Resumen de Geometric function theory in fluid mechanics

Banhirup Sengupta

• català

  Aquesta tesi està dedicada a l’estudi de determinats camps vectorials plans i les propietats rotacionals dels seus fluxos. Hem utilitzat tècniques geomètriques fortes procedents de la Teoria de Funcions Geomètriques per descriure aquests camps vectorials i els seus fluxos corresponents. És ben conegut per la teoria clàssica de Cauchy-Lipschitz que els camps vectorials de Lipschitz produeixen fluxos de Lipschitz. El teorema clàssic de Rademacher-Stepanov mostra que per a cada temps fix el mapa de flux és diferenciable en gairebé tots els punts. A continuació, ve la classe Zygmund, que cau lleugerament per sota de Lipschitz. A diferència dels camps vectorials de Lipschitz, els camps de Zygmund poden no ser diferenciables en cap moment. Per tant, en general, tampoc s’ha d’esperar que el flux d’aquest camp vectorial sigui diferenciable. Es pot pensar en camps vectorials de Sobolev que estan lluny de Lipschitz i Zygmund. Aquests camps vectorials no són continus en general. Di Perna - Lions va demostrar l’existència i la singularitat de solucions a l’equació de transport lineal per a camps vectorials de Sobolev. Entre les conseqüències de la teoria de Di Perna - Lleons, s’obté per a qualsevol camp vectorial de Sobolev l’existència i la singularitat d’un flux ben definit, que consisteix (en qualsevol moment) en automapes mesurables de l’espai, i que en general guanyarà. No seran continus, per tant tampoc seran diferenciables. Hi ha alguns casos particulars en què el flux hereta la suavitat de Sobolev del camp vectorial. Per exemple, aquest és el cas quan el camp vectorial pla apunta cap a una direcció particular tal com mostra Marconi. Malauradament, és molt probable que els camps vectorials que ens interessen no compleixin els supòsits de Marconi, com passa sovint a la mecànica de fluids. Tractant d’evitar qualsevol restricció a la direcció del camp vectorial, Clop-Jylha ha demostrat recentment que hi ha una subclasse de camps vectorials de Sobolev que no són Lipschitz, però encara el seu flux gaudeix d’una certa suavitat de Sobolev per a petits temps. Aquests són camps vectorials amb gradient en classe exponencial. Un altre exemple interessant el va donar Reimann als anys 70. Va demostrar que qualsevol camp vectorial amb diferencial simètric sense traça acotat admet un flux ben definit d’homeomorfismes continus de Holder que resulten ser regulars de Sobolev en qualsevol moment. En general, però, els camps vectorials de Sobolev no donen lloc a fluxos regulars de Sobolev (ni tan sols d’ordre fraccionari) com mostra Chemin. El paràgraf anterior indica clarament l’existència d’una bretxa entre les teories de Cauchy-Lipschitz i Di Perna-Lions. Aquesta tesi abordarà aquells camps vectorials que es troben en aquesta bretxa estreta i inestable. A la primera part de la tesi, hem proporcionat una caracterització puntual de camps vectorials amb divergència limitada i curl. Les solucions de Yudovich a les equacions d’Euler planars són un dels exemples d’aquests camps vectorials. El resultat de Clop-Jylha mostra que durant petits temps els fluxos d’Euler són de fet homeomorfismes de distorsió finita, mentre que Wolibner l’any 33 va demostrar que el flux d’Euler i el seu invers són tots dos continus de Holder. A la segona part, hem millorat els límits de la velocitat d’espiral dels homeomorfismes de distorsió finita donats per Hitruhin, amb el supòsit addicional que tenen Holder continu invers. Hem demostrat l’optimitat d’aquests límits. Com a aplicació, hem donat límits per a la velocitat en espiral dels fluxos d’Euler per a temps petits. En l’última part, hem demostrat l’existència de solucions del sistema d’equacions de Cauchy en el pla. Val la pena esmentar que tant els camps vectorials d’Euler com de Cauchy cauen en el buit que hem comentat en els paràgrafs anteriors.

• English

  This thesis is devoted to the study of certain planar vector fields and the rotational properties of their flows. We have used strong geometric techniques coming from Geometric Function Theory to describe these vector fields and their corresponding flows.

  It is well known from the classical Cauchy-Lipschitz theory that Lipschitz vector fields produce Lipschitz flows. The classical Rademacher-Stepanov Theorem shows that for every fixed time the flow map is differentiable at almost every point. Next, comes the Zygmund class, which falls slightly below Lipschitz. Zygmund vector fields are important because they are the first examples of non-Lipschitz vector fields which produce Holder continuous flows. In contrast to Lipschitz vector fields, Zygmund fields may not be differentiable at any point. So, in genral, one should not expect the flow of such vector field to be differentiable either.

  One can think of Sobolev vector fields which are away from Lipschitz and Zygmund. These vector fields are not continuous in general, when the degree of the Sobolev exponent is less than the dimension of the ambient space. Di Perna - Lions proved existence and uniqueness of solutions to Linear Transport Equationfor Sobolev vector fields. Among the consequences of Di Perna - Lions theory, one obtains for any Sobolev vector field the existence and uniqueness of a well-defined flow, which consists (at any time) of measurable self-maps of the space, and which in general won't be continuous, hence they won't be differentiable either.

