scholarly journals Instability of the abstract Rayleigh–Taylor problem and applications

2020 ◽  
Vol 30 (12) ◽  
pp. 2299-2388 ◽  
Author(s):  
Fei Jiang ◽  
Song Jiang ◽  
Weicheng Zhan
Keyword(s):  
Initial Data ◽  
Stokes Problem ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Existence Theory ◽  
Lagrangian Coordinates ◽  
Compatibility Conditions ◽  
Unstable Solutions ◽  
Taylor Problem ◽  
Proper Values

Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [Formula: see text]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates.

scholarly journals Global well-posedness for the three-dimensional incompressible viscous non-resistive MHD equations in an infinite slab

2021 ◽  
Author(s):  
Youyi Zhao
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Mhd Equations ◽  
Well Posedness ◽  
Lagrangian Coordinates ◽  
Mathematical Approach ◽  
Compatibility Conditions ◽  
Finite Height ◽  
Infinite Slab ◽  
Mhd System

In this paper, we investigate the global well-posedness of the system of incompressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no nonlinear compatibility conditions on the initial data.

scholarly journals Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Xin Liu ◽  
Edriss S. Titi
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Physical Vacuum ◽  
Well Posedness ◽  
Small Initial Data ◽  
Anelastic Equations

Abstract We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

PARAXIAL APPROXIMATION OF THE VLASOV-MAXWELL SYSTEM: LAMINAR BEAMS

1994 ◽  
Vol 04 (02) ◽  
pp. 203-221 ◽  
Author(s):  
A. NOURI
Keyword(s):  
Electromagnetic Field ◽  
Charged Particle ◽  
Initial Data ◽  
Global Existence ◽  
Local Existence ◽  
Sufficient Conditions ◽  
External Electromagnetic Field ◽  
Evolution System ◽  
Lagrangian Coordinates ◽  
Maxwell System

The Vlasov-Maxwell stationary system for charged particle laminar beams is studied with a paraxial model of approximation. It leads to a degenerate evolution system, which local existence is proved. Then, using lagrangian coordinates, with sufficient conditions on the initial data and the external electromagnetic field, it is shown that global existence is possible.

scholarly journals Method of semidiscretization in time for quasilinearintegrodifferential equations

2004 ◽  
Vol 2004 (9) ◽  
pp. 469-478 ◽  
Author(s):  
D. Bahuguna ◽  
Reeta Shukla
Keyword(s):  
Banach Space ◽  
Initial Data ◽  
Continuous Dependence ◽  
Reflexive Banach Space ◽  
Strong Solutions ◽  
Integrodifferential Equations

We consider a class of quasilinear integrodifferential equations in a reflexive Banach space. We apply the method of semidiscretization in time to establish the existence, uniqueness, and continuous dependence on the initial data of strong solutions.

scholarly journals Energy of wave and solutions of C-Holm equation under Lagrangian point of view

2018 ◽  
Vol 6 (11) ◽  
Author(s):  
Shkelqim Hajrulla ◽  
L Bezati ◽  
F Hoxha
Keyword(s):  
Energy Density ◽  
Initial Data ◽  
Coordinate Transformation ◽  
Radon Measure ◽  
Continuous Semigroup ◽  
Point Of View ◽  
Lagrangian Point ◽  
Lagrangian Coordinates ◽  
Positive Radon Measure ◽  
Holm Equation

     Abstract: We deal with the Camassa-Holm equation   possesses a global continuous semigroup of weak conservative solutions for initial data. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure µ with . The total energy is preserved by the solution.

Solutions with Singular Initial Data for a Model of Electrophoretic Separation

1990 ◽  
Vol 33 (1) ◽  
pp. 3-10 ◽  
Author(s):  
Joel D. Avrin
Keyword(s):  
Initial Data ◽  
Diffusion Coefficients ◽  
Existence Result ◽  
Strong Solutions ◽  
Approximate Solutions ◽  
Electrophoretic Separation ◽  
Small Diffusion ◽  
Singular Initial Data ◽  
Global Strong Solutions ◽  
Global Existence Result

AbstractUnique global strong solutions of a Cauchy problem arising in electrophoretic separation are constructed with arbitrary initial data in L1, thus generalizing an earlier global existence result. For small diffusion coefficients, the solutions can be viewed as approximate solutions for the corresponding zero-diffusion Riemann problem.

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