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I have the following PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u - f(x,t)$$ $$u|_{t=0} = u_0$$ over the domain $S^1 \times [0,T)$. The coefficients and $f$ are in $C^{k,\alpha}$ for some $k$ (and are $2\pi$ periodic, ignore if this doesn't make sense). Also $a$ is such that the equation is uniformly parabolic.

My questions: 1) To get an a-priori estimate for this equation

$$\lVert u \rVert_{C^{k+2, \alpha}} \leq C(\lVert f \rVert_{C^{k, \alpha}} + \lVert u_0 \rVert_{C^{k+2, \alpha}})$$

what do I do? The only thing I know of is multiplying by a test function and integrating and using Gronwall but this gives me norms in Sobolev spaces, I believe.

2) Apparently, the following is true, but I need some explanation:

There is a unique solution $u \in C^{k+2, \alpha}$. Proof: first solve the Cauchy problem in the smooth category by means of separation of variables. Then use an approximation argument coupling with the global a-priori estimate above to get the general result.

What's the Cauchy problem (Wikipedia doesn't help. It just says the domain is a manifold)? What's smooth category? What's the approximation argument thing? Sorry if these questions are stupid but I have never heard of this stuff.

Any references or help would be appreciated.

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@Andrew: we don't need a tag that specific just yet... pde on its own should be fine. –  J. M. Jul 14 '12 at 6:31
    
@TagWoh 1) Obtaining estimates in Holder spaces is somewhat labor-consuming. Here $u$ can be regarded as a periodic solution to the Cauchy problem on the line. Estimates for it can be found in books on parabolic equations, for example, Krylov N., Lectures on Elliptic and Parabolic Equations in Hölder Spaces books.google.ru/books/about/…. 2) Is the question for the same equation? Usually separation of variables is applied then coefficients don't depend on $t$. –  Andrew Jul 14 '12 at 7:21
    
May be worth moving to math overflow. –  nbubis Jul 16 '12 at 8:12
    
Thanks for the attention. @Andrew Thanks I will look at that book. Yes the 2nd question is for the same equation. –  TagWoh Jul 16 '12 at 18:08
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1 Answer 1

up vote 2 down vote accepted
+50

As Andrew commented, obtaining Hölder estimates can sometimes be labour intensive. So I'll ignore part (1) for now.

For part (2):

  • The term "Cauchy problem" is just another name for "initial(-boundary) value problem". In your case there is no boundary since the spatial domain is a closed manifold.
  • Smooth category just means that given smooth initial data, try to solve the equation with smooth solution. Usually this is obtained by separation of variables or explicitly integrating against some Green's function.
  • Approximation argument is the idea that for most reasonable function (Banach) spaces $X$ defined on a manifold $M$, the space $C^\infty_0(M)$ of smooth functions with compact support is dense in $X$ in the Banach norm. Hence we can approximate a, say, $C^{k,\alpha}$ initial data with a sequence of $C^\infty$ initial data. Now if there is a good a priori estimate available (such as the Hölder estimate you quoted in your question), then we can transfer the convergence of the initial data to a convergence of the sequence of smooth solutions in $C^{k+2,\alpha}$ norm, obtaining then a $C^{k+2,\alpha}$ solution as the limit of a sequence fo smooth solutions, which we derived from the sequence of smooth initial data approximating the $C^{k,\alpha}$ initial data.
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Thank you Willie. That's very interesting stuff, and I wonder where this sort of thing is discussed. If you can give an outline for part 1, that'd be great. Otherwise I'll try and get that book. –  TagWoh Jul 16 '12 at 18:10
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You should just look at Krylov's book. He was, after all, one of the principal players in the subject. –  Willie Wong Jul 17 '12 at 13:07
    
Smooth functions are not dense in Holder spaces. Not to nitpick, I think it is important. –  timur Aug 9 '12 at 0:21
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