RECORD
One-Dimensional Compressible Heat-Conducting Gas with Temperature-Dependent Viscosity
Analysis of PDEs · Submitted 20 May 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $|\alpha|$ is sufficiently small.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature $\theta$ and are both proportional to $h(v)\theta^{\alpha}$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $|\alpha|$ is sufficiently small. Our result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature $\theta$ and are both proportional to $h(v)\theta^{\alpha}$ for certain non-degenerate smooth function $h$. | background 0.75 | 0.40 | 0.03 | asserted 0.88 |
| s1 | We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $|\alpha|$ is sufficiently small. | result 0.95 | 0.97 | 0.87 | asserted 1.00 |
| s2 | Our result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature. | contribution 0.45 | 0.82 | 0.29 | hedged possibility 0.89 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature $\theta$ and are both proportional to $h(v)\theta^{\alpha}$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $|\alpha|$ is sufficiently small. Our result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.
Human review
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites
Builds on 1 paper
-
It is an interesting and difficult problem to extend the results in [ref] and [ref] to the non-isentropic case with transport coefficients satisfying [ref] for nonzero $α$.
Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D · Introduction · 0.90
Cites as background 1 paper
-
The existence of global strong solutions to the one-dimensional compressible Navier–Stokes equations for isentropic flows has been established in [ref] with the viscosity $μ$ given by [ref]–[ref] for $ℓ_1=0$, $0≤ℓ_2<12$, and $α=0$, and also in [ref] for the shallow water system, where the viscosity $μ$ satisfies [ref] with $h(v)=v^-1$ and $α=0$.
Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D · Introduction · 0.81
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.