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\begin{document}
\begin{center}
{\bf \Large
Note on the transformation that reduces the Burgers equation to
the heat equation.
}
\end{center}
\smallskip
We started to be interested in the question of
who first discovered the reduction
of the Burgers equation
to the heat equation (known as the Hopf-Cole transformation) since
in various recent publications it was pointed out that Hopf and Cole
were not the first to discover it.
In particular it was claimed that this transformation appeared in
the work of Forsyth at the beginning of the XX century and in the work
of Florin in the forties.
While the reference to Florin is correct, that to Forsyth
unfortunately is not. In the work of Forsyth some formulas similar
to those occurring in the Hopf-Cole transformation appear
(and this has perhaps caused some confusion), but the transformation
itself does not.
However, the theorem obtained by Forsyth is itself very interesting, and we
review it in \S1.
In \S2 we explain the transform in the multidimensional case
and in \S3 we give a review of Florin's work.
It is interesting to note that
this transformation reduces the question of uniqueness for the Burgers
equation to the Widder uniqueness theorem for the heat equation
in the class of positive solutions.
Since we have failed to find a proof of Widder's uniqueness theorem
for the multidimensional case in the literature,
we include one in the Appendix.
\smallskip
\smallskip
{\bf \S 1. Review of subsections 206 and 207
of the Forsyth's book [1] }
Consider a {\em linear} second order equation for
an unknown function $u=u(t,x)$:
\begin{equation}
\label{eq:1}
\nu(t,x)\tfrac{\partial^2 u}{\partial x^2}-\alpha(t,x)
\tfrac{\partial u}{\partial x} - \tfrac{\partial u}{\partial t}
+\gamma(t,x)u=0.
\end{equation}
By the {\em linear} substitution
\begin{equation}
\label{eq:2}
u=\lambda(t,x)v,
\end{equation}
where $\lambda=\lambda(t,x)$ is a fixed (known) function
and $v=v(t,x)$ is a new unknown function
we transform the equation (\ref{eq:1}) to the new, again {\em linear} equation
\begin{equation*}
\tilde\nu(t,x)\tfrac{\partial^2 v}{\partial x^2}-\tilde\alpha(t,x)
\tfrac{\partial v}{\partial x} - \tfrac{\partial v}{\partial t}
+\tilde\gamma(t,x)v=0,
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\tilde\nu&=\nu\,,\\
\tilde\alpha&=\alpha-\tfrac{2\nu \lambda_x}{\lambda},\\
\tilde\gamma&=\gamma-\tfrac{\alpha \lambda_x}{\lambda}-
\tfrac{\lambda_t}{\lambda}+\tfrac{\nu \lambda_{xx}}{\lambda}.
\end{aligned}
\end{equation*}
Hence the function $ I(t,x)=\nu(t,x)$
is obviously an invariant under the transformations of the type (\ref{eq:2}).
Forsyth found one more:
$$
J(t,x)=\tfrac{\partial}{\partial x}
\left(2\gamma+\nu\bigl(\tfrac{\alpha}{\nu}\bigr)_x-
\tfrac{1}{2}\tfrac{\alpha^2}{\nu}\right)-
\tfrac{\partial}{\partial t}\left(\tfrac{\alpha}{\nu}\right).
$$
Indeed, we have:
$$
2\tilde\gamma+\tilde\nu\bigl(\tfrac{\tilde\alpha}{\tilde\nu}\bigr)_x-
\tfrac{1}{2}\tfrac{(\tilde\alpha)^2}{\tilde\nu}=
2\gamma+\nu\bigl(\tfrac{\alpha}{\nu}\bigr)_x-
\tfrac{1}{2}\tfrac{\alpha^2}{\nu}-\tfrac{2\lambda_t}{\lambda},
$$
whence,
$$
\tfrac{\partial}{\partial x}
\left(2\tilde\gamma+\tilde\nu\bigl(\tfrac{\tilde\alpha}{\tilde\nu}\bigr)_x-
\tfrac{1}{2}\tfrac{(\tilde\alpha)^2}{\tilde\nu}\right)-
\tfrac{\partial}{\partial t}\left(\tfrac{\tilde\alpha}{\tilde\nu}\right)
=
\tfrac{\partial}{\partial x}
\left(2\gamma+\nu\bigl(\tfrac{\alpha}{\nu}\bigr)_x-
\tfrac{1}{2}\tfrac{\alpha^2}{\nu}\right)-
\tfrac{\partial}{\partial t}\left(\tfrac{\alpha}{\nu}\right).
$$
These invariants are found in subsection 206.
