Marko A. Rojas-Medar

On the existence and uniqueness of the stationary solution to equations of natural convection with boundary data in L2

This paper deals with the existence and uniqueness of a stationary solution to the Boussinesq system with no regular data on the boundary. We introduce very weak solutions, and we prove their existence. Using more hypotheses on the problem data, we prove the continuous dependence and uniqueness for this class of solutions.

Micropolar Fluids with Vanishing Viscosity

A study of the convergence of weak solutions of the nonstationary micropolar fluids, in bounded domains of ℝ𝑛, when the viscosities tend to zero, is established. In the limit, a fluid governed by an Euler-like system is found.

Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

Abstract In this paper, we proceed to study properties of Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions. In our previous papers (Allendes et al., 2013 [13] , Kniehl et al., 2013 [14] ), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions may be obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in d = 4 dimensions, and its analog in d = 4 − 2 e dimensions exits, too (Gonzalez and Kondrashuk, 2013 [6] ). In Allendes et al. (2013) [13] , the chain of recurrence relations for analytically regularized UD functions was obtained implicitly by comparing the left-hand side and the right-hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained using the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here, we reproduce these recurrence relations by calculating explicitly, via Barnes lemmas, the contour integrals produced by the left-hand sides of the diagrammatic relations. In this a way, we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions, which includes the MB transforms of UD functions.

A weak-Lp Prodi-Serrin type regularity criterion for the micropolar fluid equations

We investigate regularity criteria for weak solutions of the micropolar fluid equations in a bounded three-dimensional domain. We show that the solution (u, w) is strong on [0, T] if either u ∈ Ls(0, T; Lr,∞(Ω)) or ‖u‖Ls,∞(0,T;Lr,∞(Ω)) is bounded from above by a specific constant, where (3/r) + (2/s) = 1 and r > 3.

Pointwise Error Estimate for Spectral Galerkin Approximations of Micropolar Equations

ABSTRACT The goal of this article is to present pointwise time error estimates in suitable Hilbert spaces by considering spectral Galerkin approximations of the micropolar fluid model for strong solutions. In fact, we use the properties of the Stokes and Lamé operators for prove the pointwise convergence rate in the H2-norm for the ordinary velocity and microrotational velocity and the pointwise convergence rate in the L2-norm for the time-derivative of both velocities. The novelty of our method is that we do not impose any compatibility conditions in the initial data.

A weak- Prodi-Serrin type regularity criterion for a bioconvective flow

ABSTRACT We determine regularity criteria for weak solutions of a bioconvective flow in a bounded three-dimensional domain. We show that the weak solution is strong on [0, T] if either or is bounded from above by a specific constant, where and . No additional condition for is required.

On the behavior of Kazhikov-Smagulov mass diffusion model for vanishing diffusion and viscosity coefficients

We consider the motion of a viscous incompressible fluid consisting of two components with a diffusion effect obeying Fick’s law in ℝ3. We prove that there exists a small time interval where the fluid variables converge uniformly as the viscosity and the diffusion coefficient tend to zero. In the limit, we find a non-homogeneous, non-viscous, incompressible fluid governed by an Euler-like system.