Liouville type theorems for stationary Navier-Stokes equations with Lebesgue spaces of variable exponent

In this article we study some Liouville-type theorems for the stationary 3D Navier-Stokes equations. These results are related to the uniqueness of weak solutions for this system under some additional information over the velocity field, which is usually stated in the literature in terms of Lebesgue, Morrey or BMO^--1 spaces. Here we will consider Lebesgue spaces of variable exponent which will provide us with some interesting flexibility

A Lp-theory for fractional stationary Navier-Stokes equations

We consider the stationary (time-independent) Navier-Stokes equations in the whole threedimensional space, under the action of a source term and with the fractional Laplacian operator (--$\Delta$) $\alpha$/2 in the diffusion term. In the framework of Lebesgue and Lorentz spaces, we find some natural sufficient conditions on the external force and on the parameter $\alpha$ to prove the existence and in some cases nonexistence of solutions. Secondly, we obtain sharp pointwise decaying rates and asymptotic profiles of solutions, which strongly depend on $\alpha$. Finally, we also prove the global regularity of solutions. As a bi-product, we obtain some uniqueness theorems so-called Liouville-type results. On the other hand, our regularity result yields a new regularity criterion for the classical ( i.e. with $\alpha$ = 2) stationary Navier-Stokes equations. Content

Well-posedness of a nonlinear shallow water model for an oscillating water column with time-dependent air pressure

We propose in this paper a new nonlinear mathematical model of an oscillating water column. The one-dimensional shallow water equations in the presence of this device are essentially reformulated as two transmission problems: the first one is associated with a step in front of the device and the second one is related to the interaction between waves and a fixed partially-immersed structure. By taking advantage of free surface Bernoulli's equation, we close the system by deriving a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in the previous work \cite{bocchihevergara2021}. We then show that the second transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semilinear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE. Finally, the well-posedness of the transmission problem related to the wave-structure interaction in the oscillating water column is obtained as an application of the general theory.Comment: 35 pages, 1 figur

Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness

We consider here the stationary Micropolar fluid equations which are a particular generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one based mainly in the velocity field u and the other one based in the microrotation field $\omega$. We will study in this work some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establish, under some suitable decay at infinity conditions for the velocity field only, the uniqueness of the trivial solution

Asymptotic behaviour of a system modelling rigid structures floating in a viscous fluid

The PDE system introduced in Maity et al. (2019) describes the interaction of surface water waves with a floating solid, and takes into account the viscosity µ of the fluid. In this work, we study the Cummins type integro-differential equation for unbounded domains, that arises when the system is linearized around equilibrium conditions. A proof of the input-output stability of the system is given, thanks to a diffusive representation of the generalized fractional operator $\sqrt{1 + \mu s}$. Moreover, relying on Matignon (1996) stability result for fractional systems, explicit solutions are established both in the frequency and the time domains, leading to an explicit knowledge of the decay rate of the solution. Finally, numerical evidence is provided of the transition between different decay rates as a function of the viscosity $\mu$

Modelling and simulation of a wave energy converter

In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions

Relations between fractional calculus and interactions fluid-structure

We survey the literature on fractional calculus and fractional differential equations with emphasis in the context of causal distributions, which has been an area of rapid development in recent years. We examine the main concepts neccesary for the study of fractional formalism, highlighting functions expandable into fractional power series and the asymptotics results of fundamental solutions developed by Matignon. We give some new applications of this area on the asymptotic behaviour of a system modelling rigid structures floating in a viscous fluid

On a dual to the properties of Hurwitz polynomials I

In this paper we develop necessary and sufficient conditions for describe the family of anti-Hurwitz polynomials, introduced by Vergara-Hermosilla et al. in [9]. Specifically, we studied a dual version of the Theorem of Routh-Hurwitz and present explicit criteria for polynomials of low order and derivatives. Another contribution of this work is establishing a dual version of the Hermite-Biehler Theorem. To this aim, we give extensions of the boundary crossing Theorems and a zero exclusion Principle for anti-Hurwitz polynomials

On a dual to the properties of Hurwitz polynomials I

In this paper we develop necessary and sufficient conditions for describe the family of anti-Hurwitz polynomials, introduced by Vergara-Hermosilla et al. in [9]. Specifically, we studied a dual version of the Theorem of Routh-Hurwitz and present explicit criteria for polynomials of low order and derivatives. Another contribution of this work is establishing a dual version of the Hermite-Biehler Theorem. To this aim, we give extensions of the boundary crossing Theorems and a zero exclusion Principle for anti-Hurwitz polynomials