An introduction to Fourier-Besov spaces and a rescaled approach for the tridimensional Boussinesq system
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Date
2024
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Universidad Nacional de Trujillo
Abstract
En este trabajo se abordaron dos tópicos matemáticos. El primer tópico es aquel relacionado al estudio de los espacios de Fourier-Besov, es decir, su definición y algunas de sus propiedades principales. El segundo tópico trata del estudio del sistema de Boussinesq tridimensional, en el cual se probó la buena colocación global para el sistema de Boussinesq reescalado tridimensional asociado, con los parámetros de viscosidad y difusividad ambos positivos, en el contexto de los espacios de Fourier-Besov, que contienen funciones homogéneas con grado negativo. El abordaje por reescalamiento implica en reescalar ambos, la velocidad, dividiéndolo por un parámetro positivo, y la temperatura, dividiéndolo por el cuadrado del mismo parámetro, y estudiar el sistema resultante. Este abordaje permite lidiar con el término lineal en el lado derecho del sistema de Boussinesq con la finalidad de aplicar un lema de punto fijo, y también permite conocer cualitativamente el comportamiento del sistema tomando en cuenta la relación entre los dos parámetros y la velocidad inicial y temperatura inicial; por ejemplo, es posible considerar, para viscosidad suficientemente pequeña y difusividad grande, una norma de Fourier-Besov grande para la temperatura y, para difusividad suficientemente pequeña una norma de Fourier-Besov grande para la velocidad y la temperatura
Abstract This work deals with two mathematical topics. The first topic is related to the study of Fourier-Besov spaces, that is, their definition and some of their main properties. The second one deals with the tridimensional Boussinesq system in which is proved a global wellposedness result for the associated tridimensional rescaled Boussinesq system, with positive full viscosity and diffusivity parameters, in the framework of critical Fourier-Besov spaces, which allow homogeneous functions with negative degree. The rescaled approach implies to rescale both the velocity, dividing by a positive parameter, and the temperature, dividing by the square of the same parameter, and study the obtained system. This rescaled approach permits to deal with the right hand linear term for the Boussinesq system, in order to apply a fixed point lemma, and to know a qualitative behaviour of the system, according to relations between both the parameters and the initial velocity and temperature; for instance, it is possible to consider, for small enough viscosity and large diffusivity, a large enough critical Fourier-Besov norm for the initial temperature and it is also possible to consider, for small enough diffusivity and large viscosity, a large enough critical Fourier-Besov norm for both the velocity and the temperature
Abstract This work deals with two mathematical topics. The first topic is related to the study of Fourier-Besov spaces, that is, their definition and some of their main properties. The second one deals with the tridimensional Boussinesq system in which is proved a global wellposedness result for the associated tridimensional rescaled Boussinesq system, with positive full viscosity and diffusivity parameters, in the framework of critical Fourier-Besov spaces, which allow homogeneous functions with negative degree. The rescaled approach implies to rescale both the velocity, dividing by a positive parameter, and the temperature, dividing by the square of the same parameter, and study the obtained system. This rescaled approach permits to deal with the right hand linear term for the Boussinesq system, in order to apply a fixed point lemma, and to know a qualitative behaviour of the system, according to relations between both the parameters and the initial velocity and temperature; for instance, it is possible to consider, for small enough viscosity and large diffusivity, a large enough critical Fourier-Besov norm for the initial temperature and it is also possible to consider, for small enough diffusivity and large viscosity, a large enough critical Fourier-Besov norm for both the velocity and the temperature
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Keywords
Boussinesq system; Rescaled approach; Homogeneous Fourier-Besov spaces; Global existence of solutions