2011/08/03 by Duvan Henao, Henao, Duvan, Stéphane Serfaty +1
Computer Science · Engineering · Mathematics · #49K20 #74B20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1108.0939
openalex publication_date 2011/08/03 · openalex created_date 2022/09/29 · openalex updated_date 2026/07/28
We consider the minimization of \∫_\Ω ep |D u|p dd x\nin a perforated domain \Ω ep:= \Ω \∖ bigcupi=1M\nB ep( ai) of Rn, among maps u \∈ W1,p(\Ω ep,\n Rn) that are incompressible (\det D u\≡ 1), invertible, and\nsatisfy a Dirichlet boundary condition u= g on \∂ \Ω.\nIf the volume enclosed by g (\∂ \Ω) is greater than\n|\Ω|, any such deformation u is forced to map the small holes\nB ep( ai) onto macroscopically visible cavities (which do not\ndisappear as ep\→ 0). We restrict our attention to the critical exponent\np=n, where the energy required for cavitation is of the order of\n\∑i=1M vi |\log ep| and the model is suited, therefore, for an\nasymptotic analysis (v1,..., vM denote the volumes of the cavities). In the\nspirit of the analysis of vortices in Ginzburg-Landau theory, we obtain\nestimates for the "renormalized" energy \(1)/(n)\∫_\Ω ep\n|\(D u)/(\√(n-1))|p dd x - \∑i vi |\log ep|, showing its\ndependence on the size and the shape of the cavities, on the initial distance\nbetween the cavitation points a1,..., aM, and on the distance\nfrom these points to the outer boundary \∂ \Ω. Based on those\nestimates we conclude, for the case of two cavities, that either the cavities\nprefer to be spherical in shape and well separated, or to be very close to each\nother and appear as a single equivalent round cavity. This is in agreement with\nexisting numerical simulations, and is reminiscent of the interaction between\ncavities in the mechanism of ductile fracture by void growth and coalescence.\n