  There are some particular instances for which the flow inherits the Sobolev smoothness of the vector field. For instance, this is the case when the planar vector field points towards a particular direction as shown by Marconi. Unfortunately, the vector fields we are interested in will most likely not satisfy Marconi's assumptions, as happens quite often in Fluid Mechanics.

  Trying to avoid any restrictions on the direction of vector field, it was proven recently by Clop-Jylha that there is a subclasss of Sobolev vector fields which are not Lipschitz but still its flow enjoys some Sobolev smoothness for small times. These are vector fields having gradient in exponential class. Another interesting example was given by Reimann in 70s. He proved that any vector field with bounded traceless symmetric differential admits a well defined flow of Holder continuous homeomorphisms which turn out to be Sobolev regular for any time. In general, though, Sobolev vector fields do not give rise to Sobolev regular flows (not even of fractional order) as shown by Chemin.

  The above paragraph clearly indicates the existence of a gap between Cauchy-Lipschitz and Di Perna-Lions theories. This thesis is going to address those vector fields which fall in this narrow and unstable gap.

  In the first part of the thesis, we have provided a pointwise characterization of vector fields with bounded divergence and curl. Yudovich solutions to planar Euler equations are one of the examples of these vector fields. Clop-Jylha result shows that for small times Euler flows are indeed homeomorphisms of finite distortion, whereas Wolibner in '33 proved that Euler flow and its inverse are both Holder continuous.

  In the second part, we have improved the bounds for the rate of spiraling of homeomorphisms of finite distortion given by Hitruhin, with the additional assumption that they have Holder continuous inverse. We proved optimality of these bounds. As an application, we have given bounds for spiraling rate of Euler flows for small times.

  In the last part, we have proved existence of solutions to the Cauchy system of equations in the plane. It is worth mentioning that both the Euler and Cauchy vector fields fall in the gap that we have discussed about in the earlier paragraphs.

• English

  This thesis is devoted to the study of certain planar vector fields and the rotational properties of their flows. We have used strong geometric techniques coming from Geometric Function Theory to describe these vector fields and their corresponding flows. It is well known from the classical Cauchy-Lipschitz theory that Lipschitz vector fields produce Lipschitz flows. The classical Rademacher-Stepanov Theorem shows that for every fixed time the flow map is differentiable at almost every point. Next, comes the Zygmund class, which falls slightly below Lipschitz. Zygmund vector fields are important because they are the first examples of non-Lipschitz vector fields which produce Holder continuous flows. In contrast to Lipschitz vector fields, Zygmund fields may not be differentiable at any point. So, in genral, one should not expect the flow of such vector field to be differentiable either. One can think of Sobolev vector fields which are away from Lipschitz and Zygmund. These vector fields are not continuous in general, when the degree of the Sobolev exponent is less than the dimension of the ambient space. Di Perna - Lions proved existence and uniqueness of solutions to Linear Transport Equationfor Sobolev vector fields. Among the consequences of Di Perna - Lions theory, one obtains for any Sobolev vector field the existence and uniqueness of a well-defined flow, which consists (at any time) of measurable self-maps of the space, and which in general won’t be continuous, hence they won’t be differentiable either. There are some particular instances for which the flow inherits the Sobolev smoothness of the vector field. For instance, this is the case when the planar vector field points towards a particular direction as shown by Marconi. Unfortunately, the vector fields we are interested in will most likely not satisfy Marconi’s assumptions, as happens quite often in Fluid Mechanics. Trying to avoid any restrictions on the direction of vector field, it was proven recently by Clop-Jylha that there is a subclasss of Sobolev vector fields which are not Lipschitz but still its flow enjoys some Sobolev smoothness for small times. These are vector fields having gradient in exponential class. Another interesting example was given by Reimann in 70s. He proved that any vector field with bounded traceless symmetric differential admits a well defined flow of Holder continuous homeomorphisms which turn out to be Sobolev regular for any time. In general, though, Sobolev vector fields do not give rise to Sobolev regular flows (not even of fractional order) as shown by Chemin. The above paragraph clearly indicates the existence of a gap between Cauchy-Lipschitz and Di Perna-Lions theories. This thesis is going to address those vector fields which fall in this narrow and unstable gap. In the first part of the thesis, we have provided a pointwise characterization of vector fields with bounded divergence and curl. Yudovich solutions to planar Euler equations are one of the examples of these vector fields. Clop-Jylha result shows that for small times Euler flows are indeed homeomorphisms of finite distortion, whereas Wolibner in ‘33 proved that Euler flow and its inverse are both Holder continuous. In the second part, we have improved the bounds for the rate of spiraling of homeomorphisms of finite distortion given by Hitruhin, with the additional assumption that they have Holder continuous inverse. We proved optimality of these bounds. As an application, we have given bounds for spiraling rate of Euler flows for small times. In the last part, we have proved existence of solutions to the Cauchy system of equations in the plane. It is worth mentioning that both the Euler and Cauchy vector fields fall in the gap that we have discussed about in the earlier paragraphs.