ANALYSIS AND PARTIAL DERIVATIVE EQUATIONS (Q3190727): Difference between revisions
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(Created claim: summary (P836): THIS RESEARCH PROJECT IS AT THE BORDER BETWEEN ANALYSIS AND PARTIAL DERIVATIVE EQUATIONS. EULER’S EQUATION SHAPES THE EVOLUTION OF AN IDEAL AND INCOMPRESSIBLE FLUID AND HAS BEEN THE SUBJECT OF MANY RESEARCH FROM DIFFERENT POINTS OF VIEW INCLUDING PHYSICS, ENGINEERING AND MATHEMATICS. FROM THE POINT OF VIEW OF MATHEMATICS THAT THE PROBLEM IS WELL PLACED AND REGULARITY HAVE BEEN THE MAIN AXES OF STUDY. IN DIMENSION 2 THE PROBLEM IS WELL UNDERSTOOD...) |
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ANALYSIS AND PARTIAL DERIVATIVE EQUATIONS | |||
Revision as of 21:50, 12 October 2021
Project Q3190727 in Spain
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | ANALYSIS AND PARTIAL DERIVATIVE EQUATIONS |
Project Q3190727 in Spain |
Statements
33,819.5 Euro
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67,639.0 Euro
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50.0 percent
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30 December 2016
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29 June 2021
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UNIVERSIDAD AUTONOMA DE BARCELONA
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41°29'27.71"N, 2°8'15.00"E
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08266
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ESTE PROYECTO DE INVESTIGACION ESTA EN LA FRONTERA ENTRE EL ANALISIS Y LAS ECUACIONES EN DERIVADAS PARCIALES. LA ECUACION DE EULER MODELA LA EVOLUCION DE UN FLUIDO IDEAL E INCOMPRESIBLE Y HA SIDO EL OBJETO DE MUCHAS INVESTIGACIONES DESDE DIFERENTES PUNTOS DE VISTA INCLUYENDO LA FISICA, LA INGENIERIA Y LAS MATEMATICAS. DESDE EL PUNTO DE VISTA DE LA MATEMATICA QUE EL PROBLEMA ESTE BIEN PUESTO Y LA REGULARIDAD HAN SIDO LOS EJES PRINCIPALES DE ESTUDIO. EN DIMENSION 2 EL PROBLEMA ESTA BIEN ENTENDIDO, PERO MUCHAS DIFICULTADES BASICAS APARECEN EN DIMENSION 3. EN DIMENSION DOS SE PUEDE PREGUNTAR POR CUESTIONES MAS PROFUNDAS. LA ECUACION DE EULER PLANAR ES EQUIVALENTE A LA ECUACION DE LA VORTICIDAD, QUE ES UNA ECUACION DE TRANSPORTE QUE ESTABLECE QUE LA VORTICIDAD ES CONSTANTE A LO LARGO DE LAS TRAYECTORIAS. SI A TIEMPO 0 UNO TIENE VORTICIDAD 1 EN UN DOMINIO D_0 Y 0 EN EL COMPLEMENTARIO, ENTONCES LA VORTICIDAD A TIEMPO T ES 1 EN EL DOMINIO D_T Y 0 EN EL COMPLEMENTARIO. CONECER EL ¿VORTEX PATCH¿ D_T RESUELVE LA ECUACION DE LA VORTICIDAD CONSIDERADA. SIMULACIONES NUMERICAS MUESTRAN QUE, CON DOMINIOS INICIALES MUY REGULARES D_T ES UN OBJETO MUY COMPLICADO CUYA FRONTERA EMPEORA DE FORMA CASI CAOTICA. UN TEOREMA DE CHEMIN AFIRMA QUE LA FRONTERA ES REGULAR PARA TODO