16 research outputs found
Maximal regularity to the Stokes Problem with Navier boundary conditions
We prove in this paper some results on the complex and fractional powers of
the Stokes operator with slip frictionless boundary conditions involving the
stress tensor. This is fundamental and plays an important role in the
associated parabolic problem and will be used to prove maximal
regularity results for the non-homogeneous Stokes problem
Semi-group theory for the Stokes operator with Navier-type boundary conditions on -spaces
In this article we consider the Stokes problem with Navier-type boundary
conditions on a domain , not necessarily simply connected. Since under
these conditions the Stokes problem has a non trivial kernel, we also study the
solutions lying in the orthogonal of that kernel. We prove the analyticity of
several semigroups generated by the Stokes operator considered in different
functional spaces. We obtain strong, weak and very weak solutions for the time
dependent Stokes problem with the Navier-type boundary condition under
different hypothesis on the initial data and external force
. Then, we study the fractional and pure imaginary powers of
several operators related with our Stokes operators. Using the fractional
powers, we prove maximal regularity results for the homogeneous Stokes problem.
On the other hand, using the boundedness of the pure imaginary powers we deduce
maximal regularity for the inhomogeneous Stokes problem
Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum
We discuss in this short note the local-in-time strong well-posedness of the
compressible Navier-Stokes system for non-Newtonian fluids on the three
dimensional torus. We show that the result established recently by Kalousek,
V\'{a}clav, and Ne\v{c}asova in [\doi{10.1007/s00208-021-02301-8}] can be
extended to the case where vanishing density is allowed initially. Our proof
builds on the framework developed by Cho, Choe, and Kim in
[\doi{10.1016/j.matpur.2003.11.004}] for compressible Navier-Stokes equations
in the case of Newtonian fluids. To adapt their method, special attention is
given to the elliptic regularity of a challenging nonlinear elliptic system. We
show particular results in this direction, however, the main result of this
paper is proven in the general case when elliptic regularity is imposed as an
assumption. Also, we give a finite time blow-up criterion.Comment: 13 page
Semi-group theory for the Stokes and Navier-Stokes equations with Navier-type boundary conditions
Cette thĂšse est consacrĂ©e Ă l'Ă©tude thĂ©orique mathĂ©matique des Ă©quations de Stokes et de Navier-Stokes dans un domaine bornĂ© de R^3 en utilisant la thĂ©orie des semi-groupes. Trois diffĂ©rents types de conditions seront considĂ©rĂ©s : des conditions aux limites de Navier, de type-Navier et des conditions qui dĂ©pendent de la pression. Ce manuscrit est composĂ© de six chapitres. Tout d'abord nous commençons par un Ă©tat de l'art sur les Ă©quations de Navier-Stokes. Ensuite nous dĂ©montrons l'analyticitĂ© du semi-groupe de Stokes avec chacune des conditions ci-dessus. Ceci permet de rĂ©soudre le problĂšme d'Ă©volution en utilisant la thĂ©orie des semi-groupes. Nous Ă©tudions Ă©galement les puissances complexes et fractionnaires de l'opĂ©rateur de Stokes pour lesquelles nous dĂ©montrons certaines propriĂ©tĂ©s et estimations. Ces rĂ©sultats seront utilisĂ©s dans la suite pour obtenir des estimations de type L^p-L^q pour le semi-groupe de Stokes, un rĂ©sultat de rĂ©gularitĂ© L^p-L^q maximale pour le problĂšme de Stokes inhomogĂšne et des rĂ©sultats d'existence et d'unicitĂ© locale pour le problĂšme non-linĂ©aire. AprĂšs nous Ă©tudions le problĂšme d'Ă©volution de Stokes. Outre la rĂ©gularitĂ© L^p-L^q maximale, nous dĂ©montrons l'existence des solutions faibles uâL^q (0,T; W^(1,p) (Ω)), fortes uâL^q (0,T; W^(2,p) (Ω)) et trĂšs faibles uâL^q (0,T; L^p (Ω)) du problĂšme de Stokes. On termine par l'Ă©tude du problĂšme de Navier-Stokes avec chacune des conditions aux limites citĂ©es ci-dessus. Tout d'abord, en utilisant les estimations L^p-L^q on dĂ©montre l'existence d'une unique solution locale u qui vĂ©rifieuâBC([0,T_0 ); L_(Ï,Ï)^p (Ω))â©L^q (0,T_0; L_(Ï,Ï)^r (Ω)), q,r>p, 2/q+3/r=3/p.De plus, pour une donnĂ©e initiale petite, on obtient l'existence globale des solutions. Ensuite en estimant le terme non-linĂ©aire en fonction des puissances fractionnaires de l'opĂ©rateur de Stokes on dĂ©montre la rĂ©gularitĂ© 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 uâL^q (0,T; W^(1,p) (Ω)), strong uâL^q (0,T; W^(2,p) (Ω)) and very weak uâL^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 uâBC([0,T_0 ); L_(Ï,Ï)^p (Ω))â©L^q (0,T_0; L_(Ï,Ï)^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
Note on the problem of dissipative measure-valued solutions to the compressible non-Newtonian system
We introduce a dissipative measure-valued solution to the compressible non-Newtonian system. We generalized a result given by NovotnĂœ, NeÄasovĂĄ [14]. We derive a relative entropy inequality for measure-valued solution as an extension of the classical entropy inequality introduced by Dafermos [2], Mellet-Vasseur [11], Feireisl-Jin-NovotnĂœ [5]