For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.

It is known that the maximum diameter for the rupture-risk assessment of the abdominal aortic aneurysm is a generally good method, but not sufficient. Alternative features obtained with computational modeling may provide additional useful criteria. Though computational approaches are noninvasive, they are often time-consuming because of the high computational complexity. In this paper, we present a highly parallel algorithm for the numerical simulation of unsteady blood flows in the patient-specific abdominal aorta. We model the blood flow with the unsteady incompressible Navier-Stokes equations, and solve the discretized system with a highly scalable domain decomposition method. With this approach, the complete flow field can be obtained in less than an hour, instead of days with older methods. We show experimentally that the proposed method offers accurate solutions of the pressure, the velocity and the wall shear stress, and the parallel efficiency is higher than 70% on a parallel computer with more than 1,000 processor cores.

We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains and establish a unified proof that works for different regimes of hole-cell ratios, which is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and yields correctors and error estimates for vanishing hole-cell ratios. For a positive volume fraction of holes, the approach is just the standard oscillating test function method; for a vanishing volume fraction of holes, we study asymptotic behaviors of properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.

The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D*. Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D*, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.

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