For periodic initial data with density allowed to vanish initially, we establish the global existence of strong and weak solutions to the two-dimensional barotropic compressible Navier–Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is λ =ρ^β with β>4/3. These results generalize and improve the previous ones due to Vaigant–Kazhikhov [Sib. Math. J. 36 (1995) 1283–1316] who required β>3. Moreover, we also prove that the densities for both the strong and weak solutions remain bounded from above independently of time. As a consequence, it is shown that both the strong and weak solutions converge to the equilibrium state as time tends to infinity.

We successfully perform the three-dimensional tracking in a turbulent fluid flow of small axisymmetrical particles that are neutrally-buoyant and bottom-heavy, i.e., they have a non-homogeneous mass distribution along their symmetry axis. We experimentally show how a tiny mass inhomogeneity can affect the particle orientation along the preferred vertical direction and modify its tumbling rate. The experiment is complemented by a series of simulations based on realistic Navier-Stokes turbulence and on a point-like particle model that is capable to explore the full range of parameter space characterized by the gravitational torque stability number and by the particle aspect ratio. We propose a theoretical perturbative prediction valid in the high bottom-heaviness regime that agrees well with the observed preferential orientation and tumbling rate of the particles. We also show that the heavy-tail shape of the probability distribution function of the tumbling rate is weakly affected by the bottom-heaviness of the particles.

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It is shown that if a function$u$satisfies a backward parabolic inequality in an open set Ω∉$R$^{$n$+1}and vanishes to infinite order at a point ($x$_{0}·$t$_{0}) in Ω, then$u(x, t$_{0})=0 for all$x$in the connected component of$x$_{0}in Ω⌢($R$^{$n$}×{$t$_{0}}).

We study the resonances of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.