Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.Comment: 35 pages. Final revised version. To appear in Arch. Ration. Mech. Ana

On admissibility criteria for weak solutions of the Euler equations

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.Comment: 33 pages, 1 figure; v2: 35 pages, corrected typos, clarified proof

Stable ground states for the relativistic gravitational Vlasov-Poisson system

We consider the three dimensional gravitational Vlasov-Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers --which is mostly unknown-- but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved

Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients

In this paper we give an affirmative answer to an open question mentioned in [Le Bris and Lions, Comm. Partial Differential Equations 33 (2008), 1272--1317], that is, we prove the well-posedness of the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.Comment: 11 pages. The proof has been modifie

A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations

In this paper, we obtain a blow up criterion for classical solutions to the 3-D compressible Naiver-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, initial vacuum is allowed in our case.Comment: 25 page

Recent Developments in Understanding Two-dimensional Turbulence and the Nastrom-Gage Spectrum

Two-dimensional turbulence appears to be a more formidable problem than three-dimensional turbulence despite the numerical advantage of working with one less dimension. In the present paper we review recent numerical investigations of the phenomenology of two-dimensional turbulence as well as recent theoretical breakthroughs by various leading researchers. We also review efforts to reconcile the observed energy spectrum of the atmosphere (the spectrum) with the predictions of two-dimensional turbulence and quasi-geostrophic turbulence.Comment: Invited review; accepted by J. Low Temp. Phys.; Proceedings for Warwick Turbulence Symposium Workshop on Universal features in turbulence: from quantum to cosmological scales, 200

Global solutions to the three-dimensional full compressible magnetohydrodynamic flows

The equations of the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows are considered in a bounded domain. The viscosity coefficients and heat conductivity can depend on the temperature. A solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data is established

The Vlasov limit and its fluctuations for a system of particles which interact by means of a wave field

In two recent publications [Commun. PDE, vol.22, p.307--335 (1997), Commun. Math. Phys., vol.203, p.1--19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a wave type generalization of the Newtonian gravity potential, coupled in a regularized way. In the present paper the many-body dynamics of this model is studied. The Vlasov continuum limit is obtained in form equivalent to a weak law of large numbers. We also establish a central limit theorem for the fluctuations around this limit.Comment: 68 pages. Smaller corrections: two inequalities in sections 3 and two inequalities in section 4, and definition of a Banach space in appendix A1. Presentation of LLN and CLT in section 4.3 improved. Notation improve

A mechanism for the inhibition of DNA-PK-mediated DNA sensing by a virus

The innate immune system is critical in the response to infection by pathogens and it is activated by pattern recognition receptors (PRRs) binding to pathogen associated molecular patterns (PAMPs). During viral infection, the direct recognition of the viral nucleic acids, such as the genomes of DNA viruses, is very important for activation of innate immunity. Recently, DNA-dependent protein kinase (DNA-PK), a heterotrimeric complex consisting of the Ku70/Ku80 heterodimer and the catalytic subunit DNA-PKcs was identified as a cytoplasmic PRR for DNA that is important for the innate immune response to intracellular DNA and DNA virus infection. Here we show that vaccinia virus (VACV) has evolved to inhibit this function of DNA-PK by expression of a highly conserved protein called C16, which was known to contribute to virulence but by an unknown mechanism. Data presented show that C16 binds directly to the Ku heterodimer and thereby inhibits the innate immune response to DNA in fibroblasts, characterised by the decreased production of cytokines and chemokines. Mechanistically, C16 acts by blocking DNA-PK binding to DNA, which correlates with reduced DNA-PK-dependent DNA sensing. The C-terminal region of C16 is sufficient for binding Ku and this activity is conserved in the variola virus (VARV) orthologue of C16. In contrast, deletion of 5 amino acids in this domain is enough to knockout this function from the attenuated vaccine strain modified vaccinia virus Ankara (MVA). In vivo a VACV mutant lacking C16 induced higher levels of cytokines and chemokines early after infection compared to control viruses, confirming the role of this virulence factor in attenuating the innate immune response. Overall this study describes the inhibition of DNA-PK-dependent DNA sensing by a poxvirus protein, adding to the evidence that DNA-PK is a critical component of innate immunity to DNA viruses

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure