scholarly journals The Inviscid Limit Behavior for Smooth Solutions of the Boussinesq System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Junlei Zhu
Keyword(s):  
Initial Data ◽  
Convergence Rate ◽  
Convergence Result ◽  
Limit Problem ◽  
Limit Behavior ◽  
Inviscid Limit ◽  
Boussinesq System ◽  
Smooth Solutions ◽  
Viscosity Coefficients

The inviscid limit problem for the smooth solutions of the Boussinesq system is studied in this paper. We prove theHsconvergence result of this system as the diffusion and the viscosity coefficients vanish with the initial data belonging toHs. Moreover, theHsconvergence rate is given if we allow more regularity on the initial data.

The Inviscid Limit of the Fractional Complex Ginzburg–Landau Equation

2016 ◽  
Vol 17 (6) ◽  
pp. 333-341
Author(s):  
Lijun Wang ◽  
Jingna Li ◽  
Li Xia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Convergence Rate ◽  
Schrodinger Equation ◽  
Landau Equation ◽  
Limit Behavior ◽  
Inviscid Limit ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

AbstractIn this paper, the inviscid limit behavior of solution of the fractional complex Ginzburg–Landau (FCGL) equation$${\partial _t}u + (a + i\nu){\Lambda ^{2\alpha}}u + (b + i\mu){\left| u \right|^{2\sigma}}u = 0, \quad (x, t) \in {{\Cal T}^n} \times (0, \infty)$$is considered. It is shown that the solution of the FCGL equation converges to the solution of nonlinear fractional complex Schrödinger equation, while the initial data${u_0}$is taken in${L^2}, $${H^\alpha}$, and${L^{2\sigma + 2}}$as$a,\, b$tends to zero, and the convergence rate is also obtained.

scholarly journals Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

2018 ◽  
Vol 18 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Yan Yang ◽  
Yubin Yan ◽  
Neville J. Ford
Keyword(s):  
Initial Data ◽  
Convergence Rate ◽  
Homogeneous Problem ◽  
Fractional Diffusion ◽  
Nonsmooth Data ◽  
Time Stepping ◽  
Time Discretisation ◽  
Nonsmooth Initial Data ◽  
Densely Defined ◽  
Diffusion Problems

AbstractWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha [18] established an {O(k)} convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator A is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where k denotes the time step size. In this paper, we approximate the Riemann–Liouville fractional derivative by Diethelm’s method (or L1 scheme) and obtain the same time discretisation scheme as in McLean and Mustapha [18]. We first prove that this scheme has also convergence rate {O(k)} with nonsmooth initial data for the homogeneous problem when A is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretisation scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth initial data. Using this new time discretisation scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow

1997 ◽  
Vol 9 (4) ◽  
pp. 876-882 ◽  
Author(s):  
Koji Ohkitani ◽  
Michio Yamada
Keyword(s):  
Limit Behavior ◽  
Inviscid Limit

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

scholarly journals The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1195
Author(s):  
Shu Wang ◽  
Yongxin Wang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Large Amplitude ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Well Posedness ◽  
Spherical Coordinates ◽  
Smooth Solutions ◽  
Euler Systems

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

scholarly journals On the Schrödinger–Boussinesq system with singular initial data

2013 ◽  
Vol 400 (2) ◽  
pp. 487-496 ◽  
Author(s):  
Carlos Banquet ◽  
Lucas C.F. Ferreira ◽  
Elder J. Villamizar-Roa
Keyword(s):  
Initial Data ◽  
Boussinesq System ◽  
Singular Initial Data
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