In the subsection 207
Forsyth shows that
$I$ and $J$ are the full system
of invariants for
equations of type
(\ref{eq:1}) w.r.t. transformations
(\ref{eq:2}).
In other words, if we are given
%there is
two equations of the type (\ref{eq:1})
such that the functions
$I$ and $J$
for the first equations coincide with the corresponding functions
for the second equation,
%are respectively coincide,
then these equations can be transformed
one into another by a substitution (\ref{eq:2}).
We now formulate a particular case of this fact.
{\bf Theorem.} {\sl An equation $(\ref{eq:1})$
can be reduced to the form
$$
v_t=\nu(t,x) v_{xx}
$$
by the substitution $(\ref{eq:2})$
iff $J(t,x)\equiv 0$.
In the last case we have
$\lambda(t,x)=e^{\theta(t,x)}$, where
$\theta(t,x)$ is a function such that
$$
\begin{aligned}
\tfrac{\partial \theta}{\partial x} &= \tfrac{\alpha}{2 \nu},\\
\tfrac{\partial \theta}{\partial t} &= \gamma+ \tfrac{\nu}{2}
\bigl(\tfrac{\alpha}{\nu}\bigr)_x - \tfrac{1}{4}\tfrac{\alpha^2}{\nu}.
\end{aligned}
$$
}
\noindent
If $\nu=Const$, then the condition $J(t,x)=0$
is nothing but the Burgers equation for $\alpha$:
$$
\alpha_t+\alpha\alpha_x= \nu\alpha_{xx}+2\nu\gamma_x\,.
$$
We note (see. \S2), that
by the substitution $\alpha=-2\nu \left({\rm ln\,}\vf\right)_x$
(i.e. $\vf=\lambda^{-1}$)
it can be reduced to the equation:
$\vf_t -\nu \Delta \vf + \gamma\vf=0$.
{\bf Remark 1.} Forsyth used some different notations. In particular,
he normalises the coefficient in front of $u_{xx}$ to be one,
while we do this for the coefficient of
$u_t$. Consequently the form of invariants $I$ and $J$
is changed.
{\bf Remark 2.} The Burgers Equation, as a
secondary condition for fulfilling certain conditions under
some linear transformations, appears in [1] also, e.g., in exercise 3
on the page 102.
Although the Burgers equation was known to Forsyth,
he did not
investigated the possibility of reducing the
non-linear Burgers equation to the liner one.
So far the first (known) appearance %[advent]
in the mathematical
literature of such a transformation is in the paper by Florin [2].
In the next section we explain this transform and in the
third
section we give a review of what Florin actually did.
\medskip
{\bf \S 2. The substitution.}
The multidimensional Burgers equation can be written in the two
different forms:
\begin{equation}
\label{eq:bur}
\tfrac{\partial\bs{v}}{\partial t} + \nabla\tfrac{(\bs{v},\bs{v})}{2}
= \nu\Delta\bs{v}\, \quad
\text{ or } \quad\,
\tfrac{\partial\bs{v}}{\partial t} + (\bs{v},\nabla)\bs{v}
= \nu\Delta\bs{v}\,.
\end{equation}
%These equations are {\em different}.
In general $\nabla\tfrac{(\bs{v},\bs{v})}{2}\ne (\bs{v},\nabla)\bs{v}$.
However we consider the potential case only for which
%(which we consider only)
these terms coincide.
%:$\nabla (\bs{v},\bs{v})= 2(\bs{v},\nabla)\bs{v}$.
Here $\bs{v}=\bs{v}(t,\bs{x})\in \mathbb R^n$, $t\geqslant 0$,
$\bs{x}\in \mathbb R^n$.
We recall that $(\bs{v},\nabla)$ is the operator
of differentiating along the vector field
$\bs{v}$.
Since we are considering the potential case,
the initial conditions are required to be potential:
\begin{equation}
\bs{v}(0,\bs{x})=\bs{v}_0(\bs{x})=\nabla H_0(\bs{x})\,.
\label{eq:ini}
\end{equation}
We assume that the function $H_0:\mathbb R^n \to \mathbb R$ is
continuous
and that
$\liminf\limits_{|\bs{x}|\to \infty}\tfrac{H_0(\bs{x})}{|\bs{x}|^2}
\geqslant 0$.
{
\small
The last condition is required to insure the existence of a solution
for all
$t\geqslant 0$.
If, e.g.,
$\liminf\limits_{|\bs{x}|\to \infty}\tfrac{H_0(\bs{x})}{|\bs{x}|^2}
\geqslant -k$, then we can insure existence only
for $2kt<1$.
