scholarly journals Local mild solutions to three‐dimensional magnetohydrodynamic system in Morrey spaces

2020 ◽  
Author(s):  
Zirong Zeng
Keyword(s):  
Mild Solutions ◽  
Three Dimensional ◽  
Morrey Spaces ◽  
Magnetohydrodynamic System

On the motion of a rigid body with a cavity filled with a viscous liquid

2012 ◽  
Vol 142 (2) ◽  
pp. 391-423 ◽  
Author(s):  
Ana L. Silvestre ◽  
Takéo Takahashi
Keyword(s):  
Rigid Body ◽  
Viscous Liquid ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Mild Solutions ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Coupled System ◽  
Analytic Approach ◽  
Navier Stokes

We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier–Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Kato's scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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