Renata Bunoiu

In this paper we consider the steady incompressible flow of a Newtonian fluid injected in a thin two-dimensional T-like shaped structure and also subjected to an external force. We study the asymptotic behavior of such a problem when the small parameter describing the thickness of the branches of the structure vanishes. We derive the limit problem satisfied by the limit velocity and the limit pressure. We obtain a flux coupling condition due to the injection. We prove that the limit pressure is defined and continuous on the T-graph underpinning the initial thin structure. If the external force depends only on the variable along the branches, we explicitly calculate the solution to the limit problem. Moreover, if the external force is zero, we obtain a Kirchhoff type relation for the derivatives of the pressure at the junction of the T-graph.

We study the asymptotic behavior of a sign-changing transmission problem, stated in a symmetric oscillating domain obtained by gluing together a positive and a negative material, separated by an imperfect and rapidly oscillating interface. The interface separating the two heterogeneous materials has a periodic microstructure and is a small perturbation of a flat interface. The solution of the transmission problem is continuous and its flux has a jump on the oscillating interface. Under certain conditions on the properties of the two materials, we derive the limit problem and we prove the convergence result. The T-coercivity method is used to handle the lack of coercivity for both the microscopic and the macroscopic limit problems. For more information see https://ejde.math.txstate.edu/Volumes/2024/60/abstr.html

We study the asymptotic behavior of a sign-changing transmission problem, stated in a symmetric oscillating domain obtained by gluing together a positive and a negative material, separated by an imperfect and rapidly oscillating interface. The interface separating the two heterogeneous materials has a periodic microstructure and is a small perturbation of a flat interface. The solution of the transmision problem is continuous and its flux has a jump on the oscillating interface. Under certain conditions on the properties of the two materials, we derive the limit problem and we prove the convergence result. The T-coercivity method is used to handle the lack of coercivity for both the microscopic and the macroscopic limit problems.

Knowledge in mathematics (definitions, theorems, proofs, etc.) is usually expressed in a way that combines natural language and mathematical expressions (e.g. equations). Using an ontology formalism such as OWL DL is well-suited for formalizing the natural language part, but complex mathematical expressions can be better handled by symbolic computation systems. We examine this representation issue and propose an original extension of OWL DL by call formulas, i.e., formulas from which assertions can be drawn thanks to calls to external functions. Using this formalism makes it possible to classify a mathematical problem defined by its relations to instances and classes and by some mathematical expressions: if a theorem for solving this problem is represented in the knowledge base, it can be retrieved, and thus, the problem can be solved by applying this theorem. We describe an inference algorithm and discuss its properties as well as its limitations. Indeed, the proposed extension, algorithm, and implementation represent a first step towards a combined formalism for representing mathematical knowledge, with some open issues regarding the representation of more complex problems: the resolution of multiscale, multiphysics cases in physics are foreseen.

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We consider the steady Bingham flow in a two-dimensional thin Y-like shaped structure, with no-slip boundary conditions and under the action of given external forces. After passage to the limit with respect to a small parameter related to the thickness of the domain, we obtain three uncoupled problems. Each of these problems describes an anisotropic flow, corresponding to a lower-dimensional “Bingham-like” constitutive law. These results are in accordance with the engineering models.

We study the homogenization of a double porosity diffusion problem in a thin highly heterogeneous composite medium formed by two materials separated by an imperfect interface, where the solution and its flux exhibit jumps. By applying homogenization techniques specific to the thin periodic domain under study, one derives the limit problem.

We study a scalar problem in thin periodic composite media formed by two materials, a classical one and a metamaterial (also known as negative material). By applying T-coercivity methods and homogenization techniques specific to the thin periodic domains under consideration, for two geometric settings, we derive the homogenized limit problems, which both exhibit dimension-reduction effects.

In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn-Hilliard-Navier-Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is suitable for the passage to the limit.