Global existence of solutions to resonant system of isentropic gas dynamics

2011 ◽  
Vol 12 (5) ◽  
pp. 2802-2810 ◽  
Author(s):  
Yun-guang Lu
Keyword(s):  
Global Existence ◽  
Gas Dynamics ◽  
Existence Of Solutions ◽  
Resonant System ◽  
Global Existence Of Solutions ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas

scholarly journals Soluciones globales para sistema isotérmico con fuente

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Xian Ting Wang
Keyword(s):  
Global Existence ◽  
Gas Dynamics ◽  
Short Note ◽  
Global Solutions ◽  
Momentum Equation ◽  
Global Existence Of Solutions ◽  
Existence And Stability ◽  
Isothermal System ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas

In this short note, we are concerned with the global existence of solutions to the isothermal system with source, where the inhomogeneous terms f(x,t,ρ,u)=b(x,t)ρ+(a′(x)/a(x))*ρu^2+α(x,t)ρu|u| are appeared in the momentum equation. Our work extended the results in the previous papers “Resonance for the Isothermal System of Isentropic Gas Dynamics” (Proc. A.M.S.139(2011),2821-2826), “Global Existence and Stability to the Polytropic Gas Dynamics with an Outer Force” (Appl. Math. Let-ters, 95(2019), 35-40) and “Existence of Global Solutions for Isentropic GasFlow with Friction” (Nonlinearity, 33(2020), 3940-3969), where the global solution was obtained for the source f(x,t,ρ,u)=(a′(x)/a(x))*ρu^2, f(x,t,ρ,u)=b(x,t)ρ, f(x,t,ρ,u)=α(x,t)ρu|u| respectively.

Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit

1996 ◽  
Vol 126 (6) ◽  
pp. 1309-1340 ◽  
Author(s):  
M. Slemrod
Keyword(s):  
Equation Of State ◽  
Gas Dynamics ◽  
Existence Of Solutions ◽  
Piston Problem ◽  
Search For Self ◽  
Isentropic Gas Dynamics ◽  
Self Similar ◽  
Isentropic Gas ◽  
Image Position ◽  
Viscous Limit

This paper proves the existence of solutions to the spherical piston problem for isentropic gas dynamics with equation of state p(p) = Apγ, γ≧ 1. The method of analysis is to replace the usual viscosity ε with εt, thus permitting a search for self-similar viscous limits. The main result of the paper is that self-similar viscous limits are proved to exist and converge to a solution to the piston problem whenN = 1, 2, 3 space dimensions.

scholarly journals Global Existence of Solutions for a Nonstrictly Hyperbolic System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
De-yin Zheng ◽  
Yun-guang Lu ◽  
Guo-qiang Song ◽  
Xue-zhou Lu
Keyword(s):  
Global Existence ◽  
Existence Of Solutions ◽  
Bounded Function ◽  
Compensated Compactness ◽  
Resonant System ◽  
Existence Of Weak Solutions ◽  
Global Existence Of Solutions ◽  
Flux Approximation ◽  
The Cauchy Problem ◽  
Uniformly Bounded

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly boundedL∞estimatesz(ρδ,ε,uδ,ε)≤B(x)andw(ρδ,ε,uδ,ε)≤βwhena(x)is increasing (similarly,w(ρδ,ε,uδ,ε)≤B(x)andz(ρδ,ε,uδ,ε)≤βwhena(x)is decreasing) for theε-viscosity andδ-flux approximation solutions of nonhomogeneous, resonant system without the restrictionz0(x)≤0orw0(x)≤0as given in Klingenberg and Lu (1997), wherezandware Riemann invariants of nonhomogeneous, resonant system;B(x)>0is a uniformly bounded function ofxdepending only on the functiona(x)given in nonhomogeneous, resonant system, andβis the bound ofB(x). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

scholarly journals Global existence of solutions for a multi-phase flow: A drop in a gas-tube

2016 ◽  
Vol 13 (02) ◽  
pp. 381-415
Author(s):  
Debora Amadori ◽  
Paolo Baiti ◽  
Andrea Corli ◽  
Edda Dal Santo
Keyword(s):  
Global Existence ◽  
Existence Of Solutions ◽  
Inviscid Fluid ◽  
Phase Flow ◽  
Lagrangian Coordinates ◽  
Large Initial Data ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
Multi Phase Flow ◽  
Multi Phase

In this paper we study the flow of an inviscid fluid composed by three different phases. The model is a simple hyperbolic system of three conservation laws, in Lagrangian coordinates, where the phase interfaces are stationary. Our main result concerns the global existence of weak entropic solutions to the initial-value problem for large initial data.

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