Weak Solutions of a Phase-field Model with Convection for Solidification of an Alloy |
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a.e. in Q ac² Banach space binary alloy bounded Lipschitz continuous bounded sets C₂ Carman-Kozeny cm|² compact embedding compact support constant C₁ convection converges defined denote dødt existence of weak existence result exists a constant Fernando Torres flag manifolds free boundary value Gronwall's inequality H(Nº H¹(N H¹(Nº H²(Nº Hölder's and Young's Hölder's inequality infer José Luiz Boldrini k(fº K₁ K₁(ps K₂Acm K₂M L²(N L²(Nº L²(Q L²(Qº Lemma Lipschitz continuous function LP(Q m² dxdt mathematical mathematical analysis modified Navier-Stokes equations multiplying mushy region Navier-Stokes equations norm Numbers obtain غ open set P² dx Phase-Field Model Proof ps(v Quasi-Newton Methods regularized problem Rojas-Medar Santos satisfying sequence solidification process strongly in L²(Qʻ t₁ Tx(x V(Nº W₁ weak solutions weakly in L²(0 Young's inequalities Әп дп