In the study of nonlinear PDEs, I often work on conservation laws and a lot of time I work on the two problems given bellow:

$$(1) \hspace{1cm} \begin{cases} u_t+(f_{1}(u))_x=\lambda \cdot g(u) \\[2ex] u(x,0)=h_{1}(x) \end{cases} $$

$$(2) \hspace{1cm} \begin{cases} u_t+(f_{2}(u))_x=0 \\[2ex] u(x,0)=h_{2}(x) \end{cases} $$

Here u $\in \mathbb{R}^n$ and $\lambda$ is a constant. Besides $u$, the functions $g,f_1, f_2$ could also depend of $x$ and/or $t$.

A long time ago one professor of mine told me that he read somewhere that

there is a way of transforming problem $(1)$ to the problem $(2)$ for some special cases of functions $g,f_1, f_2$. And vice versa.

But he can't remember where he has read that.

It would be very useful if we can switch between problems $(1)$ and $(2)$ from time to time.

So in one moment you work on a problem that has a source term $\lambda \cdot g(u)$, i.e. problem $(1)$. In the other moment you use change of coordinates (or something similar) and now you work with a problem with no source term and with different initial condition (and possibly different flux too), i.e. problem $(2)$.

Of course I am pretty sure that some general transformation doesn't exist.

In the problem $(1)$ source is time dependent and in the problem $(2)$ new initial data are not time dependent. But

  • maybe source could go partially in the flux and partially into the initial data or something similar.
  • Or maybe this kind of transformation exists for some functions $g,f_1, f_2, h_1, h_2$ with the special properties.
  • Or maybe it exists for some other kind of PDE problems - that don't deal with conservation laws.

The only example of that transformation that I know would be this:

if $g(u)=(w(u))_x$ (assuming that function $w$ could be found), then the source term would be absorbed in the flux term. So we would get the problem $(2)$ form where $f_2 (u) = f_1 (u) - \lambda \cdot w(u)$ and $h_2(x)=h_1 (x).$

So if anyone knows any reference in the literature that deals with this issue share it. Of course you could share your ideas or answers too. This question intrigues me from the moment the professor told me that.

  • $\begingroup$ such a transformation from eq. 1 to eq. 2 seems unlikely to exist, at least not for a general source: equation 2 has the conservation law that $\int u dx$ is time independent; in equation 1 it is not, because of the source. $\endgroup$ – Carlo Beenakker Dec 7 '18 at 15:17
  • $\begingroup$ @CarloBeenakker: I am skeptical of existence of such transformation for a general source too. But I hope it exists for some special source term. In problem (1) I could use different kinds of source terms (the one written above is the most general one I work with). $\endgroup$ – Mark Dec 7 '18 at 15:26
  • $\begingroup$ @WillieWong: In general I am pretty sure too that it is not possible. Also your simple example explains it good. But I was wondering is there some paper that studies some conservation laws type system or equation, where authors transformed problem with some kind of source term (any type of source term) to the problem with zero source term (but now they have different initial conditions and/or flux). Or vice versa. I guess that the fact that the professor said it to me that he saw something similar intrigues me. $\endgroup$ – Mark Dec 7 '18 at 19:20
  • $\begingroup$ since you are doing general change of coordinates, are you going to insist that $g$ and $f_1$, $f_2$ are only functions of the unknown, or can they also depend on the space-time coordinates $t$ and $x$? If they are allowed to depend also on $t$ and $x$ I think what you want may be possible. $\endgroup$ – Willie Wong Dec 7 '18 at 19:40
  • $\begingroup$ @WillieWong: They could also depend of $t$ and $x$ (in the problem I am currently working on, they depend only on $u$ - that's why I write it that way above - I probably should explain it more clearer). You could use any type of source term, initial condition and flux. $\endgroup$ – Mark Dec 7 '18 at 19:53

Here's one possible solution, but this may or may not be what your professor had in mind.

Since $\lambda$ is a constant, we can ignore it by absorbing it into $g$.

Assume $u$ is scalar (takes value in $\mathbb{R}^1$).

Assume that $g$ is a function of $u$ only.

Let $G(s; u_0)$ be the solution to $\partial_s G = g(G)$ with initial data $G(0;u_0) = u_0$.

Suppose for every fixed $s$, the two functions $f(s;j)$ and $F(s;k)$ are related by $$(\partial_k F)(s;G(s;j)) = \partial_jf(s;j). \tag{*}$$

Then if $v(t,x)$ is a solution to the homogeneous equation

$$ \partial_t v + \partial_x [f(t;v)] = 0 $$

then the function $u(t,x) = G(t; v(t,x))$ satisfies

$$ \partial_t u + \partial_x [F(t; u)] = g(u) $$

You even have that $u(0,x) = v(0,x)$.

In particular, you can definitely convert from (1) to (2) in this setting, just by solving two ODEs (for $G$ and for $f$).

When $u$ is not a scalar (has multiple components), the replacement for (*) may not be integrable, giving an obstruction to the sort of transformation you seek using this naive method. (Roughly, the equation is $ \partial f = (\partial G)^{-1} \cdot [(\partial F)\circ G] \cdot (\partial G)$.)

  • $\begingroup$ In the case of equations, the addition of $t$ in the fluxes solved the problem very elegantly. For systems, when $u$ has multiple components such as 2 or 3 this construction gets complicated (it would depend of the concrete system in question and of course of solving $(*)$). Still, this is the first example I've seen with transformation between $(1)$ and $(2)$. Thanks for that. Also I talked with my professor yesterday. He said to me that he didn't have this in mind. :( Who knows on what he was thinking of. :) $\endgroup$ – Mark Dec 9 '18 at 12:58

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