2014/03/31 by Amal Alphonse, Charles M. Elliott, Björn Stinner · 2 citations
Mathematics · #math.AP #msc:35K90 #msc:46E35
paper · pdf · doi:10.4171/pm/1955
38 pages. Minor typos corrected
arxiv created 2015/07/10 · arxiv updated 2015/07/13
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.