac{1}{K(\xi)} \exp\left[-\int_0^\xi \frac{\eta\,d\eta}{2K(\eta)}\right]d\xi = \int_{-\infty}^{0}\frac{s}{\xi}\exp\left[-\frac{s^2}{4}\right]ds > \]
\[ > \int_{-\infty}^{0}\exp\left[-\frac{s^2}{4}\right]ds = \int_{-\infty}^{0}\exp\left[-\frac{\xi^2}{4}\right]d\xi; \]
thus it is proved that \(B<1\).
It is seen from this that the curve \(u = u(\xi), w = w(\xi)\) can have common points with the line \(u=w\), distinct from \((+1,+1)\), only for \(\xi < -1\).
Defining the function \(v(\xi)\) as a solution of the equation
\[ \frac{dv}{dt}=\frac{\partial}{\partial x}\left[K(-\xi)\frac{\partial v}{\partial x}\right] \]
with initial data \(v|_{t=0}=\operatorname{sign} x\), we find that \(v(\xi)=-u(-\xi)\), and thereby establish that the curve \(v=v(\xi), w=w(\xi)\) can intersect the line \(v=w\) only at points with \(\xi>1\) and at the point \(v=w=-1\). Hence it is clear that in the space \((u,v,w)\) the curve \(u=u(\xi), v=v(\xi), w=w(\xi)\) intersects the line \(u=v=w\) only when \(u=v=w=\pm 1\). Let us also note that \(u=u(\xi), v=v(\xi), w=w(\xi)\) are strictly monotone functions.
Define along the curve \(u=u(\xi), v=v(\xi), w=w(\xi)\) the coefficients
\[ \begin{aligned} a(u,v,w)&=K(\xi),\\ b(u,v,w)&=K(-\xi),\\ c(u,v,w)&=1. \end{aligned} \]
On the line \(u=v=w\) put \(a=b=c=1\). After this we extend \(a,b,c\) to the whole space \((u,v,w)\). These extended \(a,b,c\) may be regarded as greater than \(1/2\), not exceeding \(1\), and sufficiently smooth. The possibility of a smooth extension is ensured by the smoothness of the curve \(u=u(\xi), v=v(\xi), w=w(\xi)\). This smoothness can be achieved by choosing a smooth \(f(\xi)\).
The system of equations
\[ \begin{aligned} \frac{\partial u}{\partial t}&=\frac{\partial}{\partial x}\left[a\frac{\partial u}{\partial x}\right],\\ \frac{\partial v}{\partial t}&=\frac{\partial}{\partial x}\left[b\frac{\partial v}{\partial x}\right],\\ \frac{\partial w}{\partial t}&=\frac{\partial}{\partial x}\left[c\frac{\partial w}{\partial x}\right] \end{aligned} \]
with initial data
\[ u|_{t=0}=v|_{t=0}=w|_{t=0}=\operatorname{sign} x \]
has the solutions:
\[ \begin{aligned} 1)\quad \begin{cases} u=u(\xi),\\ v=v(\xi),\\ w=w(\xi); \end{cases} \qquad 2)\quad \begin{cases} u=u(\xi),\\ v=u(\xi),\\ w=u(\xi). \end{cases} \end{aligned} \]
Received
15 II 1962
CITED LITERATURE
- S. K. Godunov, DAN, 136, No. 6, 1281 (1961).