59 research outputs found
Integrability and regularity of 3D Euler and equations for uniformly rotating fluids
AbstractWe consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ(t, x1, x2) + V(t, x1, x2, x3) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical 3). Here r is a remainder of order Ro12a where Roa = (H0U0(Щ0L20) is the anisotropic Rossby number (H0—height, L0—horizontal length scale, Щ0—angular velocity of background rotation, U0—horizontal velocity scale); Roa = (H0L0)Ro where H0L0 is the aspect ratio and Ro = U0(Щ0L0) is a Rossby number based on the horizontal length scale L0. The vector field V(t, x1, x2, x3) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Щ0t, τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e3 and velocity Ū3(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x1, x2, x3) and 2D turbulence for vertically averaged fields Ū(t, x1, x2) if the anisotropic Rossby number Roa is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T∗ of regular solutions to 3D Euler equations (T∗ → +∞, as Roa → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case
Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model
The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental
models that describe plasma turbulence. The model also appears as a simplified
reduced Rayleigh-B\'enard convection model. The mathematical analysis the
Hasegawa-Mima equation is challenging due to the absence of any smoothing
viscous terms, as well as to the presence of an analogue of the vortex
stretching terms. In this paper, we introduce and study a model which is
inspired by the inviscid Hasegawa-Mima model, which we call a
pseudo-Hasegawa-Mima model. The introduced model is easier to investigate
analytically than the original inviscid Hasegawa-Mima model, as it has a nicer
mathematical structure. The resemblance between this model and the Euler
equations of inviscid incompressible fluids inspired us to adapt the techniques
and ideas introduced for the two-dimensional and the three-dimensional Euler
equations to prove the global existence and uniqueness of solutions for our
model. Moreover, we prove the continuous dependence on initial data of
solutions for the pseudo-Hasegawa-Mima model. These are the first results on
existence and uniqueness of solutions for a model that is related to the
three-dimensional inviscid Hasegawa-Mima equations
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
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Attractors for the Kuramoto-Sivashinsky equations
In this paper, we address the question of constructing an upper bound of the Hausdorff and Fractal dimensions d/sub H/(x) and d/sub F/(x) for attractors X of the kuramoto-Sivanshinsky equation
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Traveling-wave solutions to reaction-diffusion systems modeling combustion
We consider the deflagration-wave problem for a compressible reacting gas, with species involved in a single-step chemical reaction. In the limit of small Mach numbers, the one-dimensional traveling-wave problem reduces to a system of reaction-diffusion equations. Thermomechanical coefficients are temperature-dependent. Existence is proved by first considering the problem in a bounded domain, and taking an infinite-domain limit. In the singular limit of high activation energy in the Arrhenius exponential term, we prove strong convergence to a limiting free-boundary problem (discontinuity of the derivatives on the free boundary)
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