Weak and Strong Solutions of the Navier-Stokes Initial Value Problem

Giga, Yoshikazu

doi:10.2977/prims/1195182014

Cited by 35 publications

(20 citation statements)

References 25 publications

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“…To reflect this invariance one assigns scaling dimensions as given in [9], [21]. For example we assign dimension 2 to time variable and dimension 1 to spatial variable.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

Giga

Saal

2012

J. Math. Fluid Mech.

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ABSTRACT. In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.

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“…To reflect this invariance one assigns scaling dimensions as given in [9], [21]. For example we assign dimension 2 to time variable and dimension 1 to spatial variable.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

Giga

Saal

2012

J. Math. Fluid Mech.

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“…Under some conditions on the initial data and the external force, the result was extended to a global solution. A generalization in terms of theory can be found in [23,24]. In the last decades, existence and uniqueness results of solutions to the stochastic Navier-Stokes equations have been studied extensively.…”

Section: Introductionmentioning

confidence: 99%

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Benner

Trautwein

2019

Mathematische Nachrichten

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We consider the controlled stochastic Navier-Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier-Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists. K E Y W O R D SLévy process, local mild solution, stochastic Navier-Stokes equations, stochastic optimal control M S C ( 2 0 1 0 ) 93E20, 49J20, 35Q30 1444

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“…Many methods for these equations to find weak solutions have been developed [4][5][6], but still is not clear that such systems have unique solutions. There was proved in [7,8] that if there exists a classical solution in a connected subset of ´[ ] t T , 3 0 then it is also a Leray-Hopf weak solution [4,9,10]. It is also proved that if there exists a Leray-Hopf weak solutions in ´[ ] t T , 3 0 , it is a unique solution.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for restricted incompressible Navier–Stokes equations with Dirichlet boundary conditions

García-Casado

2019

J. Phys. Commun.

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This paper exposes how to obtain a relation that have to be hold for all free-divergence velocity fields that evolve according to Navier-Stokes equations. However, checking the violation of this relation requires a huge computational effort. To circumvent this problem it is proposed an additional ansatz to free-divergence Navier-Stokes fields. This makes available six degrees of freedom which can be tuned. When they are tuned adequately, it is possible to find finite L 2 norms of the velocity field for volumes of  3 and for Î ¥. In particular, the kinetic energy of the system is bounded when the field components u i are class C 3 functions on ´¥ [ ) t , 3 0 that hold Dirichlet boundary conditions. This additional relation lets us conclude that Navier-Stokes equations with no-slip boundary conditions have not unique solution. Moreover, under a given external force the kinetic energy can be computed exactly as a funtion of time.

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Weak and Strong Solutions of the Navier-Stokes Initial Value Problem

Cited by 35 publications

References 25 publications

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Exact solutions for restricted incompressible Navier–Stokes equations with Dirichlet boundary conditions

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