Strong solutions of the Boussinesq system in exterior domains

Analysis ◽  
2015 ◽  
Vol 35 (3) ◽  
Author(s):  
Christian Komo
Keyword(s):  
Weak Solution ◽  
Strong Solution ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Boussinesq Equations ◽  
Boussinesq System ◽  
Uniqueness Criterion ◽  
Serrin’S Condition ◽  
Instationary Boussinesq Equations

AbstractWe consider the instationary Boussinesq equations in a smooth three-dimensional exterior domain. A strong solution is a weak solution such that the velocity field additionally satisfies Serrin's condition. The crucial point in this concept of a strong solution is the fact that we have required no additional integrability condition for the temperature. We present a sufficient criterion for the existence of such a strong solution. Further we will characterize the class of initial values that allow the existence of such a strong solution in a sufficiently small interval. Finally, we will obtain a uniqueness criterion for weak solutions of the Boussinesq equations which is based on the identification of a weak solution with a strong solution.

scholarly journals The Cauchy problem and decay rates for strong solutions of a Boussinesq system

2005 ◽  
Vol 2005 (7) ◽  
pp. 757-766 ◽  
Author(s):  
Francisco Guillén González ◽  
Márcio Santos da Rocha ◽  
Marko Rojas Medar
Keyword(s):  
Heat Transfer ◽  
Cauchy Problem ◽  
Viscous Fluid ◽  
Strong Solutions ◽  
Boussinesq Equations ◽  
Decay Rates ◽  
Boussinesq System ◽  
Three Dimension ◽  
The Cauchy Problem ◽  
Derivatives Of

The Boussinesq equations describe the motion of an incompressible viscous fluid subject to convective heat transfer. Decay rates of derivatives of solutions of the three-dimension-al Cauchy problem for a Boussinesq system are studied in this work.

HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

2015 ◽  
Vol 10 (3) ◽  
pp. 2825-2833
Author(s):  
Achala Nargund ◽  
R Madhusudhan ◽  
S B Sathyanarayana
Keyword(s):  
Homotopy Analysis Method ◽  
Degrees Of Freedom ◽  
Initial Approximation ◽  
Boussinesq Equations ◽  
Convergent Series ◽  
Boussinesq System ◽  
Analysis Method ◽  
Analytical Technique ◽  
Homotopy Analysis ◽  
Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysis method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-Sykes plot for the region of convergence. We have applied Pade for the HAM series to identify the singularity and reflect it in the graph. The HAM is a analytical technique which is used to solve non-linear problems to generate a convergent series. HAM gives complete freedom to choose the initial approximation of the solution, it is the auxiliary parameter h which gives us a convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

The eruptive regime of mass-transfer-driven Rayleigh–Marangoni convection

2016 ◽  
Vol 791 ◽  
Author(s):  
Thomas Köllner ◽  
Karin Schwarzenberger ◽  
Kerstin Eckert ◽  
Thomas Boeck
Keyword(s):  
Concentration Distribution ◽  
Marangoni Convection ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Navier Stokes ◽  
Flow Structures ◽  
Rayleigh Convection ◽  
Marangoni Effects ◽  
Validation Experiments

The transfer of an alcohol, 2-propanol, from an aqueous to an organic phase causes convection due to density differences (Rayleigh convection) and interfacial tension gradients (Marangoni convection). The coupling of the two types of convection leads to short-lived flow structures called eruptions, which were reported in several previous experimental studies. To unravel the mechanism underlying these patterns, three-dimensional direct numerical simulations and corresponding validation experiments were carried out and compared with each other. In the simulations, the Navier–Stokes–Boussinesq equations were solved with a plane interface that couples the two layers including solutal Marangoni effects. Our simulations show excellent agreement with the experimentally observed patterns. On this basis, the origin of the eruptions is explained by a two-step process in which Rayleigh convection continuously produces a concentration distribution that triggers an opposing Marangoni flow.

scholarly journals The Yang-Mills heat equation with finite action in three dimensions

2022 ◽  
Vol 275 (1349) ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Three Dimensions ◽  
Dimensional Manifold ◽  
Uniqueness Of Solutions ◽  
Reconstruction Procedure ◽  
Content Type ◽  
Yang Mills ◽  
Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

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