Partial Regularity of Weak Solutions to the Navier—Stokes Equations in the Class $ L^{\infty}(0,T;\, L^3(\Omega)^3) $

We show that if v is a weak solution to the Navier—Stokes equations in the class L^￥(0,T; L³(W)³) L^{\infty}(0,T;\, L^3(\Omega)^3) then the set of all possible singular points of v in W \Omega , at every time t₀ ? (0,T) t_0\in(0,T) , is at most finite and we also give the estimate of the number of the singular points.     

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier—Stokes equations

We prove a criterion of local Hölder continuity for suitable weak solutions to the Navier—Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields.     

Functional and measure-valued solutions of the euler equations for flows of incompressible fluids

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under very weak a priori estimates on approximate solution sequences of the equation; an estimate uniform in L _loc ¹ together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L _loc ² available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution.     

Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis

This paper is concerned with the time fractional Sharma–Tasso–Olver (FSTO) equation, Lie point symmetries of the FSTO equation with the Riemann–Liouville derivatives are considered. By using the Lie group analysis method, the invariance properties of the FSTO equation are investigated. In the sense of point symmetry, the vector fields of the FSTO equation are presented. And then, the symmetry reductions are provided. By making use of the obtained Lie point symmetries, it is shown that this equation can transform into a nonlinear ordinary differential equation of fractional order with the new independent variable ξ=xt ^?α/3. The derivative is an Erdélyi–Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions to the FSTO equation are given.     

Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u _j of the velocity field u is determined by the scalar θ through \({u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}\) , where \({\mathcal{R}}\) is a Riesz transform and Λ = (?Δ)^1/2. The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound \({\|\nabla u||_{L^\infty}}\) for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log(I + log(I ? Δ))) ^γ with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ ^β with 0 ≦ β ≦ 1 is also obtained.     

Stable Manifolds for Impulsive Equations Under Nonuniform Hyperbolicity

For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C ¹ outside the jumping times, we show that the invariant manifolds are also of class C ¹ outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.     

Local Existence and Blow-up Criterion of the Inhomogeneous Euler Equations

We prove local in time existence and uniqueness of solution of the inhomogeneous Euler equations in the critical Besov spaces. We also obtain the finite time blow-up criterion of the local solution.     

A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum

We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcys law. In this paper, we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data. We prove that any L weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges, strongly in L^p with decay rates, to matching Barenblatts profile of the porous medium equation. The density function tends to the Barenblatts solution of the porous medium equation while the momentum is described by Darcys law.This revised version was published in April 2005. The volume number has now been inserted into the citation line.     

A simplified v2–f model for near‐wall turbulence

A simplified version of the v²–f model is proposed that accounts for the distinct effects of low‐Reynolds number and near‐wall turbulence. It incorporates modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional v²–f, it requires one additional equation (i.e. the elliptic equation for the elliptic relaxation parameter f_µ) to be solved in conjunction with the k–ε model. The scaling is evaluated from k in collaboration with an anisotropic coefficient C_v and f_µ. Consequently, the model needs no boundary condition on and avoids free stream sensitivity. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2007 John Wiley & Sons, Ltd.     

Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.     

Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient

In this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values. Geometrically, this can be interpreted as finding a minimal Lipschitz extension. Due to the weak convexity of the functional associated to this problem, solutions are generally nonunique. By adopting G. Aronsson's notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation. In fact, we actually prove a weak maximum principle for this partial differential equation, which in some sense is the Euler equation for the minimization problem. This is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity. To overcome this difficulty we use the weak solutions of M. G. Crandall and P.-L. Lions, also known as viscosity solutions, in conjunction with some arguments using integration by parts.     

A Global Existence Result for a Zero Mach Number System

In this paper we study the global-in-time existence of weak solutions to a zero Mach number system that derives from the Navier–Stokes–Fourier system, under a special relationship between the viscosity coefficient and the heat conductivity coefficient. Roughly speaking, this relation implies that the source term in the equation for the newly introduced divergence-free velocity vector field vanishes. In dimension two, thanks to a local-in-time existence result of a unique strong solution in critical Besov spaces given by Danchin and Liao (Commun Contemp Math 14:1250022, 2012), for arbitrary large initial data, we show that this unique strong solution exists globally in time, as a consequence of a weak-strong uniqueness argument.     

Comparative study of Euler/Euler and Euler/Lagrange approaches simulating evaporation in a turbulent gas–liquid flow

This study deals with the Reynolds‐averaged Navier–Stokes simulation of evaporation in a turbulent gas–liquid flow in a three‐dimensional duct, focussing on the results obtained by a four‐equation turbulence model within the framework of the Euler/Euler approach for multiphase flow calculations: in addition to the two‐equation k?ε model describing the turbulence of the continuous (C) phase, the computational model employs transport equations for the turbulence kinetic energy of the disperse (D) phase and for the velocity covariance q=〈{u}^D{u}^C〉^D. In the present study, the evaporation model according to Abramzon and Sirignano (Int. J. Heat Mass Transfer 1989; 32 :1605–1618) has been extended by introducing an additional transport equation for a newly defined quantity ā, defined as the phase‐interface surface fraction. This allows the change in the drop diameter to be quantified in terms of a probability density function. The source term in the equation describing the dynamics of the volumetric fraction of the dispersed phase α^D is related to the evaporation time scale τ_Γ. The performance of the new model is evaluated by performing a comparative analysis of the results obtained by simulating a polydispersed spray in a three‐dimensional duct configuration with the results of the Euler/Lagrange calculations performed in parallel. Prior to these calculations, some selected (solid) particle‐laden flow configurations were computationally examined with respect to the validation of the background, four‐equation, eddy‐viscosity‐based turbulence model. Copyright © 2008 John Wiley & Sons, Ltd.     

Oriented Shadowing Property and $$\Omega $$-Stability for Vector Fields

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.     

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.     

On the Euler–Poincaré Equation with Non-Zero Dispersion

We consider the Euler–Poincaré equation on ${\mathbb{R}^d, \, d \geqq 2}$ R d , d ≧ 2 . For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu (Commun Math Phys 314:671–687, 2012). Our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the Euler–Poincaré flow. In particular we show an abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.     

Numerical solution of a first-order conservation equation by a least square method

Least square methods have been frequently used to solve fluid mechanics problems. Their specific usefulness is emphasized for the solution of a first-order conservation equation. On the one hand, the least square formulation embeds the first-order problem into equivalent second-order problem, better adapted to discretization techniques due to symmetry and positive-definiteness of the associated matrix. On the other hand, the introduction of a least square functional is convenient for finite element applications. This approach is applied to the model problem of the conservation of mass (the unknown is the density ρ) in a nozzle with a specified velocity field (u, v), possibly including jumps along lines simulating shock waves. This represent a preliminary study towards the solution of the steady Euler equations. A finite difference and a finite element method are presented. The choice of the finite difference scheme and of a continuous finite element representation for the groups of variables (ρu, ρv) is discussed in terms of conservation of mass flux. Results obtained with both methods are compared in two numerical tests with the same mesh system.     

On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations

Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.