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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?

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up vote 5 down vote accepted

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.

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Thanks for your answer! :-) –  Darry Jun 7 '12 at 7:06
    
If a solution is in $C^k(0,T;H^m(\Omega))$,can I call it smooth solution? –  Darry Jun 7 '12 at 7:09
    
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 7 '12 at 8:43
    
Thank you very much! –  Darry Jun 7 '12 at 9:13
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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.

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Thanks for your help!:-) –  Darry Jun 7 '12 at 7:10
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