Non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid

Jianwei Dong; Junhui Zhu; Litao Zhang

Displaying similar documents to “Non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid”

Global existence of smooth solutions for the compressible viscous fluid flow with radiation in $ℝ^{3}$

Hyejong O, Hakho Hong, Jongsung Kim (2023)

Applications of Mathematics

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This paper is concerned with the 3-D Cauchy problem for the compressible viscous fluid flow taking into account the radiation effect. For more general gases including ideal polytropic gas, we prove that there exists a unique smooth solutions in $[0, \infty)$ , provided that the initial perturbations are small. Moreover, the time decay rates of the global solutions are obtained for higher-order spatial derivatives of density, velocity, temperature, and the radiative heat flux.

Global existence of solutions for incompressible magnetohydrodynamic equations

Wisam Alame, W. M. Zajączkowski (2004)

Applicationes Mathematicae

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Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_{p}^{2, 1} (Ω \times (0, T))$ and the pressure q satisfies $\nabla q \in L_{p} (Ω \times (0, T))$ for p ≥ 7/3.

The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

Igor Rodnianski, Jared Speck (2013)

Journal of the European Mathematical Society

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In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in $1 + 3$ spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state $p = c^{2} ρ, 0 < c^{2} < 1 / 3$ , the background metric + fluid...

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

On the global existence for a regularized model of viscoelastic non-Newtonian fluid

Ondřej Kreml, Milan Pokorný, Pavel Šalom (2015)

Colloquium Mathematicae

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We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like ${μ (D) \sim | D |}^{p - 2}$ (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.

On the local strong solutions for a system describing the flow of a viscoelastic fluid

Ondřej Kreml, Milan Pokorný (2009)

Banach Center Publications

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We consider a model for the viscoelastic fluid which has recently been studied in [4] and [1]. We show the local-in-time existence of a strong solution to the corresponding system of partial differential equations under less regularity assumptions on the initial data than in the above mentioned papers. The main difference in our approach is the use of the $L^{p}$ theory for the Stokes system.

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Asymptotic dynamics in double-diffusive convection

Mikołaj Piniewski (2008)

Applicationes Mathematicae

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We consider the double-diffusive convection phenomenon and analyze the governing equations. A system of partial differential equations describing the convective flow arising when a layer of fluid with a dissolved solute is heated from below is considered. The problem is placed in a functional analytic setting in order to prove a theorem on existence, uniqueness and continuous dependence on initial data of weak solutions in the class $([0, \infty); H) \cap L ²_{l o c} (ℝ^{+}; V)$ . This theorem enables us to show that the infinite-dimensional...

Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle

Reinhard Farwig (2005)

Banach Center Publications

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Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved $L^{q}$ -estimates of second order derivatives uniformly in the angular and translational velocities,...

Optimal Convective Heat-Transport

Josef Dalík, Oto Přibyl (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The one-dimensional steady-state convection-diffusion problem for the unknown temperature $y (x)$ of a medium entering the interval $(a, b)$ with the temperature $y_{min}$ and flowing with a positive velocity $v (x)$ is studied. The medium is being heated with an intensity corresponding to $y_{max} - y (x)$ for a constant $y_{max} > y_{min}$ . We are looking for a velocity $v (x)$ with a given average such that the outflow temperature $y (b)$ is maximal and discuss the influence of the boundary condition at the point $b$ on the “maximizing” function $v (x)$ . ...

Uniform analytic-Gevrey regularity of solutions to a semilinear heat equation

Todor Gramchev, Grzegorz Łysik (2008)

Banach Center Publications

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We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation $\partial_{t} u - Δ u = u^{M}$ . The approach is based on suitable iterative fixed point methods in $L^{p}$ based Banach spaces with anisotropic Gevrey norms with respect to the time and the space variables. We also construct explicit solutions uniformly analytic in t ≥ 0 and x ∈ ℝⁿ for some conservative nonlinear terms with symmetries.

Free convection flow of water at $4^{\circ} C$ past an infinite porous plate with constant suction

V. M. Soundalgekar (1975)

Matematički Vesnik

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Steady-state buoyancy-driven viscous flow with measure data

Tomáš Roubíček (2001)

Mathematica Bohemica

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Steady-state system of equations for incompressible, possibly non-Newtonean of the $p$ -power type, viscous flow coupled with the heat equation is considered in a smooth bounded domain $Ω \subset ℝ^{n}$ , $n = 2$ or 3, with heat sources allowed to have a natural $L^{1}$ -structure and even to be measures. The existence of a distributional solution is shown by a fixed-point technique for sufficiently small data if $p > 3 / 2$ (for $n = 2$ ) or if $p > 9 / 5$ (for $n = 3$ ).

Global existence and $L_{p}$ decay estimate of solution for Cahn-Hilliard equation with inertial term

Hongmei Xu, Qi Li (2021)

Applications of Mathematics

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The Cauchy problem of the Cahn-Hilliard equation with inertial term in multi space dimension is considered. Based on detailed analysis of Green’s function, using fixed-point theorem, we get the global existence in time of classical solution with large initial data. Furthermore, we get $L_{p}$ decay rate of the solution.

Numerical comparison of unsteady compressible viscous flow in convergent channel

Pořízková, Petra, Kozel, Karel, Horáček, Jaromír

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This study deals with a numerical solution of a 2D flows of a compressible viscous fluids in a convergent channel for low inlet airflow velocity. Three governing systems – Full system, Adiabatic system, Iso-energetic system $b a s e d o n t h e N a v i e r - S t o k e s e q u a t i o n s f o r l a m i n a r f l o w a r e t e s t e d . T h e n u m e r i c a l s o l u t i o n i s r e a l i z e d b y f i n i t e v o l u m e m e t h o d a n d t h e p r e d i c t o r - c o r r e c t o r M a c C o r m a c k s c h e m e w i t h J a m e s o n a r t i f i c i a l v i s c o s i t y u s i n g a g r i d o f q u a d r i l a t e r a l c e l l s . T h e u n s t e a d y g r i d o f q u a d r i l a t e r a l c e l l s i s c o n s i d e r e d i n t h e f o r m o f c o n s e r v a t i o n l a w s u s i n g A r b i t r a r y L a g r a n g i a n - E u l e r i a n m e t h o d . T h e n u m e r i c a l r e s u l t s, a c q u i r e d f r o m a d e v e l o p e d p r o g r a m, a r e p r e s e n t e d f o r i n l e t v e l o c i t y$ u=4.12 ms-1 $a n d R e y n o l d s n u m b e r R e =$ 4 103 $.$