We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ɛΔ3. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ɛ > 0.
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We show global existence for a class of models of fluids that change their properties depending on the concentration of a chemical. We allow that the stress tensor in (t, x) depends on the velocity and concentration at other points and times. The example we have in mind foremost are materials with memory.
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The effects of rainfall speed, VT, and meridional/vertical moisture gradients on the meridional moisture transport are examined in the context of mid-latitude baroclinic waves. These effects are investigated in an idealized model that can be solved analytically. The model is systematically derived in a precipitating quasi-geostrophic limit, starting from a moist atmospheric model with minimal representation of cloud microphysics. Single-phase dynamics are considered, with a comparison of three cases: unsaturated, saturated with VT = 0, and saturated with VT > 0. The Eady problem for linear baroclinic waves is analyzed, with modifications to incorporate moisture. As a preliminary step, the moist waves are shown to have properties consistent with prior studies, including larger growth rates and smaller spatial scales in the saturated cases in comparison to the classic dry Eady problem. Then, in addition, it is shown that the meridional moisture flux, as a function of height, has a mid-tropospheric maximum in the case of VT = 0, and a maximum in the lower troposphere or at the surface for sufficiently large values of VT. These results for different VT values are discussed in the context of meridional moisture transport in observational data.
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We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ϕ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.
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We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.
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In this paper we prove the existence of a global φ-attractor in the weak topology of the natural phase space for the family of multi-valued processes generated by solutions of a nonautonomous modified 3D Bénard system in unbounded domains for which Poincaré inequality takes place.
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