Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density

We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. Various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded. Another longstanding problem is whether a rigorous proof could be provided for the inviscid limit of the multidimensional compressible Navier-Stokes to Euler equations with large initial data. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit even when the total initial-energy is unbounded.Comment: 57 page

Global solutions of the compressible euler-poisson equations with large initial data of spherical symmetry

We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self-consistent gravitational field for gaseous stars. A fundamental unsolved problem is whether the density of the global solution forms a delta measure (i.e., concentration) at the origin. To solve this problem, we develop a new approach for the construction of approximate solutions as the solutions of an appropriately formulated free boundary problem for the compressible Navier-Stokes-Poisson equations with a carefully adapted class of degenerate density-dependent viscosity terms, so that a rigorous convergence proof of the approximate solutions to the corresponding global solution of the compressible Euler-Poisson equations with large initial data of spherical symmetry can be obtained. Even though the density may blow up near the origin at a certain time, it is proved that no delta measure (i.e., concentration) in space-time is formed in the vanishing viscosity limit for the finite-energy solutions of the compressible Euler-Poisson equations for both gaseous stars and plasmas in the physical regimes under consideration

Global Solutions of the Compressible Euler-Poisson Equations with Large Initial Data of Spherical Symmetry

We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self-consistent gravitational field for gaseous stars. A fundamental unsolved problem is whether the density of the global solution forms concentration to become a delta measure at the origin. To solve this problem, we develop a new approach for the construction of approximate solutions as the solutions of an appropriately formulated free boundary problem for the compressible Navier-Stokes-Poisson equations with a carefully adapted class of degenerate density-dependent viscosity terms, so that a rigorous convergence proof of the approximate solutions to the corresponding global solution of the compressible Euler-Poisson equations with large initial data of spherical symmetry can be obtained. Even though the density may blow up near the origin at certain time, it is proved that no concentration (delta measure) is formed in the vanishing viscosity limit for the finite-energy solutions of the compressible Euler-Poisson equations for both gaseous stars and plasmas in the physical regimes under consideration.Comment: 58 page

Global Solutions of the Compressible Euler-Poisson Equations for Plasma with Doping Profile for Large Initial Data of Spherical Symmetry

We establish the global-in-time existence of solutions of finite relative-energy for the multidimensional compressible Euler-Poisson equations for plasma with doping profile for large initial data of spherical symmetry. Both the total initial energy and the initial mass are allowed to be {\it unbounded}, and the doping profile is allowed to be of large variation. This is achieved by adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the inviscid limit of global weak solutions of the Navier-Stokes-Poisson equations with the density-dependent viscosity terms to the corresponding global solutions of the Euler-Poisson equations for plasma with doping profile can be established. New difficulties arise when tackling the non-zero varied doping profile, which have been overcome by establishing some novel estimates for the electric field terms so that the neutrality assumption on the initial data is avoided. In particular, we prove that no concentration is formed in the inviscid limit for the finite relative-energy solutions of the compressible Euler-Poisson equations with large doping profiles in plasma physics.Comment: 42 page

Temperature Effects on the Unsaturated Permeability of the Densely Compacted GMZ01 Bentonite under Confined Conditions

International audienceIn this study, temperature controlled soil-water retention tests and unsaturated hydraulic conductivity tests for densely compacted Gaomiaozi bentonite - GMZ01 (dry density of 1.70 Mg/m3) were performed under confined conditions. Relevant soil-water retention curves (SWRCs) and unsaturated hydraulic conductivities of GMZ01 at temperatures of 40°C and 60°C were obtained. Based on these results as well as the previously obtained results at 20°C, the influence of temperature on water-retention properties and unsaturated hydraulic conductivity of the densely compacted Gaomiaozi bentonite were investigated. It was observed that: (i) water retention capacity decreases as temperature increases, and the influence of temperature depends on suction; (ii) for all the temperatures tested, the unsaturated hydraulic conductivity decreases slightly in the initial stage of hydration; the value of the hydraulic conductivity becomes constant as hydration progresses and finally, the permeability increases rapidly with suction decreases as saturation is approached; (iii) under confined conditions, the hydraulic conductivity increases as temperature increases, at a decreasing rate with temperature rise. It was also observed that the influence of temperature on the hydraulic conductivity is quite suction-dependent. At high suctions (s > 60 MPa), the temperature effect is mainly due to its influence on water viscosity; by contrast, in the range of low suctions (s < 60 MPa), the temperature effect is related to both the water viscosity and the macro-pores closing phenomenon that is supposed to be temperature dependent

Global finite-energy solutions of the compressible Euler–Poisson equations for general pressure laws with large initial data of spherical symmetry

We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler– Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier– Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in L p, and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum (ρ = 0) and far-field (ρ = ∞) are carefully analyzed. Owing to the generality of the pressure law, only the W−1,p loc -compactness of weak entropy dissipation measures with p ∈ [1, 2) can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the L p compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features

Changes of mitochondrial pathway in hypoxia/reoxygenation induced cardiomyocytes apoptosis.

The role of mitochondrial apoptotic pathway in cardiomyocytes subjected to hypoxia/reoxygenation(H/R) was studied. Cultured cardiomyocytes from neonatal Sprague-Dawley rats were exposed to hyoxia/reoxygenation, the apoptotic cardiomyocytes were stained with Annexin-V-FITC, Hoechst 33342 and TUNEL assay. Mitochondrial transmembrane potential of cardiomyocytes was assessed by JC-1 under fluorescence microscope, the expressions of bcl-2, bax, cytochrome c, apoptosis-induced factor (AIF), and caspase-3 were tested by western-blot. Our data showed apoptosis of cardiomyocytes was significantly increased during H/R, accompanied by translocation of bax to mitochondria, release of cytochrome c and AIF to cytosol. The results indicate that the mitochondrial-mediated apoptotic pathway is initiated as a result of H/R