163 research outputs found

    On the well-posedness theory of compressible Navier-Stokes system and related topics.

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    Yu, Rongfeng.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 57-63).Abstracts in English and Chinese.Introduction --- p.3Chapter 1 --- Preliminaries --- p.11Chapter 1.1 --- Notations and function spaces --- p.11Chapter 1.2 --- Some useful inequalities --- p.12Chapter 1.3 --- Fundamental lemmas --- p.15Chapter 2 --- Compressible Navier-Stokes Equations for Quantum Fluids --- p.16Chapter 2.1 --- Background --- p.17Chapter 2.2 --- Derivation of model --- p.17Chapter 3 --- Global Weak Solutions to Barotropic Navier-Stokes Equations for Quantum Fluids --- p.22Chapter 3.1 --- Reformulation and main results --- p.23Chapter 3.2 --- Construction of approximate solutions --- p.27Chapter 3.3 --- A priori estimates --- p.39Chapter 3.4 --- Proof of Theorem 3.1.6 --- p.40Chapter 4 --- Global Existence and Large Time Behavior of Weak Solutions to Quantum Navier-Stokes-Poisson Equa-tions --- p.46Chapter 4.1 --- Global existence of weak solutions --- p.47Chapter 4.2 --- Large time behavior --- p.50Chapter 5 --- Discussions and Future Work --- p.55Bibliography --- p.5

    Evolution equations in physical chemistry

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    textWe analyze a number of systems of evolution equations that arise in the study of physical chemistry. First we discuss the well-posedness of a system of mixing compressible barotropic multicomponent flows. We discuss the regularity of these variational solutions, their existence and uniqueness, and we analyze the emergence of a novel type of entropy that is derived for the system of equations. Next we present a numerical scheme, in the form of a discontinuous Galerkin (DG) finite element method, to model this compressible barotropic multifluid. We find that the DG method provides stable and accurate solutions to our system, and that further, these solutions are energy consistent; which is to say that they satisfy the classical entropy of the system in addition to an additional integral inequality. We discuss the initial-boundary problem and the existence of weak entropy at the boundaries. Next we extend these results to include more complicated transport properties (i.e. mass diffusion), where exotic acoustic and chemical inlets are explicitly shown. We continue by developing a mixed method discontinuous Galerkin finite element method to model quantum hydrodynamic fluids, which emerge in the study of chemical and molecular dynamics. These solutions are solved in the conservation form, or Eulerian frame, and show a notable scale invariance which makes them particularly attractive for high dimensional calculations. Finally we implement a wide class of chemical reactors using an adapted discontinuous Galerkin finite element scheme, where reaction terms are analytically integrated locally in time. We show that these solutions, both in stationary and in flow reactors, show remarkable stability, accuracy and consistency.Chemistry and Biochemistr

    A dynamo driven by zonal jets at the upper surface: Applications to giant planets

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    We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 1000. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.Comment: published in Icaru

    Large-Time Behavior of the 2D Compressible Navier-Stokes System in Bounded Domains with Large Data and Vacuum

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    The large time behavior of the unique strong solution to the barotropic compressible Navier-Stokes system is studied with large external forces and initial data, where the shear viscosity is a positive constant and the bulk one is proportional to a power of the density. Some uniform estimates on the Lp-norm of the density are established, and then deduce that the density converges to its steady state in Lp-spaces, which transforms the large external force into a small one in some sense. Moreover, to deal with the obstacles brought by boundary, the conformal mapping and the pull back Green function are applied to give a point-wise representation of the effective viscous flux, and then make use of slip boundary conditions to cancel out the singularity.Comment: 55 page
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