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Discrete & Continuous Dynamical Systems - S

2014 , Volume 7 , Issue 5

Issue on new developments in mathematical theory of fluid mechanics

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New developments in mathematical theory of fluid mechanics
Eduard Feireisl , Šárka Nečasová , Reimund Rautmann and  Werner Varnhorn
2014, 7(5): i-ii doi: 10.3934/dcdss.2014.7.5i +[Abstract](31) +[PDF](86.2KB)
Mathematical theory of fluid mechanics is a field with a rich long history and active present. The volume collects selected contributions of distinguished experts in various domains ranging from modeling through mathematical analysis to numerics and practical implementations related to real world problems.

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Some uniqueness result of the Stokes flow in a half space in a space of bounded functions
Ken Abe
2014, 7(5): 887-900 doi: 10.3934/dcdss.2014.7.887 +[Abstract](35) +[PDF](434.9KB)
This paper presents a uniqueness theorem for the Stokes equations in a half space in a space of bounded functions. The Stokes equations is well understood for decaying velocity as $|x|\to\infty$, but less known for non-decaying velocity even for a half space. This paper presents a uniqueness theorem on $L^{\infty}(\mathbb{R}_+^n)$ for unbounded velocity as $t\downarrow 0$. Under suitable sup-bounds both for velocity and pressure gradient, a uniqueness theorem for non-decaying velocity is proved.
Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces
Chérif Amrouche , Mohamed Meslameni and  Šárka Nečasová
2014, 7(5): 901-916 doi: 10.3934/dcdss.2014.7.901 +[Abstract](52) +[PDF](496.4KB)
In this work, we study the linearized Navier-Stokes equations in $\mathbb{R}^3$, the Oseen equations. We are interested in the existence and the uniqueness of generalized and strong solutions in $L^p$-theory which makes analysis more difficult. Our approach rests on the use of weighted Sobolev spaces.
Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model
Xiaoyun Cai , Liangwen Liao and  Yongzhong Sun
2014, 7(5): 917-923 doi: 10.3934/dcdss.2014.7.917 +[Abstract](32) +[PDF](328.9KB)
We establish the global-in-time existence of strong solution to the initial-boundary value problem of a 2-D Kazhikov-Smagulov type model for incompressible nonhomogeneous fluids with mass diffusion for arbitrary size of initial data.
Flow-plate interactions: Well-posedness and long-time behavior
Igor Chueshov , Irena Lasiecka and  Justin Webster
2014, 7(5): 925-965 doi: 10.3934/dcdss.2014.7.925 +[Abstract](48) +[PDF](775.0KB)
We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent results on existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting sets corresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to a PDE system with delay.
Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions
Paul Deuring , Stanislav Kračmar and  Šárka Nečasová
2014, 7(5): 967-979 doi: 10.3934/dcdss.2014.7.967 +[Abstract](38) +[PDF](437.8KB)
We consider Leray solutions of the Oseen system with rotational terms, in an exterior domain. Such solutions are characterized by square-integrability of the gradient of the velocity and local square-integrability of the pressure. In a previous paper, we had shown a pointwise decay result for a slightly stronger type of solution. Here this result is extended to Leray solutions. We thus present a second access to this result, besides the one in G. P. Galdi, M. Kyed, Arch. Rat. Mech. Anal., 200 (2011), 21-58.
Thermo-visco-elasticity for the Mróz model in the framework of thermodynamically complete systems
Piotr Gwiazda , Filip Z. Klawe and  Agnieszka Świerczewska-Gwiazda
2014, 7(5): 981-991 doi: 10.3934/dcdss.2014.7.981 +[Abstract](21) +[PDF](355.2KB)
We derive a thermomechanical model for the evolution of the visco-elastic body subject to the action of external forces. The presented framework captures elastic and visco-elastic deformation as well as thermal effects occurring in the material. Consequently we couple the momentum balance with the heat equation. The system is supplemented by the constitutive relation for the Cauchy stress tensor and visco-elastic strain tensor. As an example exhibiting the proposed approach one can mention the Mróz model. We establish existence of solutions to the quasi-static version of the derived system.
Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates
Trygve K. Karper
2014, 7(5): 993-1023 doi: 10.3934/dcdss.2014.7.993 +[Abstract](26) +[PDF](472.7KB)
We construct a new finite difference method for the flow of ideal viscous isentropic gas in one spatial dimension. For the continuity equation, the method is a standard upwind discretization. For the momentum equation, the method is an uncommon upwind discretization, where the moment and the velocity are solved on dual grids. Our main result is convergence of the method as discretization parameters go to zero. Convergence is proved by adapting the mathematical existence theory of Lions and Feireisl to the numerical setting.
