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1937-1632

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1937-1179

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## Discrete & Continuous Dynamical Systems - S

October 2013 , Volume 6 , Issue 5

Issue on Recent Progress in Mathematical Fluid Dynamics: Vorticity, Rotation, Symmetry and Regularity of Fluid Motion

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2013, 6(5): i-iii
doi: 10.3934/dcdss.2013.6.5i

*+*[Abstract](1736)*+*[PDF](92.3KB)**Abstract:**

The papers in this special volume come from the authors' presentations at the conference ''Vorticity, Rotation and Symmetry (II) -- Regularity of Fluid Motion'', which was held in ''Centre International de Rencontres Mathématiques (CIRM)'' in Luminy (Marseille, France) from May 23 to May 27, 2011.

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2013, 6(5): 1113-1137
doi: 10.3934/dcdss.2013.6.1113

*+*[Abstract](2438)*+*[PDF](494.0KB)**Abstract:**

We consider the Navier-Stokes equations with pressure boundary conditions in the case of a bounded open set, connected of class $\mathcal{C}^{\,1,1}$ of $\mathbb{R}^3$. We prove existence of solution by using a fixed point theorem over the type-Oseen problem. This result was studied in [5] in the Hilbertian case. In our study we give the $L^p$-theory for $1< p <\infty$.

On the blow-up problem for the Euler equations and
the Liouville type results in the fluid equations

2013, 6(5): 1139-1150
doi: 10.3934/dcdss.2013.6.1139

*+*[Abstract](2103)*+*[PDF](374.6KB)**Abstract:**

In this paper we briefly review recent results mostly by the author related to the blow-up problem of the 3D Euler equations and the Liouville type results in the various equations of the fluids.

2013, 6(5): 1151-1162
doi: 10.3934/dcdss.2013.6.1151

*+*[Abstract](1851)*+*[PDF](390.0KB)**Abstract:**

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a straightforward framework for studying time-averages of dynamical systems whose solutions exhibit fast oscillatory behaviors. Time integration averages out the oscillatory part of the solution that is caused by the large skew-symmetric operator. Then, the time-average of the solution stays close to the kernel of this operator. The key assumption in this framework is that the inverse of the large operator is a bounded mapping between certain Hilbert spaces modular the kernel of the operator itself. This assumption is verified for several examples of time-dependent PDEs.

2013, 6(5): 1163-1172
doi: 10.3934/dcdss.2013.6.1163

*+*[Abstract](1836)*+*[PDF](380.5KB)**Abstract:**

This note is dedicated to a few questions related to the divergence equation which have been motivated by recent studies concerning the Neumann problem for the Laplace equation or the (evolutionary) Stokes system in domains of $\mathbb{R}^n.$ For simplicity, we focus on the classical Sobolev spaces framework in bounded domains, but our results have natural and simple extensions to the Besov spaces framework in more general domains.

2013, 6(5): 1173-1191
doi: 10.3934/dcdss.2013.6.1173

*+*[Abstract](2350)*+*[PDF](459.7KB)**Abstract:**

We are interested in regularity results, up to the boundary, for the second derivatives of the solutions of some nonlinear systems of partial differential equations with $p$-growth. We choose two representative cases: the ''full gradient case'', corresponding to a $p$-Laplacian, and the ''symmetric gradient case'', arising from mathematical physics. The domain is either the ''cubic domain'' or a bounded open subset of $\mathbb{R}^3$ with a smooth boundary. Depending on the model and on the range of $p$, $p<2$ or $p>2$, we prove different regularity results. It is worth noting that in the full gradient case with $p<2$ we cover the singular case and obtain $W^{2,q}$-global regularity results, for arbitrarily large values of $q$. In turn, the regularity achieved implies the Hölder continuity of the gradient of the solution.

2013, 6(5): 1193-1213
doi: 10.3934/dcdss.2013.6.1193

*+*[Abstract](2447)*+*[PDF](465.7KB)**Abstract:**

The global existence of weak solutions is proved for the problem of the motion of several rigid bodies either in a non-Newtonian fluid of power law type or in a barotropic compressible fluid, under the influence of gravitational forces.

2013, 6(5): 1215-1224
doi: 10.3934/dcdss.2013.6.1215

*+*[Abstract](2365)*+*[PDF](409.4KB)**Abstract:**

We present a uniqueness theorem for backward asymptotically almost periodic solutions to the Navier-Stokes equations in $3$-dimensional unbounded domains. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in $BC(-\infty,T;L^{3}_w)$ within the class of solutions which have sufficiently small $L^{\infty}( L^{3}_w)$-norm. In this paper, we show that a small backward asymptotically almost periodic solution in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$ is unique within the class of all backward asymptotically almost periodic solutions in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$.

