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In this paper, we study the Cauchy problem of the classical incompressible Navier--Stokes equations and the parabolic-elliptic Keller--Segel system in the framework of the Fourier--Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier--Besov spaces for the heat equation. Such estimates are fundamental for solving certain PDE's of parabolic type. As an applications, we prove global well-posedness in variable Fourier--Besov spaces for the 3D classical incompressible Navier--Stokes equations and the 3D parabolic-elliptic Keller--Segel system., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:2410.00060

In this paper, we are concerned with the well-posed issues of the fractional dissipative system in the framework of the Fourier--Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier--Besov spaces for the fractional heat equation. Such estimates are fundamental for solving certain dissipative PDE's of fractional type. As an applications, we prove global well-posedness in variable Fourier--Besov spaces for the 3D generalized incompressible Navier--Stokes equations and the 3D fractional Keller--Segel system., Comment: 21 pages. arXiv admin note: text overlap with arXiv:2405.01209

We consider a viscous compressible barotropic flow in the interval $[0,\pi]$ with homogeneous Dirichlet boundary conditions for the flow velocity and a constant rest state as initial data. Given two sufficiently close subintervals $I=[\alpha_1,\alpha_2]$ and $J=[\beta_1,\beta_2]$ of $(0,1)$, a nonempty open set $\omega \subset (1,\pi)$, and $T>0$, we construct an external force $f$ supported in $\omega$ acting on the momentum equation such that the corresponding flow map moves the fluid particles initially occupying $I$ exactly onto $J$ in time $T$., Comment: 16 pages

In this paper, we are mainly concerned with the well-posed problem of the fractional Keller--Segel system in the framework of variable Lebesgue spaces. Based on carefully examining the algebraical structure of the system, we reduced the fractional Keller--Segel system into the generalized nonlinear heat equation to overcome the difficulties caused by the boundedness of the Riesz potential in a variable Lebesgue spaces, then by mixing some structural properties of the variable Lebesgue spaces with the optimal decay estimates of the fractional heat kernel, we were able to establish two well-posedness results of the fractional Keller--Segel system in this functional setting., Comment: arXiv admin note: substantial text overlap with arXiv:2404.05127

In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations. In a first theorem, we prove the existence and uniqueness of global-in-time mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{ \frac{nb}{2\alpha - \langle 1 \rangle_\gamma} } (\mathbb{R}^n,L^\infty([0,T[ ))$. On the other hand, by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$.

In this paper, we are mainly concerned with the well-posedness of the dissipative surface quasi-geostrophic equation in the framework of variable Lebesgue spaces. Based on some analytical results developed in the variable Lebesgue spaces and the $L^{p}$-$L^{q}$ decay estimates of the fractional heat kernel, we establish the local existence and regularity of solutions to the 2D dissipative surface quasi-geostrophic equation in the variable Lebesgue space., Comment: 11 pages

It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters $\gamma_1$ and $\gamma_2$ involved in the non-linear terms verify $ \gamma_1=\frac{\gamma_1}{2}$ or $\gamma_2=0$. In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard) problem. Motivated by this fact, in this work we consider a non-local version of the Kuramoto-Velarde equation. This equation allows us to apply a Fourier-based method and, within the framework $\gamma_2\neq \frac{\gamma_1}{2}$ and $\gamma_2\neq 0$, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm., Comment: 12 pages

In this work we study the 3D Navier-Stokes equations, under the action of an external force and with the fractional Laplacian operator $(-\Delta)^{\alpha}$ in the diffusion term, from the point of view of variable Lebesgue spaces. Based on decay estimates of the fractional heat kernel we prove the existence and uniqueness of mild solutions on this functional setting. Thus, in a first theorem we obtain an unique local-in-time solution in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$. As a bi-product, in a second theorem we prove the existence of an unique global-in-time solution in the mixed-space $\mathcal{L}^{p(\cdot)}_{\frac{3}{2\alpha -1}}(\mathbb{R}^3,L^\infty([0,T[))$.

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.

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. Contents

In this article we study some problems related to the incompressible 3D Navier-Stokes equations from the point of view of Lebesgue spaces of variable exponent. These functional spaces present some particularities that make them quite different from the usual Lebesgue spaces: indeed, some of the most classical tools in analysis are not available in this framework. We will give here some ideas to overcome some of the difficulties that arise in this context in order to obtain different results related to the existence of mild solutions for this evolution problem.

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.

We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device is reformulated as a transmission problem related to the interaction between waves and a fixed partially-immersed structure. By imposing the conservation of the total fluid-OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in \cite{bocchihevergara2021}. We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear 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., Comment: 35 pages, 1 figure

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., Comment: 16 pages

We analyze the dynamics of the COVID-19 epidemic taking into account the role of the unreported cases. After a first section in which we deal with a framework of very slow test capacity, we turn to the model recently introduced/implemented by Liu, Magal, Seydi and Webb. First, we prove some basic structural results for the corresponding ODE, as for instance the convergence of S(t) to a positive limit. These are similar to those of the classical SIR model, although the maxima of the corresponding curves are not necessarily unique. Finally, we implement the model -- but with a variable transmission rate -- in the Chilean context. A key parameter adjustment (namely, the fraction of unreported cases) is done via an argument using mortality rates. We conclude with several conclusions and lines of future research., Comment: 22 pages