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I have multiple questions relating to Galerkin Method and the Semigroup Method of proving existence of solutions to PDEs.

  1. In the Galerkin Method, we decide on a function space, find eigenfunctions for it and then convert the PDE into a system of ODEs, establish existence of solution and pass to limit using weak convergence. Semigroup method, on the other hand seems like a very formalized method of solving. On a level of perception, this looks just like ODEs in infinite dimensions. But, is there any direct and mathematical relationship between the two methods?

  2. I read in some informal notes that behind the machinery, the Semigroup method amounts to taking Laplace transform of the time variable and then doing inversion. Is there a reference that amplifies this relationship between Laplace transformation and Semigroups?

  3. If semigroup method is Laplace transform on the time variable, could we say that the Galerkin method is Fourier Transform on the space variable?

  4. Galerkin Method uses compactness results in an essential way and therefore bounded domains become important. Does compactness not enter into the semigroup method at all?

  5. What are the advantages/disadvantages of each method?

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  • In the context of 2 (relationship between Laplace transformation and Semigroups), Arendt's book can be of interest. The following result is presented in section 3.1:

Given a closed operator $A$ on a Banach space $X$, the abstract Cauchy problem $u'(t)=Au(t)$ is mildly well-posed (i.e., for each initial data there exists a unique mild solution) if and only if the resolvent of $A$ is a Laplace transform; and this in turn is the same as saying that $A$ generates a $C_0$-semigroup.

  • A brief answer for 5 is given in section 7.4.3 of Evans' book:

The semigroup theory requires that the differential operator be independent of $t$. The Galerkin method works without this restriction. On the other hand, semigroup theory constructs at the outset a more regular solution than that produced by the Galerkin technique, at least until we develop regularity theory.

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