TIEMPO. HAY DOMINIOS CONCRETOS, LLAMADOS V-STATES, TALES QUE D_T ES UNA ROTACION DEL DOMINIO INICIAL D_0 ALREDEDOR DE SU CENTRO DE MASAS CON VELOCIDAD ANGULAR CONSTANTE. EL UNICO V-STATE SIMPLEMENTE CONEXO CONOCIDO DE FORMA EXPLICITA ES LA ELIPSE Y EL UNICO DOBLEMENTE CONEXO ES EL ANILLO, PERO SE PUEDE DEMOSTRAR, UTILIZANDO TECNICAS DE BIFURCACION QUE EXISTEN MUCHOS OTROS V-STATES, TANTO SIMPLEMENTE CONEXOS COMO DOBLEMENTE CONEXOS. NOSOTROS DESCUBRIMOS RECIENTEMENTE UN HECHO SORPRENDENTE EN RELACION A LOS V-STATES DOBLEMENTE CONEXOS: LA REGION ENTRE DOS ELIPSES ES UN V-STATE SI Y SOLAMENTE SI ES UNA ANILLO. NOS PROPONEMOS CONSTRUIR V-STATES, CON CIERTAS PROPIEDADES SIN UTLIZAR BIFURCACION. _x000D_ LA ECUACION DE LA AGREGACION HA SIDO MUY ESTUDIADA ULTIMAMENTE. ESTA ECUACION DESCRIBE VORTICES DE DENSIDAD EN CIERTOS SUPERCONDUCTORES Y LA QUIMIOTAXIS BACTERIANA. LA ECUACION DE LA AGREGACION ES UNA ECUACION DE FLUJO GRADIENTE. SE PUEDE DEMOSTRAR NUMERICAMENTE QUE LOS ¿PATCHES¿ DE LA AGREGACION COLAPSAN EN TIEMPO 1 A UN CONJUNTO LLAMADO ESQUELETO. NOS GUSTARIA TRATAR EL PROBLEMA DE ENTENDER LA ESTRUCTURA DEL ESQUELETO, EMPEZANDO POSIBLEMENTE CON CUESTIONES SENCILLAS, POR EJEMPLO QUE EL ESQUELETO ES UN CONJUNTO DE AREA 0. TAMBIEN INTENTAREMOS ESTUDIAR LA RECTIFICABILIDAD O SU DIMENSION DE HAUSDORFF._x000D_ UNA TERCERA LINEA DE INVESTIGACION ESTA RELACIONADA CON OPERADORES ELIPTICOS TIPO DIVERGENCIA ASOCIADOS A MATRICES ELIPTICAS A(X). EN EL CASO NO UNIFORMEMENTE ELIPTICO, SI CONSIDERAMOS ECUACIONES DEGENERADAS, UN CAMBIO DE VARIABLES CUASICONFORME TRANSFORMA EL PROBLEMA DEGENERADO EN UN PROBLEMA UNIFORMEMENTE ELIPTICO, QUE DEBE SER RESUELTO EN CIERTOS ESPACIOS L^2 CON PESOS. ESTO NOS LLEVA A LOS ESPACIOS CON OSCILACION MEDIA ACOTADA Y A LOS PESOS DE MUCKENHOUPT DONDE APARECEN NUEVAS CUESTIONES. NOS GUSTARIA RESOLVER ESTAS NUEVAS CUESTIONES Y OBTENER CONCLUSIONES RELACIONADAS CON ECUACIONES ELIPTICAS DEGENERADAS._x000D_ ADEMAS QUEREMOS ESTUDIAR PROPIEDADES DE DIFERENCIABILIDAD DE LOS POTENCIALES DE MEDIDAS FINITAS EN EL SENTIDO DE LA MEDIA CAPACITARIA, CON INTENCION DE PODERLO APLICAR A LAS EDP. (Spanish)
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THIS RESEARCH PROJECT IS AT THE BORDER BETWEEN ANALYSIS AND PARTIAL DERIVATIVE EQUATIONS. EULER’S EQUATION SHAPES THE EVOLUTION OF AN IDEAL AND INCOMPRESSIBLE FLUID AND HAS BEEN THE SUBJECT OF MANY RESEARCH FROM DIFFERENT POINTS OF VIEW INCLUDING PHYSICS, ENGINEERING AND MATHEMATICS. FROM THE POINT OF VIEW OF MATHEMATICS THAT THE PROBLEM IS WELL PLACED AND REGULARITY HAVE BEEN THE MAIN AXES OF STUDY. IN DIMENSION 2 THE PROBLEM IS WELL UNDERSTOOD, BUT MANY BASIC DIFFICULTIES APPEAR IN DIMENSION 3. IN DIMENSION TWO YOU CAN ASK ABOUT DEEPER QUESTIONS. THE EQUATION OF EULER