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In this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if ∥ ∇ μ (ρ) ∥ L ∞ (0 , T ; L p) is bounded. Second, we prove the strong solution exists globally (in time) only if ∥ ∇ μ (ρ 0) ∥ L p is suitably small, even the presence of vacuum is permitted. [ABSTRACT FROM AUTHOR]

As is well known, fixed point theorems and problems play important roles in differential equations, mathematical programming, control, and so on. In this paper, by providing some unboundedness conditions and by using some inequality techniques, we can give constructive proofs of the existence of fixed points on unbounded non-convex sets via homotopy methods. [ABSTRACT FROM AUTHOR]