Submitted by MARCELO ADRIANO DOS SANTOS BONGARTI null (bongartimarcelo@yahoo.com.br) on 2016-03-28T13:33:27Z No. of bitstreams: 1 Dissertação Final Final_Marcelo Bongarti.pdf: 945971 bytes, checksum: da9a2828878f6d2c197507ad078d999d (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-03-29T13:51:32Z (GMT) No. of bitstreams: 1 bongarti_mas_me_sjrp.pdf: 945971 bytes, checksum: da9a2828878f6d2c197507ad078d999d (MD5) / Made available in DSpace on 2016-03-29T13:51:32Z (GMT). No. of bitstreams: 1 bongarti_mas_me_sjrp.pdf: 945971 bytes, checksum: da9a2828878f6d2c197507ad078d999d (MD5) Previous issue date: 2016-02-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Sejam X um espaço de Banach,\alpha um número complexo tal que Re\alpha > 0 e A um operador linear fechado, não negativo, com domínio e imagem em X. O objetivo deste trabalho é definir o objeto A^\alpha de modo que as propriedades de potência de números complexos sejam preservadas, ou seja, (i) A ^\alpha A^\beta = A^(\alpha+\beta) ; (aditividade) (ii) A^1 = A; (iii) (A^\alpha )^\beta = A (quando o primeiro membro faz sentido). Como aplicação da teoria, caracterizamos o dom ínio da potência fracionária de um operador de nido matricialmente a partir da seguinte Equação Diferencial Parcial abstrata em espaço de Hilbert, prototipo utilizado para modelar sistemas elásticos com forte (ou estrutural) amortecimento: x '' + A^\alpha x' + Ax = 0; 0 < \alpha <= 1; com A sendo um operador positivo e autoadjunto. / Let X be a Banach space, \alpha a complex number such that Re \alpha > 0 and A a non-negative closed linear operator with domain and range in X. The purpose of this work is to de fine the object A^\alpha in a way that the properties of powers of complex numbers be preserved, i.e, (i) A ^\alpha A^\beta = A^(\alpha+\beta) ; (additivity) (ii) A^1 = A; (iii) (A^\alpha )^\beta = A (when the fi rst member makes sense). As an application of theory, we characterized the domain of fractional power of a matrix-valued operator from the abstract Partial Di erential Equation in Hilbert space, prototype used to model elastic systems with strong/structural damping: x'' + A^\alpha x' + Ax = 0; 0<\alpha <= 1; with A being a positive self-adjoint operator.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T15:40:30Z No. of bitstreams: 1 arquivototal.pdf: 1244185 bytes, checksum: d60955ede563305b1f641dd53d7154a7 (MD5) / Made available in DSpace on 2017-08-17T15:40:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1244185 bytes, checksum: d60955ede563305b1f641dd53d7154a7 (MD5) Previous issue date: 2017-03-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thisworkwestudytheasymptoticbehaviorofthesolutionsofpartialdi erential equations slowlynon-dissipative,wewillgiveabriefhistoricaloverviewonthetheme, wewillpresentanintroductiontothetheoryofsemigroupsofboundedlinearoperators in Banachspaces,fractionalpowersofsectoroperatorsandresultsontheexistenceand uniqueness ofsolutionsofabstractCauchysemilinearproblemsoftheparabolictype. / Neste trabalhoestudamosocomportamentoassint oticodassolu c~oesdeequa c~oes diferenciais parciaislentamenten~aodissipativas,daremosumbreveapanhadohist orico sobre otema,apresentaremosumaintrodu c~ao ateoriadesemigruposdeoperadores lineares limitadosemespa cosdeBanach,pot^enciasfracion ariasdeoperadoressetoriais e resultadossobreexist^enciaeunicidadedesolu c~oesdeproblemasabstratosdeCauchy semilineares dotipoparab olico.

