Global solutions of the non-linear problem describing Joule's heating in three space dimensions

✍ Scribed by Marian Bień

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 203 KB
• Volume: 28
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.559

✦ Synopsis

The existence of global-in-time weak solutions to the Joule problem modelling heating or cooling in a current and heat conductive medium is proved via the Faedo -Galerkin method. The existence proof entails some a priori estimates that together with the monotonicity and compactness methods make up a main tool to prove the desired result. Under appropriate hypotheses on the data, it will be shown the boundedness in L∞(Q T ) of the absolute temperature of the medium and of the t-derivative of this temperature, which is achieved by means of the Gagliardo -Nirenberg theorem, the Sobolev embedding theorem and the method of Stampacchia. The paper is some extension of our investigation initiated in (Math.