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The Navier-Stokes equation driven by heat conduction is studied. It is proven that if the driving force is small then the solutions of the Navier-Stokes equation are ultimately regular. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl numer close to one, we prove the ultimate existence and regularity of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal $\mathcal B$-attractor. Examples of simple $\mathcal B$-attractors from pattern formation are given and a method to study their instabilities proposed.
The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5
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We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.
Varistor ceramics are very heterogeneous nonlinear conductors, used in devices to protect electrical equipment against voltage surges in power lines. The fine structure in the material induces highly oscillating coefficients in the elliptic electrostatic equation as well as in the Maxwell equations. We suggest how the properties of ceramic varistors can be simulated by solving the homogenized problems, i.e. the corresponding homogenized elliptic problem and the homogenized Maxwell equations. The fine scales in the model yield local equations coupled with the global homogenized equations. Lower and upper bounds are also given for the overall electric conductivity of varistor ceramics. These two bounds are associated with two types of failures in varistor ceramics. The upper bound corresponds to thermal heating and the puncture failure due to localization of strong currents. The lower bound corresponds to fracturing of the varistor, due to charge build up at the grain boundaries resulting in stress caused by the piezoelectric property of the varistor.
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