9
$\begingroup$

The equation \begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \varphi(x), \end{gather} is called Kolmogorov partial differential equation.

I know about a result which states, that this equation has a unique, at most polynomially growing viscosity solution under some assumptions including continuity for $\varphi$ and local Lipschitz continuity for $\sigma, \mu$.

I have seen several solution-existence theorems of this type associated with PDEs and I am confused by the emphasis placed on "at most polynomial growth" of solutions seemingly required for uniqueness in several cases. I am fairly new to the theory of partial differential equations and I wonder why we only seem to care about solutions which grow at most polynomially.

What about other solutions which might exist and do not fulfill this growth requirement?

Are at most polynomially growing solutions the only ones which are interesting in applications?

I see no reason to single out at most polynomially growing solutions and give them special attention if there might be many other solutions; it seems like a rather arbitrary property to me. So why are we satisfied with existence theorems like the one about Kolmogorov PDEs above?

If someone could shed some light on what makes solutions of this type particularly interesting, that would be fantastic!

Kind regards,

Joker

$\endgroup$
4
+50
$\begingroup$

It is well-known (see Evans Ch.2) that solutions of parabolic equations are not unique without some growth conditions at $\infty$. For the linear heat equation, it is required that $|u(x,t)|\leq e^{A|x|^2}$, so polynomial and exponential growth is allowed. So it is not unusual to restrict to, say, polynomial growth solutions, since this would imply existence and uniqueness. Functions with polynomial growth constitute a very wide class of functions in the context of solving PDEs. Most of the time, one has to place far stronger growth conditions, say bounded, or linear growth, to get uniqueness of viscosity solutions (in the nonlinear setting).

ADDITION: When the growth bounds are violated, solutions are not unique. For the heat equation, there are infinitely many solutions that violate the growth condition $|u(x,t)|\leq e^{A|x|^2}$. These solutions are so large as $x\to \infty$ that heat flows very quickly from $\infty$ in towards the origin and the solution exhibits finite time blow up. So the solutions are poorly behaved and do not represent the physical phenomenon being modeled (e.g., heat flow), so are normally discarded as being "non-physical".

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks for your answer. Are there any other reasons except convenient existence/uniqueness-results for focussing our attention to polynomial-growth solutions? Are they in some way the "natural" solutions? $\endgroup$ – Joker123 Apr 5 '19 at 15:19
  • 1
    $\begingroup$ They are very poorly behaved and often "non-physical". I added a paragraph to the answer. $\endgroup$ – Jeff Apr 6 '19 at 2:38
  • $\begingroup$ Thank you. Very interesting. An almost philosophical question: do you know if there is any systematic way to decide a priori (i.e. without an experiment) whether a solution of a PDE modelling some physical phenomenon can actually be instantiated and observed in a real system? $\endgroup$ – Joker123 Apr 6 '19 at 20:39
  • $\begingroup$ Uniqueness and continuous dependence on initial/boundary data (or stability) are good indications. So if we perturb the initial or boundary data by a small amount, the solution should change by a correspondingly small amount. If you do not have uniqueness or stability, probably you are not capturing the physics properly. $\endgroup$ – Jeff Apr 8 '19 at 3:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.