Asymptotic dynamics in double-diffusive convection

Mikołaj Piniewski

Displaying similar documents to “Asymptotic dynamics in double-diffusive convection”

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

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It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...

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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Porous medium equation and fast diffusion equation as gradient systems

Samuel Littig, Jürgen Voigt (2015)

Czechoslovak Mathematical Journal

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We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u} - Δ u^{m} = f$ , with $m \in (0, \infty)$ , can be modeled as a gradient system in the Hilbert space $H^{- 1} (Ω)$ , and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $Ω \subseteq ℝ^{n}$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

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 and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents

Aya Khaldi, Amar Ouaoua, Messaoud Maouni (2022)

Mathematica Bohemica

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We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms $u_{t} - M (\int_{Ω} {| \nabla u |}^{2} d x) Δ u + {| u |}^{m (x) - 2} u_{t} = {| u |}^{r (x) - 2} u .$ We prove with suitable assumptions on the variable exponents $r (\cdot),$ $m (\cdot)$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.

On the global existence for the Muskat problem

Peter Constantin, Diego Córdoba, Francisco Gancedo, Robert M. Strain (2013)

Journal of the European Mathematical Society

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The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^{2} (ℝ)$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ${∥f∥}_{1} \leq 1 / 5$ . Previous results...

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.

On estimation of diffusion coefficient based on spatio-temporal FRAP images: An inverse ill-posed problem

Kaňa, Radek, Matonoha, Ctirad, Papáček, Štěpán, Soukup, Jindřich

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We present the method for determination of phycobilisomes diffusivity (diffusion coefficient $D$ ) on thylakoid membrane from fluorescence recovery after photobleaching (FRAP) experiments. This was usually done by analytical models consisting mainly of a simple curve fitting procedure. However, analytical models need some unrealistic conditions to be supposed. Our method, based on finite difference approximation of the process governed by the Fickian diffusion equation and on the minimization...

A second order unconditionally positive space-time residual distribution method for solving compressible flows on moving meshes

Dobeš, Jiří, Deconinck, Herman

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A space-time formulation for unsteady inviscid compressible flow computations in 2D moving geometries is presented. The governing equations in Arbitrary Lagrangian-Eulerian formulation (ALE) are discretized on two layers of space-time finite elements connecting levels $n$ , $n + 1 / 2$ and $n + 1$ . The solution is approximated with linear variation in space (P1 triangle) combined with linear variation in time. The space-time residual from the lower layer of elements is distributed to the nodes at level...

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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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.