2013, 6(5): 1225-1236
doi: 10.3934/dcdss.2013.6.1225

*+*[Abstract](1737)*+*[PDF](399.5KB)**Abstract:**

The aim of this note is to present a new approach to linear and nonlinear instability of the Ekman spiral, the famous stationary geostrophic solution of the 3D Navier-Stokes equations in a rotating frame. As former approaches to the Ekman boundary layer problem, our result is based on the numerical existence of an unstable wave perturbation for Reynolds numbers large enough derived by Lilly in [15]. By the fact that this unstable wave is tangentially nondecaying at infinity, however, standard approaches (e.g. by cut-off techniques) to instability in standard function spaces (e.g. $L^p$) remain a technical and intricate issue. In spite of this fact, we will present a rather short proof of linear and nonlinear instability of the Ekman spiral in $L^2$. The results are based on a recently developed general approach to rotating boundary layer problems, which relies on Fourier transformed vector Radon measures, cf.[11].

2013, 6(5): 1237-1257
doi: 10.3934/dcdss.2013.6.1237

*+*[Abstract](2300)*+*[PDF](511.1KB)**Abstract:**

We consider the two-dimensional motion of a Navier-Stokes liquid in the whole plane, under the action of a time-periodic body force $F$ of period $T$, and tending to a prescribed nonzero constant velocity at infinity. We show that if the magnitude of $F$, in suitable norm, is sufficiently small, there exists one and only one corresponding time-periodic flow of period $T$ in an appropriate function class.

2013, 6(5): 1259-1275
doi: 10.3934/dcdss.2013.6.1259

*+*[Abstract](2148)*+*[PDF](427.1KB)**Abstract:**

In this paper we prove that the $L^p$ realisation of a system of Laplace operators subjected to mixed first and zero order boundary conditions admits a bounded $H^{\infty}$-calculus. Furthermore, we apply this result to the Magnetohydrodynamic equation with perfectly conducting wall condition.

2013, 6(5): 1277-1289
doi: 10.3934/dcdss.2013.6.1277

*+*[Abstract](2722)*+*[PDF](356.4KB)**Abstract:**

We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.

2013, 6(5): 1291-1306
doi: 10.3934/dcdss.2013.6.1291

*+*[Abstract](2355)*+*[PDF](438.8KB)**Abstract:**

The paper concerns theory of anisotropic Orlicz spaces and its applications in continuum mechanics. Our main motivations are e.g. flow of non-Newtonian fluid and response of inelastic materials with non-standard growth conditions of the Cauchy stress tensor. The set of basic definitions and theorems with proofs is presented. We prove the existence of a weak solutions to the generalized Stokes system. Overview covering recent results in the referred topic is given.

2013, 6(5): 1307-1313
doi: 10.3934/dcdss.2013.6.1307

*+*[Abstract](2256)*+*[PDF](320.5KB)**Abstract:**

Consider the set of equations describing Oldroyd-B fluids with finite Weissenberg numbers in exterior domains. In this note, we describe the main ideas of the proofs of two recent results on global existence for this set of equations on exterior domains subject to Dirichlet boundary conditions. The methods described here are quite different from the techniques used in the Lagrangian approach which is often used in the case of infinite Weissenberg numbers.

2013, 6(5): 1315-1322
doi: 10.3934/dcdss.2013.6.1315

*+*[Abstract](2049)*+*[PDF](334.2KB)**Abstract:**

The Oseen problem arises as the linearization of a steady-state Navier-Stokes flow past a translating body. If the body, in addition to the translational motion, is also rotating, the corresponding linearization of the equations of motion, written in a frame attached to the body, yields the Oseen system with extra terms in the momentum equation due to the rotation. In this paper, the effect these rotation terms have on the asymptotic structure at spatial infinity of a solution to the system is studied. A mapping property of the whole space Oseen operator with rotation is identified from which asymptotic properties of a solution can be derived. As an application, an asymptotic expansion of a steady-state, linearized Navier-Stokes flow past a rotating and translating body is established.

2013, 6(5): 1323-1342
doi: 10.3934/dcdss.2013.6.1323

*+*[Abstract](2104)*+*[PDF](492.4KB)**Abstract:**

We study the Stokes initial boundary value problem with an initial data in a Lorentz space. We develop a suitable technique able to solve the problem and to prove the semigroup properties of the resolving operator in the case of the

*''limit exponents''*. The results are a completion of the ones related to the usual $L^p$-theory, of the ones already known and they are also useful tool to study some questions related to the Navier-Stokes equations.