PLANAR IS EQUIVALENT TO THE EQUATION OF VORTEX, WHICH IS AN EQUATION OF TRANSPORT THAT ESTABLISHES THAT VORTEX IS CONSTANT ALONG TRAJECTORIES. IF AT TIME 0 ONE HAS VORTICITY 1 IN A DOMAIN D_0 AND 0 IN THE COMPLEMENTARY, THEN THE VORTEX TO TIME T IS 1 IN THE DOMAIN D_T AND 0 IN THE COMPLEMENTARY. CONCEIVING THE PATCHVORTEX D_T SOLVES THE EQUATION OF THE CONSIDERED VORTEX. NUMERIC SIMULATIONS SHOW THAT, WITH VERY REGULAR INITIAL DOMAINS D_T IS A VERY COMPLICATED OBJECT WHOSE BORDER BECOMES ALMOST CHAOTIC. A CHEMIN THEOREM STATES THAT THE BORDER IS REGULAR FOR ALL TIMES. THERE ARE SPECIFIC DOMAINS, CALLED V-STATES, SUCH THAT D_T IS A ROTATION OF THE INITIAL DOMAIN D_0 AROUND ITS CENTER OF MASSES WITH CONSTANT ANGULAR VELOCITY. THE ONLY V-STATE SIMPLY KNOWN EXPLICITLY IS THE ELLIPSE AND THE ONLY DOUBLY RELATED ONE IS THE RING, BUT IT CAN BE SHOWN, USING BIFURCATION TECHNIQUES, THAT THERE ARE MANY OTHER V-STATES, BOTH SIMPLY RELATED AND DOUBLY RELATED. WE RECENTLY DISCOVERED A SURPRISING FACT IN RELATION TO THE DOUBLY RELATED V-STATES: THE REGION BETWEEN TWO ELLIPSES IS A V-STATE IF AND ONLY IF IT IS A RING. WE INTEND TO BUILD V-STATES, WITH CERTAIN PROPERTIES WITHOUT USING BIFURCATION. _x000D_ the Aggregation Equation HAS VERY STUDY ULTIMATE. THIS EQUATION DESCRIBES DENSITY VORTICES IN CERTAIN SUPERCONDUCTORS AND BACTERIAL CHEMOTAXIS. THE EQUATION OF AGGREGATION IS AN EQUATION OF GRADIENT FLOW. IT CAN BE SHOWN NUMERICALLY THAT THE PATCHES OF THE AGGREGATION COLLAPSE IN TIME 1 TO A SET CALLED SKELETON. WE WOULD LIKE TO DEAL WITH THE PROBLEM OF UNDERSTANDING THE STRUCTURE OF THE SKELETON, POSSIBLY STARTING WITH SIMPLE QUESTIONS, FOR EXAMPLE THAT THE SKELETON IS A SET OF AREA 0. We will also consider the rectability or your dimension of HAUSDORFF._x000D_ A third line of research is related to elliptic OPERATORS TYPE DIVERGENCE ASSOCIED TO MATRICES Elliptic A(X). IN THE NOT UNIFORMLY ELLIPTICAL CASE, IF WE CONSIDER DEGENERATED EQUATIONS, A CHANGE OF QUASI-CONFORMING VARIABLES TRANSFORMS THE DEGENERATED PROBLEM INTO A UNIFORMLY ELLIPTIC PROBLEM, WHICH MUST BE SOLVED IN CERTAIN SPACES L^2 WITH WEIGHTS. THIS LEADS US TO THE SPACES WITH LIMITED MEAN OSCILLATION AND TO THE WEIGHTS OF MUCKENHOUPT WHERE NEW QUESTIONS APPEAR. We would like to take care of these new questions and agree to conclude agreements related to degenerated elliptical ECUs._x000D_ ADEMAS WE WANT TO STUDY PRODUCTS OF Differentiabilities OF Finite Measurements IN THE SETTING OF THE MEDIA, WITH INTENCE OF APPLICATION TO THE EDPs. (English)
12 October 2021
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Cerdanyola del Vallès
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Identifiers
MTM2016-75390-P
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