The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

2017 ◽  
Vol 147 (6) ◽  
pp. 1243-1277 ◽  
Author(s):  
Eiji Onodera
Keyword(s):  
Fourth Order ◽  
Energy Method ◽  
Initial Value Problem ◽  
Local Existence ◽  
Three Dimensional ◽  
Closed Curve ◽  
Initial Value ◽  
Curve Flow ◽  
Local Existence And Uniqueness ◽  
Loss Of Derivatives

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

scholarly journals A Singular Initial-Value Problem for Second-Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Second Order ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Local Existence And Uniqueness ◽  
Singular Initial Value Problem

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

scholarly journals Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev space

1997 ◽  
Vol 40 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Nakao Hayashi ◽  
Hitoshi Hirata
Keyword(s):  
Sobolev Space ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Initial Value ◽  
Hyperbolic Case ◽  
Local Existence And Uniqueness ◽  
Usual Sobolev Space

We study the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness H5/2 with a condition such that the L2 norm of the data is sufficiently small.

MATHEMATICAL ANALYSIS OF A REACTIVE-DIFFUSIVE MODEL OF THE DISPERSAL OF A CHEMICAL TRACER WITH NONLINEAR CONVECTION

1995 ◽  
Vol 05 (01) ◽  
pp. 29-46 ◽  
Author(s):  
STEVE COHN ◽  
J. DAVID LOGAN
Keyword(s):  
Initial Value Problem ◽  
Local Existence ◽  
Heteroclinic Orbit ◽  
Traveling Wave Solutions ◽  
Sobolev Embedding ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Contraction Mapping Theorem ◽  
Weakly Nonlinear ◽  
Local Existence And Uniqueness

We formulate and analyze a nonlinear reaction-convection-diffusion system that models the dispersal of solutes, or chemical tracers, through a one-dimensional porous medium. A similar set of model equations also arises in a weakly nonlinear limit of the combustion equations. In particular, we address two fundamental questions with respect to the model system: first, the existence of wavefront type traveling wave solutions, and second, the local existence and uniqueness of solutions to the pure initial value problem. The solution to the wavefront problem is obtained by showing the existence of a heteroclinic orbit in a two-dimensional phase space. The existence argument for the initial value problem is based on the contraction mapping theorem and Sobolev embedding. In the final section we prove non-negativity of the solution.

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.

scholarly journals A PROBABILISTIC REPRESENTATION FOR THE VORTICITY OF A THREE-DIMENSIONAL VISCOUS FLUID AND FOR GENERAL SYSTEMS OF PARABOLIC EQUATIONS

2005 ◽  
Vol 48 (2) ◽  
pp. 295-336 ◽  
Author(s):  
Barbara Busnello ◽  
Franco Flandoli ◽  
Marco Romito
Keyword(s):  
Viscous Fluid ◽  
Parabolic Equations ◽  
Stokes Equations ◽  
Local Existence ◽  
Three Dimensional ◽  
Lagrangian Formalism ◽  
Probabilistic Representation ◽  
Local Existence And Uniqueness ◽  
Linear Parabolic Equations ◽  
General Systems

AbstractA probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

scholarly journals Revisiting the characteristic initial value problem for the vacuum Einstein field equations

2020 ◽  
Vol 52 (10) ◽  
Author(s):  
David Hilditch ◽  
Juan A. Valiente Kroon ◽  
Peng Zhao
Keyword(s):  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Initial Value ◽  
Bootstrap Argument ◽  
Null Hypersurfaces ◽  
Characteristic Initial Value Problem ◽  
Local Existence Of Solutions

AbstractUsing the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

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