On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids

2006 ◽  
Vol 29 (16) ◽  
pp. 1883-1906 ◽  
Author(s):  
Joachim Naumann
Keyword(s):  
Weak Solutions ◽  
Viscous Fluids ◽  
Stationary Motion ◽  
Heat Conducting ◽  
Existence Of Weak Solutions ◽  
Incompressible Viscous Fluids

scholarly journals Global Existence of the Three Dimensional Heat-conductive Incompressible Viscous Fluids

2018 ◽  
Vol 179 ◽  
pp. 01001
Author(s):  
Jie Wu
Keyword(s):  
Cauchy Problem ◽  
Heat Conductivity ◽  
A Priori Estimates ◽  
Three Dimensional ◽  
Global Solutions ◽  
Local Solution ◽  
Viscous Fluids ◽  
Heat Conducting ◽  
The Cauchy Problem ◽  
Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heatconducting incompressible viscous fluids in ℝ3. About the heat-conducting incompressible viscous fluids, there are many mathematical researchers study the variants systems when the viscosity and heat-conductivity coefficient are positive. For the heat-conductive system, it is difficulty to get the better regularity due to the gradient of velocity of fluid own the higher order term. It is hard to control it. In order to get its global solutions, we must obtain the a priori estimates at first, then using fixed point theorem, it need the mapping is contracted. We can get a local solution, then applying the criteria extension. We can extend the local solution to the global solutions. For the two dimensional case, the Gagliardo-Nirenberg interpolation inequality makes use of better than the three dimensional situation. Thus, our problem will become more difficulty to handle. In this paper, we assume the coefficient of viscosity is a constant and the coefficient of heat-conductivity satisfying some suitable conditions. We show that the Cauchy problem has a global-in-time strong solution (u,θ) on ℝ3 ×(0, ∞).

On weak solutions to a dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases

2020 ◽  
Vol 30 (08) ◽  
pp. 1517-1553
Author(s):  
Young-Sam Kwon ◽  
Antonin Novotny ◽  
C. H. Arthur Cheng
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Open Problem ◽  
Type System ◽  
Convergent Subsequence ◽  
Original System ◽  
Heat Conducting ◽  
Existence Of Weak Solutions ◽  
Essential Step ◽  
Mathematical Literature

In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential step to the proof of the existence of weak solutions. This is the first result of this type in the mathematical literature. Nevertheless, the construction of weak solutions to this system however remains still an (difficult) open problem.

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