A linear condition determining local or global existence for nonlinear problems

2013 ◽  
Vol 11 (8) ◽  
Author(s):  
John Neuberger ◽  
John Neuberger ◽  
James Swift
Keyword(s):  
American Mathematical Society ◽  
Local Existence ◽  
Nonlinear Problems ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Local Existence And Uniqueness ◽  
Linear Condition ◽  
Nonlinear Autonomous System ◽  
Necessary And Sufficient ◽  
Future Work

AbstractGiven a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This work is the second part of a program, the first part being [Neuberger J.W., How to distinguish local semigroups from global semigroups, Discrete Contin. Dyn. Syst. (in press), available at http://arxiv.org/abs/1109.2184]. Future work points to a distant goal for problems as in [Fefferman C.L., Existence and Smoothness of the Navier-Stokes Equation, In: The Millennium Prize Problems, Clay Mathematics Institute, Cambridge/American Mathematical Society, Providence, 2006, 57–67].

Strong solutions to the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system

2019 ◽  
Vol 16 (04) ◽  
pp. 701-742 ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Stokes System ◽  
Initial Density ◽  
Local Existence ◽  
Strong Solutions ◽  
Incompressible Fluids ◽  
Navier Stokes ◽  
Initial Value ◽  
Two Phase ◽  
Density Dependent ◽  
Local Existence And Uniqueness

We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local existence and uniqueness of strong solutions to the initial value problem in a bounded domain, when the initial density function enjoys a positive lower bound.

GENESIS OF A SOURCE/SINK LINE INTO A SINGULAR UPDRAFT

1998 ◽  
Vol 08 (03) ◽  
pp. 431-444 ◽  
Author(s):  
JOËL CHASKALOVIC
Keyword(s):  
Mathematical Models ◽  
Existence And Uniqueness ◽  
Vortex Line ◽  
Stokes Equations ◽  
Local Existence ◽  
Fluid Motion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Local Existence And Uniqueness ◽  
Source Sink

Mathematical models applied to tornadoes describe these kinds of flows as an axisymmetric fluid motion which is restricted for not developing a source or a sink near the vortex line. Here, we propose the genesis of a family of a source/sink line into a singular updraft which can modeled one of the step of the genesis of a tornado. This model consists of a three-parameter family of fluid motions, satisfying the steady and incompressible Navier–Stokes equations, which vanish at the ground. We establish the local existence and uniqueness for these fields, at the neighborhood of a nonrotating singular updraft.

THE CLASSICAL INCOMPRESSIBLE NAVIER-STOKES LIMIT OF THE BOLTZMANN EQUATION

1991 ◽  
Vol 01 (02) ◽  
pp. 235-257 ◽  
Author(s):  
CLAUDE BARDOS ◽  
SEIJI UKAI
Keyword(s):  
Boltzmann Equation ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Navier Stokes Equation ◽  
Sharp Estimates ◽  
The Cauchy Problem ◽  
Global Strong Solutions ◽  
Necessary And Sufficient

The convergence hypothesis of Bardos, Golse, and Levermore,1 which leads to the incompressible Navier-Stokes equation as the limit of the scaled Boltzmann equation, is substantiated for the Cauchy problem with initial data small but independent of the Knudsen number ε. The uniform (in ε) existence of global strong solutions and their strong convergence as ε→0 are proved. A necessary and sufficient condition for the uniform convergence up to t=0, which implies the absence of the initial layer, is also established. The proof relies on sharp estimates of the linearized operators, which are obtained by the spectral analysis and the stationary phase method.

APPEARANCE OF A SOURCE/SINK LINE INTO A SWIRLING VORTEX

2003 ◽  
Vol 13 (01) ◽  
pp. 121-142 ◽  
Author(s):  
J. CHASKALOVIC ◽  
A. CHAUVIÈRE
Keyword(s):  
Vortex Line ◽  
Stokes Equations ◽  
Local Existence ◽  
Fluid Motion ◽  
Vertical Shear ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Mature Phase ◽  
Local Existence And Uniqueness ◽  
Source Sink

Several mathematical models applied to tornadoes consist of exact and axisymmetric solutions of the steady and incompressible Navier–Stokes equations. These models studied by Serrin,9 Goldshtik and Shtern8 describe families of fluid motions vanishing at the ground and are restricted not to develop a source nor a sink near the vortex line. Therefore, Serrin showed that the flow patterns of the resulting velocity field may have some realistic characteristics to model the mature phase of the lifetime of a tornado, in comparison with atmospheric observations. On the other hand, no reason has been given to motivate the restriction of the absence of a source/sink vortex line. Therefore, we present here the construction and the analysis of a fluid motion driven by the vertical shear near the ground, the rate of the azimuthal rotation and by the intensity of a central source/sink line. We prove the local existence and uniqueness of a family of fluid motions, leading to the genesis of such source/sink lines inside a non-rotating updraft which does not develop, before perturbation, a source nor a sink.

scholarly journals Strong solutions in the dynamical theory of compressible fluid mixtures

2015 ◽  
Vol 25 (07) ◽  
pp. 1217-1256 ◽  
Author(s):  
Matthias Kotschote ◽  
Rico Zacher
Keyword(s):  
Local Existence ◽  
Strong Solutions ◽  
Dynamical Theory ◽  
Uniqueness Result ◽  
Navier Stokes ◽  
Mixing Ratio ◽  
Strongly Coupled ◽  
Transition Layers ◽  
Fluid Mixtures ◽  
Local Existence And Uniqueness

In this paper we investigate the compressible Navier–Stokes–Cahn–Hilliard equations (the so-called NSCH model) derived by Lowengrub and Truskinovsky. This model describes the flow of a binary compressible mixture; the fluids are supposed to be macroscopically immiscible, but partial mixing is permitted leading to narrow transition layers. The internal structure and macroscopic dynamics of these layers are induced by a Cahn–Hilliard law that the mixing ratio satisfies. The PDE constitute a strongly coupled hyperbolic–parabolic system. We establish a local existence and uniqueness result for strong solutions.

Divergent expansion, Borel summability and three-dimensional Navier–Stokes equation

2008 ◽  
Vol 366 (1876) ◽  
pp. 2775-2788 ◽  
Author(s):  
Ovidiu Costin ◽  
Guo Luo ◽  
Saleh Tanveer
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Local Existence ◽  
Three Dimensional ◽  
Nonlinear Pdes ◽  
Navier Stokes ◽  
Borel Summability ◽  
Numerical Computations ◽  
Navier Stokes Equation ◽  
Existence Proofs ◽  
General Data

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier–Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/ t or some positive power of 1/ t ). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally. Moreover, computation of the solution of the IE over an interval [0, p 0 ] provides sharper control of its p →∞ behaviour. Preliminary numerical computations give encouraging results.

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