Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

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On non-resistive limit of 1D MHD equations with no vacuum at infinity

Advances in Nonlinear Analysis ◽

10.1515/anona-2021-0209 ◽

2021 ◽

Vol 11 (1) ◽

pp. 702-725

Author(s):

Zilai Li ◽

Huaqiao Wang ◽

Yulin Ye

Keyword(s):

Cauchy Problem ◽

A Priori ◽

Strong Solutions ◽

Mhd Equations ◽

Well Posedness ◽

Large Initial Data ◽

One Dimensional ◽

The Cauchy Problem ◽

Global Strong Solutions ◽

The One

Abstract In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

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Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation

Mathematical Notes ◽

10.1134/s0001434619110130 ◽

2019 ◽

Vol 106 (5-6) ◽

pp. 800-813

Author(s):

S. A. Sergeev

Keyword(s):

Cauchy Problem ◽

Wave Equation ◽

Finite Difference ◽

Initial Data ◽

Difference Scheme ◽

One Dimensional ◽

The Cauchy Problem ◽

The One ◽

Localized Initial Data ◽

Dimensional Wave

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Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-01957-z ◽

2021 ◽

Vol 60 (2) ◽

Author(s):

Xin Zhong

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Heat Conducting ◽

Well Posedness ◽

Magnetohydrodynamic Equations ◽

Large Initial Data ◽

Nonhomogeneous Heat

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On global weak solutions to the Cauchy problem for the Navier–Stokes equations with largeL3-initial data

Nonlinear Analysis ◽

10.1016/j.na.2016.01.018 ◽

2017 ◽

Vol 154 ◽

pp. 269-296 ◽

Cited By ~ 9

Author(s):

G. Seregin ◽

V. Šverák

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Weak Solutions ◽

Stokes Equations ◽

Navier Stokes ◽

Global Weak Solutions ◽

Navier Stokes Equations ◽

The Cauchy Problem

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Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data

Journal of Differential Equations ◽

10.1016/0022-0396(92)90042-l ◽

1992 ◽

Vol 95 (1) ◽

pp. 33-74 ◽

Cited By ~ 59

Author(s):

David Hoff

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Stokes Equations ◽

Navier Stokes ◽

Well Posedness ◽

Navier Stokes Equations ◽

Discontinuous Initial Data ◽

The Cauchy Problem

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On the Cauchy problem for the one-dimensional heat equation

Mathematics of Computation ◽

10.1090/s0025-5718-1963-0162064-8 ◽

1963 ◽

Vol 17 (83) ◽

pp. 257-257 ◽

Cited By ~ 19

Author(s):

F. Ginsberg

Keyword(s):

Cauchy Problem ◽

Heat Equation ◽

One Dimensional ◽

The Cauchy Problem ◽

The One

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Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2019292 ◽

2019 ◽

Vol 39 (11) ◽

pp. 6713-6745 ◽

Cited By ~ 2

Author(s):

Xin Zhong ◽

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Boussinesq Equations ◽

Well Posedness ◽

Two Dimensional ◽

Large Initial Data ◽

Density Dependent ◽

The Cauchy Problem

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Well-posedness of the linear heat equation with a second order memory term

ITM Web of Conferences ◽

10.1051/itmconf/20203403011 ◽

2020 ◽

Vol 34 ◽

pp. 03011

Author(s):

Constantin Niţă ◽

Laurenţiu Emanuel Temereancă

Keyword(s):

Initial Data ◽

Heat Equation ◽

Unique Solution ◽

Source Term ◽

Memory Term ◽

Well Posedness ◽

Dimensional Torus ◽

One Dimensional ◽

Linear Heat ◽

The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

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Global Analysis for Rough Solutions to the Davey-Stewartson System

Abstract and Applied Analysis ◽

10.1155/2012/578701 ◽

2012 ◽

Vol 2012 ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

Han Yang ◽

Xiaoming Fan ◽

Shihui Zhu

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Nonlinear Equation ◽

Global Solution ◽

Global Analysis ◽

Space Time ◽

Well Posedness ◽

The Cauchy Problem ◽

New Space ◽

Time Bounds

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

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