What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory? Especially，the difference for the evolution equations? If a solution is in $C^k(0,T;H^m(\Omega))$，can I call it smooth solution?
Asked
Active
Viewed 1.2k times
2 Answers
14
A smooth solution is infinitely differentiable. A classical solution is a solution which is differentiable as many times as needed if you want to plug the function into the PDE (for example, if the PDE contains the term $u_{xxxx}$, then the fourth derivate $u_{xxxx}$ must exist in order for $u$ to be a classical solution).
In particular, every smooth solution is a solution in the classical sense. But for the unidirectional wave equation $u_x + u_t = 0$, any function of the form $u(x,t)=f(x-t)$ where $f$ is only (say) twice differentiable, is a classical solution which is not smooth.
Hans Lundmark
- 53,395
-
Thanks for your answer! :-) – Darry Jun 07 '12 at 07:06
-
If a solution is in $C^k(0,T;H^m(\Omega))$，can I call it smooth solution? – Darry Jun 07 '12 at 07:09
-
Well, "smooth" usually means $C^{\infty}$, but in some contexts I guess it might mean "smooth enough". As long as you define your terminology you can call it whatever you like. ;-) – Hans Lundmark Jun 07 '12 at 08:43
4
A classical solution is a function that solves the PDE in the usual sense, ie. $x'=x, x(0)=1 \implies x(t)=e^t$. You can also have weak solutions, which is a variant of the equation with integrals, and is equivalent to the original equation if the solution you are looking at is a classical solution. You can also have solutions as distributions. Look up weak and distribution solutions of Laplace's equation as an example. A smooth solution is one with infinitely many derivatives. A smooth solution is classical, but a classical solution may not be smooth.
JLA
- 6,442
- 2
- 24
- 40