In this article,we study a coupled Allen-Cahn-Navier-Stokes model in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter.We present an equivalent weak formulation for the model, and we prove a new regularity result for the weak solutions.
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In this paper we consider the following nonlinear wave equation (1) utt-uxx = f(x,t,u,ux,ut), x is an element of (0,l), 0 < t < T, (2) ux(0,t)-hou(0,t)=go(t), u(1,t) = g1(t), (3) u(x,0) = uo(x), ut((x,0) = u1x).
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In this article we focus on the global well-posedness of the differential equation u [...] in Omega x(O, T), where j' denotes the derivative of a C1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with non-degenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obt ains existence, regularity or finite time blow up of solutions. More specifically, when p [is less than or equal to] m + k global existence of generalized solutions in H1 x L2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H2 x H1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.
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