Cette thèse est consacrée à l'étude théorique mathématique des équations de Stokes et de Navier-Stokes dans un domaine borné de R^3 en utilisant la théorie des semi-groupes. Trois différents types de conditions seront considérés : des conditions aux limites de Navier, de type-Navier et des conditions qui dépendent de la pression. Ce manuscrit est composé de six chapitres. Tout d'abord nous commençons par un état de l'art sur les équations de Navier-Stokes. Ensuite nous démontrons l'analyticité du semi-groupe de Stokes avec chacune des conditions ci-dessus. Ceci permet de résoudre le problème d'évolution en utilisant la théorie des semi-groupes. Nous étudions également les puissances complexes et fractionnaires de l'opérateur de Stokes pour lesquelles nous démontrons certaines propriétés et estimations. Ces résultats seront utilisés dans la suite pour obtenir des estimations de type L^p-L^q pour le semi-groupe de Stokes, un résultat de régularité L^p-L^q maximale pour le problème de Stokes inhomogène et des résultats d'existence et d'unicité locale pour le problème non-linéaire. Après nous étudions le problème d'évolution de Stokes. Outre la régularité L^p-L^q maximale, nous démontrons l'existence des solutions faibles u∈L^q (0,T; W^(1,p) (Ω)), fortes u∈L^q (0,T; W^(2,p) (Ω)) et très faibles u∈L^q (0,T; L^p (Ω)) du problème de Stokes. On termine par l'étude du problème de Navier-Stokes avec chacune des conditions aux limites citées ci-dessus. Tout d'abord, en utilisant les estimations L^p-L^q on démontre l'existence d'une unique solution locale u qui vérifieu∈BC([0,T_0 ); L_(σ,τ)^p (Ω))∩L^q (0,T_0; L_(σ,τ)^r (Ω)), q,r>p, 2/q+3/r=3/p.De plus, pour une donnée initiale petite, on obtient l'existence globale des solutions. Ensuite en estimant le terme non-linéaire en fonction des puissances fractionnaires de l'opérateur de Stokes on démontre la régularité de la solution. / This thesis is devoted to the mathematical theoretical study of the Stokes and Navier-Stokes equations in a bounded domain of R^3 using the semi-group theory. Three different types of boundary conditions will be considered: Navier boundary conditions, Navier-type boundary conditions and boundary condition involving the pressure. This manuscript contains six chapters. We prove first the analyticity of the Stokes semi-group with each of the boundary conditions stated above. This allows us to solve the time dependent Stokes problem using the semi-group theory. We will study also the complex and fractional powers of the Stokes operator for which we prove some properties and estimations. These results will be used in the sequel to prove an estimate of type L^p-L^q for the Stokes semigroup, as well as the maximal L^p-L^q regularity for the inhomogeneous Stokes problem and an existence result for the non-linear problem. Next we study the time dependent Stokes problem, besides the maximal L^p-L^q regularity, we prove the existence of weak u∈L^q (0,T; W^(1,p) (Ω)), strong u∈L^q (0,T; W^(2,p) (Ω)) and very weak u∈L^q (0,T; L^p (Ω)) solutions to the Stokes problem. We end with the study of the Navier-Stokes problem. First using the L^p-L^q estimate for the Stokes semi-group we prove the existence of a unique local in time mild solution for the Navier-Stokes problem that verifies u∈BC([0,T_0 ); L_(σ,τ)^p (Ω))∩L^q (0,T_0; L_(σ,τ)^r (Ω)), q,r>p, 2/q+3/r=3/p.Furthermore, for some initial data the solution is global in time. Finally, by estimating the non-linear term as a function of the fractional powers of the Stokes operator we prove that the solution is regular.

<p>Doktorska disertacija je posvećena primeni teorije polugrupa operatora na re&scaron;avanje dve klase Cauchy-jevih početnih problema. U prvom delu smo<br />ispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom koji<br />generi&scaron;e <em>C</em>₀&minus;polugrupu i linearnim ograničenim operatorom kombinovanim<br />sa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-It&ocirc;-ovom<br />haos ekspanzijom. Dokazali smo postojanje i jedinstvenost re&scaron;enja ove klase<br />SPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod po<br />vremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepene<br /><em>C</em>-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnim<br />prostorima. Kompleksne stepene operatora smo posmatrali kao integralne<br />generatore uniformno ograničenih analitičkih <em>C</em>-regularizovanih rezolventnih<br />familija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema vi&scaron;3eg ili necelog reda.</p> / <p>The doctoral dissertation is devoted to applications of the theory<br />of semigroups of operators on two classes of Cauchy problems. In the first<br />part, we studied parabolic stochastic partial differential equations (SPDEs),<br />driven by two types of operators: one linear closed operator generating a<br /><em>C</em>₀&minus;semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-It&ocirc;<br />chaos expansions. We proved existence and uniqueness of solutions for this<br />class of SPDEs. In particular, we also treated the stationary case when the<br />time-derivative is equal to zero. In the second part, we constructed com-plex powers of <em>C</em>&minus;sectorial operators in the setting of sequentially complete<br />locally convex spaces. We considered these complex powers as the integral<br />generators of equicontinuous analytic <em>C</em>&minus;regularized resolvent families, and<br />incorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.</p>