# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020289

## Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation

 1 Università degli Studi di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, 41125 Modena, Italy 2 Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany

* Corresponding author

Received  April 2020 Revised  October 2020 Published  December 2020

Fund Project: The work of E. Ipocoana is supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), by MIUR through the project FFABR (M. Eleuteri) and by the University of Modena and Reggio Emilia through the project FAR2017 "Equazioni differenziali: problemi evolutivi, variazionali ed applicazioni" (S. Gatti). A. Zafferi acknowledges the funding by the DFG through grant CRC1114 "Scaling Cascades in Complex Systems", Project Number 235221301, Project (C09) "Dynamics of rock dehydration on multiple scales"

The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial $L^{\infty}$ estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.

Citation: Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020289
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