Existence and decay of solutions of the 2D QG equation in the presence of an obstacle
Leonardo Kosloff and  Tomas Schonbek
2014, 7(5): 1025-1043 doi: 10.3934/dcdss.2014.7.1025 +[Abstract](28) +[PDF](428.0KB)
We continue the study initiated in [16] of dissipative differential equations governing fluid motion in the presence of an obstacle, in which the dissipative term is given by the Laplacian, or a fractional power of the Laplacian. Our main tools are the Ikebe-Ramm transform, and the localized version of the fractional Laplacian due to Caffarelli and Silvestre [5] as improved by Stinga and Torrea [21]. We give applications to the problem of existence of weak solutions of the two dimensional dissipative quasi-geostrophic equation and the decay of these solutions in the $L^2$-norm.
Stokes and Navier-Stokes equations with perfect slip on wedge type domains
Siegfried Maier and  Jürgen Saal
2014, 7(5): 1045-1063 doi: 10.3934/dcdss.2014.7.1045 +[Abstract](25) +[PDF](529.4KB)
Well-posedness of the Stokes and Navier-Stokes equations subject to perfect slip boundary conditions on wedge type domains is studied. Applying the operator sum method we derive an $\mathcal{H}^\infty$-calculus for the Stokes operator in weighted $L^p_\gamma$ spaces (Kondrat'ev spaces) which yields maximal regularity for the linear Stokes system. This in turn implies mild well-posedness for the Navier-Stokes equations, locally-in-time for arbitrary and globally-in-time for small data in $L^p$.
A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium
Anna Marciniak-Czochra and  Andro Mikelić
2014, 7(5): 1065-1077 doi: 10.3934/dcdss.2014.7.1065 +[Abstract](93) +[PDF](362.4KB)
We present modeling of an incompressible viscous flow through a fracture adjacent to a porous medium. A fast stationary flow, predominantly tangential to the porous medium is considered. Slow flow in such setting can be described by the Beavers-Joseph-Saffman slip. For fast flows, a nonlinear filtration law in the porous medium and a non- linear interface law are expected. In this paper we rigorously derive a quadratic effective slip interface law which holds for a range of Reynolds numbers and fracture widths. The porous medium flow is described by the Darcy law. The result shows that the interface slip law can be nonlinear, independently of the regime for the bulk flow. Since most of the interface and boundary slip laws are obtained via upscaling of complex systems, the result indicates that studying the inviscid limits for the Navier-Stokes equations with linear slip law at the boundary should be rethought.
Approximate solutions to a model of two-component reactive flow
Piotr Bogusław Mucha , Milan Pokorný and  Ewelina Zatorska
2014, 7(5): 1079-1099 doi: 10.3934/dcdss.2014.7.1079 +[Abstract](45) +[PDF](440.4KB)
We consider a model of motion of binary mixture, based on the compressible Navier-Stokes system. The mass balances of chemically reacting species are described by the reaction-diffusion equations with generalized form of multicomponent diffusion flux. Under a special relation between the two density dependent viscosity coefficients and for singular cold pressure we construct the weak solutions passing through several levels of approximation.
Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow
Reimund Rautmann
2014, 7(5): 1101-1109 doi: 10.3934/dcdss.2014.7.1101 +[Abstract](40) +[PDF](185.6KB)
For the transition from slip- to no-slip fluid flow, we establish lower and upper bounds to the resulting change of the $L^2$-norm of the vorticity. Moreover we present a transport-diffusion splitting scheme, built up solely by a transport step and subsequent diffusion step (without any additional vorticity creation operator as introduced in former studies by Lighthill, Marsden, and Chorin), the splitting scheme being consistent with the Navier-Stokes equations with no-slip condition.
On one multidimensional compressible nonlocal model of the dissipative QG equations
Shu Wang , Zhonglin Wu , Linrui Li and  Shengtao Chen
2014, 7(5): 1111-1132 doi: 10.3934/dcdss.2014.7.1111 +[Abstract](22) +[PDF](495.4KB)
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations and discuss the effect of the sign of initial data on the wellposedness of this model. First, we prove the existence and uniqueness of local smooth solutions for the Cauchy problem for the model with the nonnegative initial data, which seems to imply that whether the well-posedness of this model holds or not depends heavily upon the sign of the initial data even for the subcritical case. Secondly, for the sub-critical case $1<\alpha\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Next, we prove the global existence of the weak solution for $0<\alpha\le 2$ and $\nu>0$. Finally, for the sub-critical case $1<\alpha\leq 2$, we establish $H^\beta(\beta\geq 0)$ and $L^p(p\geq 2)$ decay rates of the smooth solution as $t\to\infty$. A inequality for the Riesz transformation is also established.

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