A diffused interface with the advection term in a Sobolev space
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Calculus of Variations
A diffused interface with the advection term in a Sobolev space Yoshihiro Tonegawa1 · Yuki Tsukamoto1 Received: 31 March 2019 / Accepted: 31 August 2020 © The Author(s) 2020
Abstract We study the asymptotic limit of diffused surface energy in the van der Waals–Cahn–Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min–max method. Mathematics Subject Classification 49Q20
1 Introduction The object of study in this paper is the energy functional appearing in the van der Waals– Cahn–Hillard theory [4,7], ε|∇u|2 W (u) E ε (u) = + , (1.1) 2 ε where u : ⊂ Rn→ R (n ≥ 2) is the normalized density distribution of two phases of a material, |∇u|2 = nk=1 (∂u/∂ xk )2 and W : R → [0, ∞) is a double-well potential with two global minima at ±1. In the thermodynamic context, W corresponds to the Helmholtz free energy density and the typical example is W (u) = (1 − u 2 )2 . When the positive parameter ε is small relative to the size of the domain and E ε (u) is bounded, it is expected that u is close to +1 or −1 on most of while a spatial change between ±1 occurs within a hypersurface-like region of O(ε) thickness which we may call the diffused interface of u. In this case, the quantity E ε (u) is expected to be proportional to the surface area of the diffused
Communicated by A. Neves. The first author is partially supported by JSPS KAKENHI Grant Numbers (A) 25247008 and (S) 26220702.
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Yoshihiro Tonegawa [email protected] Yuki Tsukamoto [email protected]
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Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan 0123456789().: V,-vol
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interface. Due to the importance of the surface area in calculus of variations, it is interesting to investigate the validity of such expectation and other salient properties of E ε . In this direction, there have been a number of works studying the asymptotic behavior of E ε as ε → 0+ under various assumptions. For the energy minimizers with appropriate side conditions, it is well-known that it -converges to the area functional of the limit interface [10,12–14,20]. On the other hand, due in part to the non-convex nature of the functional, there may exist multiple and even infinite number of critical points of E ε different from the energy minimizers. For general critical points, Hutchinson and the first author [8] proved that the limit is an integral stationary varifold [1]. For general stable critical points, the first author and Wickramasekera [26] proved that the limit is an embedded real-analytic minimal hypersurface except for a closed singular set of codimension seven. More recently, Guaraco [6] showed that a uniform Morse index bound is sufficient to conclude the same regularity for n ≥ 3 and gave a new proof of Alm
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