An abstract framework for parabolic PDEs on evolving spaces Amal Alphonse
[email protected] Charles M. Elliott
[email protected] Bjorn¨ Stinner
[email protected] Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom Abstract 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 mate- rial 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. We finish by applying this theory to a surface heat equa- tion, a bulk equation, a novel coupled bulk-surface system and a dynamic boundary condition problem for a moving domain. 2010 Mathematics Subject Classification: Primary 35K05, 35K90; Secondary 46E35. Keywords: parabolic equations on moving domains, advection-diffusion on evolving surfaces Acknowledgements Two of the authors (A.A. and C.M.E.) were participants of the Isaac Newton Institute programme Free Boundary Problems and Related Topics (January – July 2014) when this article was completed. A.A. was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grant EP/H023364/1 within the MASDOC Centre for Doctoral Training. 1 2 A.