Example: Let $\bs{v}_0(\bs{x})=-2k\bs{x}$, then
$\bs{v}(t,\bs{x})=\frac{-2k\bs{x}}{1-2kt}$ and
the solution fails to exist for $2kt\geqslant 1$.
The necessary and sufficient condition for
global existence
of the solution is the
convergence
of the integral
(\ref{eq:sol-heat}) for each $t>0$ and $\bs{x}\in
\mathbb R^n$.
}
We define $\vf_0(\bs{x})=\exp\left(\frac{H_0(\bs{x})}{-2\nu}\right)$.
Let $\vf(t,\bs{x})$ is the solution of the Cauchy problem
for the heat equation:
\begin{equation}
\vf_t=\nu\Delta\vf\,, \qquad \vf(0,\bs{x})=\vf_0(\bs{x})\,,
\label{eq:tepl}
\end{equation}
i.e.
\begin{equation}
\label{eq:sol-heat}
\vf(t,\bs{x})=\frac{1}{(4\pi t\nu)^{n/2}}\int_{\mathbb R^n}
\vf_0(z)\exp\left(\tfrac{-(x-z)^2}{4t\nu}\right) d \bs{z}\,.
\end{equation}
We note that (\ref{eq:tepl}) has no more than one solution in the
class of positive functions. In [5, chpt 8, \S2]
this is proven for the 1D case.
However the proof can be easily extended to the multidimensional case
without essential changes (see Appendix).
As a corollary we get that for
{\em any}
initial conditions the Burgers equation admits no more than one
solution.
{\bf Theorem.} {\sl
With our notations, the function
$\bs{v}=-2\nu\nabla\, {\rm ln\,} \vf = - 2 \nu \frac{\nabla \vf}{\vf}$
is a solution of the Cauchy problem (\ref{eq:bur}), (\ref{eq:ini}).}
{\bf Proof.} Indeed:
$$
\tfrac{\partial\bs{v}}{\partial t} + \nabla\tfrac{(\bs{v},\bs{v})}{2}
- \nu\Delta\bs{v} = -2\nu\nabla\left(\tfrac{\vf_t-\nu\Delta\vf}{\vf}\right)\,.
\eqno \square
$$
The uniqueness result for Burgers equation follows from the Widder's
uniqueness theorem and the observation:
{\sl
Let a potential vector field $\bs{v}$ satisfies the (\ref{eq:bur}).
Take any potential $H_0$ of the initial state, i.e.,
$\bs{u}(0,\cdot)=\nabla H_0$. Then there exists a positive function
$\vf$ such that
$$
\vf_t=\nu\Delta\vf\,, \qquad
\vf(0,\bs{x})=\exp\left(\tfrac{H_0(\bs{x})}{-2\nu}\right)\quad
\text{ and }
\quad
\bs{v}=-2\nu\nabla\, {\rm ln\,} \vf\,.
$$
}
It is interesting to note that for the uniqueness theorem for the
non-linear Burgers
equation no assumptions on growth of the initial conditions should be
made.
Whereas for the linear heat equation we should put some restrictions, e.g.,
the Tikhonov growth restrictions.
\medskip
{\bf \S 3 Florin's work.}
If a potential vector field satisfies (\ref{eq:bur})
then there exists a potential $H$
(i.e. $\bs{v}=\text{grad}\,H$) that satisfies the following equation:
$$
\tfrac{\partial H}{\partial
t}+\tfrac{1}{2}(\text{grad}\,H,\text{grad}\,H)=
\nu \Delta H.
$$
In the late forties V. A. Florin, considering the problem of
consolidation of wet soil,
arrived at the equation (the 3D case):
$$ \tfrac{\partial H}{\partial
t}+\alpha(\text{grad}\,H)^2+\beta(\text{grad}\,H,\text{grad}\,\psi)+
\delta\nabla^2H+\tfrac{\partial
F}{\partial t}=0,
\eqno [2, \text{ eq. (17)}]
$$
where $H$ is the hydrostatic pressure,
$\alpha,\beta,\delta,\psi$, and $\partial
F/\partial t$ are certain functions of position and time.
In [2] on the page 1392,
under the assumption $\alpha/\delta=const$,
Florin makes the substitution
$$
H=\tfrac{\delta}{\alpha} {\rm ln\,}(\vf+C)+D,
\eqno [2, \text{ eq. (18)}]
$$
where $C$ and $D$ are some constants, and reduces the equation above
to the linear equation:
$$
\tfrac{\partial \vf}{\partial t}+
\beta(\text{grad}\,\vf,\text{grad}\,\psi)
+ \delta \nabla^2\vf + \tfrac{\alpha}{\delta}(\vf+C)
\tfrac{\partial F}{\partial t}=0.