2013, 6(5): 1343-1353
doi: 10.3934/dcdss.2013.6.1343

*+*[Abstract](1559)*+*[PDF](336.0KB)**Abstract:**

We consider capillary laminar fluid motions on an inclined plane and study spatially periodic surface waves with fixed periodicity on the line of maximum slope $\alpha_1$ and in the horizontal direction $\alpha_2$. Actually, we provide a sufficient condition on Reynolds and Weber numbers, and on the inclination angle, named condition (C), in order that the Poiseuille flow $(v_b,p_b,\Gamma_b)$ with upper flat free boundary $\Gamma_b$ and with periodicity conditions on the plane, is nonlinearly stable. Under condition (C), the perturbed surface $\Gamma_t$ is bounded for all time, and the free boundary Poiseuille flow is stable.

2013, 6(5): 1355-1369
doi: 10.3934/dcdss.2013.6.1355

*+*[Abstract](2870)*+*[PDF](431.2KB)**Abstract:**

We present here different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in $\mathbb{R}^3$, such as Dirichlet, Neumann or Hodge boundary conditions. We first study the linear Stokes operator associated to the boundary conditions. Then we show how the properties of the operator lead to local solutions or global solutions for small initial data.

2013, 6(5): 1371-1390
doi: 10.3934/dcdss.2013.6.1371

*+*[Abstract](2047)*+*[PDF](471.6KB)**Abstract:**

Starting from Prandtl's (1945) turbulence model, we consider two systems of PDEs for the scalar functions $u$ and $k$ which characterize the stationary turbulent pipe-flow. This system is completed by a homogeneous Dirichlet condition on $u$, and homogeneuos Neumann or mixed boundary conditions on $k$, respectively. For these boundary value problems we prove the existence of weak solutions $(u,k)$ such that $k>0$ on a set of positive measure.

2013, 6(5): 1391-1400
doi: 10.3934/dcdss.2013.6.1391

*+*[Abstract](2259)*+*[PDF](386.8KB)**Abstract:**

We formulate a criterion which guarantees a local regularity of a suitable weak solution $v$ to the Navier--Stokes equations (in the sense of L. Caffarelli, R. Kohn and L. Nirenberg [3]). The criterion shows that if $(x_0,t_0)$ is a singular point of solution $v$ then the $L^3$--norm of $v$ concentrates in an amount greater than or equal to some $\epsilon>0$ in an arbitrarily small neighbourhood of $x_0$ at all times $t$ in some left neighbourhood of $t_0$. As a partial result, we prove that a localized solution satisfies the strong energy inequality.

2013, 6(5): 1401-1407
doi: 10.3934/dcdss.2013.6.1401

*+*[Abstract](2084)*+*[PDF](364.0KB)**Abstract:**

We consider the incompressible Navier--Stokes equations in the entire three-dimensional space. Assuming additional regularity on the components of the vector field $\partial_3$

**u**we show intermediate anisotropic regularity results between the results by Kukavica and Ziane [5] and by Cao and Titi [3]and improve the result from the paper by Penel and Pokorný [9].

2013, 6(5): 1409-1415
doi: 10.3934/dcdss.2013.6.1409

*+*[Abstract](2392)*+*[PDF](359.3KB)**Abstract:**

The Cauchy problem of the Euler equations is considered with initial data with possibly less regularity. The time-local existence and the uniqueness of strong solutions were established by Pak-Park, when the initial velocity is in the Besov space $B^1_{\infty, 1}$. By treating non-decaying initial data, we are able to discuss the propagation of almost periodicity. It is also proved that if the initial data are real analytic, then the solutions become necessarily real analytic in space variables with an explicit convergence rate of the radius in Taylor's expansion. This result comes from the calculation of higher order derivatives, inductively.

2013, 6(5): 1417-1425
doi: 10.3934/dcdss.2013.6.1417

*+*[Abstract](1991)*+*[PDF](355.7KB)**Abstract:**

In the present paper we provide the decomposition and local estimates for the pressure function for the non-stationary flow of incompressible non-Newtonian fluids in Orlicz spaces. We show that this method can be applied to prove the existence of weak solutions to the problem of motion of one or several rigid bodies in a non-Newtonian incompressible fluid with growth conditions given by an $N$-function.

2013, 6(5): 1427-1455
doi: 10.3934/dcdss.2013.6.1427

*+*[Abstract](2289)*+*[PDF](449.6KB)**Abstract:**

We consider the motion of incompressible viscous non-homogene-ous fluid described by the Navier-Stokes equations in a bounded cylinder $\Omega$ under boundary slip conditions. Assume that the $x_3$-axis is the axis of the cylinder. Let $\varrho$ be the density of the fluid, $v$ -- the velocity and $f$ the external force field. Assuming that quantities $\nabla\varrho(0)$, $\partial_{x_3}v(0)$, $\partial_{x_3}f$, $f_3|_{\partial\Omega}$ are sufficiently small in some norms we prove large time regular solutions such that $v\in H^{2+s,1+s/2}(\Omega\times(0,T))$, $\nabla p\in H^{s,s/2}(\Omega\times(0,T))$, $½ < s < 1$ without any restriction on the existence time $T$. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.

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