JournalspmVol. 72, No. 1pp. 1–46

An abstract framework for parabolic PDEs on evolving spaces

  • Amal Alphonse

    University of Warwick, Coventry, UK
  • Charles M. Elliott

    University of Warwick, Coventry, UK
  • Björn Stinner

    University of Warwick, Coventry, United Kingdom
An abstract framework for parabolic PDEs on evolving spaces cover

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Abstract

We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial di¤erential 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.

Cite this article

Amal Alphonse, Charles M. Elliott, Björn Stinner, An abstract framework for parabolic PDEs on evolving spaces. Port. Math. 72 (2015), no. 1, pp. 1–46

DOI 10.4171/PM/1955