\eqno [2, \text{ eq. (19)}]
$$
Later, this transformation\footnote{for the case of constant
coefficients}
was independently rediscovered by
E. Hopf [3] and J. Cole [4].
\medskip
{\bf Appendix. The uniqueness theorem for the heat equation in the class of
non-negative functions.}
For the 1D case the uniqueness theorem was proved by Widder [5], [6].
We reproduce his proof for the multidimensional case.
{\bf Theorem.} {\sl Suppose that a non-negative continuous
function $u=u(t,\bs{x})$ is defined on the strip
$[0,T)\times \mathbb R^n$, and suppose that for $t>0$
%Χ ΠΟΜΟΣΕ $(0,T)\times \mathbb R^n$
this function is $C^{1,2}$-smooth and satisfies the equation
$u_t=\Delta u$.
Then for any
$t\in(0,T)$ and $\bs{x} \in \mathbb R^n$
we have:
\begin{equation}
\label{eq:7hh}
u(t,\bs{x})=\int_{\mathbb R^n} G(t,\bs{x}-\bs{\xi})u(0,\bs{\xi})
d\bs{\xi},
\end{equation}
where $G(t,\bs{y})=\frac{1}{(4\pi t)^{n/2}}e^{-|\bs{y}|^2/4t}$ is the
heat
kernel.
}
This theorem states, in particular,
the convergence of the integral in the r.h.s. of (\ref{eq:7hh}).
{\bf Proof. } In order to obtain (\ref{eq:7hh}) we show that two
opposite inequalities hold in (\ref{eq:7hh}):
1. ($\geqslant$)
For any fixed $A>0$
consider the function:
$$
v_A(t,\bs{x}) = u(t,\bs{x})-\int_{|\bs{\xi}|0$ we have
$v_A(t,\bs{x})\geqslant -\ve$.
To show this we apply the maximum principle for function
$v_A$ in the cylinder
$[0,T)\times \{|\bs{x}|\leqslant B\}$ for sufficiently large $B$,
noting that, $v_A\geqslant -\ve$ on the boundary.
2. ($\leqslant$)
We have proved that the integral
on the r.h.s. of (\ref{eq:7hh}) converges and, therefore,
defines some function $v$ such that $\bar u=u-v$ is a non-negative solution
of the heat equation with zero initial state.
It remains to prove that a non-negative solution $u$ of the heat equation
with zero initial state is zero.
Without loss of generality we can assume that $u_t\geqslant 0$.
Indeed, otherwise we consider the function $\tilde u(t,\bs{x})=\int_0^t
u(\tau,\bs{x})d\tau$.
Now we prove that the function $u$ satisfies the conditions of
the Tikhonov uniqueness theorem
(see. e.g., [7], section 3.4).
Let $\delta >0$ and $t+\delta0$ the function $u$ is bounded in
$[0,T-\ve]\times\{|\bs{x}|\leqslant1\}$ we see that with
a suitable constant $C_2$ we have the inequality
$$
u(t,\bs{x}) \leqslant C_2 e^{2|\bs{x}|^2/\ve},
$$
in the strip $[0,T-\ve]\times \mathbb R^n$. \hfill $\square$
\medskip
The author is grateful to Professor S.~Kuksin for his friendly support.
\medskip
{\bf References}
[1] A. R. Forsyth; {\em Theory of differential equations.} Vol 6.
Cambridge Univ. Press 1906.
[2] V. A. Florin; {\em Some of the simplest nonlinear problems
arising in the consolidation of wet soil. }
Izvestiya Akad. Nauk SSSR. Otd. Tehn. Nauk {\bf 1948}, no 9 (1948),
1389--1402.
[3] E. Hopf;
%Eberhard Hopf
The partial differential equation $u\sb t+uu\sb x=\mu u\sb {xx}$.
Comm. Pure Appl. Math. {\bf 3}, (1950), 201--230.
[4] J. D. Cole;
%Cole, Julian D.
On a quasi-linear parabolic equation occurring in aerodynamics.
Quart. Appl. Math. {\bf 9}, (1951), 225--236.
[5] D. V. Widder; {\em The heat equation.}
Academic Press, 1975.
[6] D. V. Widder; {\em Positive temperatures on an infinite rod.}
Trans. Amer. Math. Soc.
{\bf 55}, (1944) 85--95.
[7] E. M. Landis; {\em Second order equations of elliptic and parabolic
type} Translations of mathematical monographs, Vol. {\bf 171}, 1998.
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