On flow of a Navier–Stokes fluid in curved pipes. Part I: Steady flow

We consider the steady, fully developed motion of a Navier–Stokes fluid in a curved pipe of cross-section $D$ under a given axial pressure gradient $G$. We show that, if $G$ is constant, this problem has a smooth steady solution, for arbitrary values of the Dean’s number $\kappa$, for $D$ of arbitrary shape and for any curvature ratio $\delta$ of the pipe. This solution is also unique for $\kappa$ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if $\kappa =0$. Finally, we show the same properties for the approximations to the Navier–Stokes equations called “Dean’s equations” and provide a rigorous way in which solutions to the full Navier–Stokes equations approach those to this approximation in the limit of $\delta \to 0$.     

Navier–Stokes equations with nonhomogeneous boundary conditions in a convex bi-dimensional domain

We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR², with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle.     

Quadratic Hamilton–Poisson Systems on $\mathfrak{se}(1,1)^{*}_{-}$: The Inhomogeneous Case

The blow-up of smooth solution to the isentropic compressible Navier-Stokes-Poisson (NSP) system on \(\mathbb{R}^{d}\) is studied in this paper. We obtain that if the initial density is compactly supported, the spherically symmetric smooth solution to the NSP system on \(\mathbb{R}^{d}\ (d\geq 2)\) blows up in finite time. In the case \(d=1\), if \(2\mu +\lambda >0\), then the NSP system only exits a zero smooth solution on ? for the compactly supported initial density.     

Large time behavior of the isentropic compressible Navier–Stokes–Maxwell system

In this paper, we study the large time behavior of the isentropic compressible Navier–Stokes–Maxwell system introduced by Jiang and Li (Nonlinearity 25(6):1735–1752, 2012) in the whole space \({{\mathbb{R}}^3}\) when the initial data are a small perturbation of some given constant state. We obtain the desired result through taking the refined analysis on the time decay property and Green’s function of the linearized system. Moreover, we also obtain the optimal time rate of the solution.     

Dirac flow on the 3-sphere

We illustrate some well-known facts about the evolution of the 3-sphere (S³, g) generated by the Ricci flow. We define the Dirac flow and study the properties of the metric \(\bar g = dt^2 + g(t)\), where g(t) is a solution of the Dirac flow. In the case of a metric g conformally equivalent to the round metric on S³ the metric \(\bar g\) is of constant curvature. We study the properties of solutions in the case when g depends on two functional parameters. The flow on differential 1-forms whose solution generates the Eguchi–Hanson metric was written down. In particular cases we study the singularities developed by these flows.     

Variation of Laplace spectra of compact “nearly” hyperbolic surfaces

We use the real analyticity of the Ricci flow with respect to time proved by B. Kotschwar to extend a result of P. Buser, namely, we prove that the Laplace spectra of negatively curved compact orientable surfaces having the same genus $\gamma \ge 2$, the same area and the same curvature bounds vary in a “controlled way”, of which we give a quantitative estimate in our main theorem. The basic technical tool is a variational formula that provides the derivative of an eigenvalue branch under the normalized Ricci flow. In a related manner, we also observe how the above-mentioned real analyticity result can lead to unexpected conclusions concerning the spectral properties of generic metrics on a compact surface of genus $\gamma \ge 2$.     

Free boundary problem for the equations of spherically symmetric motion of compressible gas with density-dependent viscosity

We consider a free boundary problem for the equations of spherically symmetric motion of a isentropic gas with a density-dependent viscosity , where and λ are positive constants. We prove that the problem admits a weak solution provided that 0 < λ < 1/4.      

Steady flow for incompressible fluids in domains with unbounded curved channels

We give an overview on the solution of the stationary Navier-Stokes equations for non newtonian incompressible fluids established by G. Dias and M.M. Santos (Steady flow for shear thickening fluids with arbitrary fluxes, J. Differential Equations 252 (2012), no. 6, 3873-3898), propose a definition for domains with unbounded curved channels which encompasses domains with an unbounded boundary, domains with nozzles, and domains with a boundary being a punctured surface, and argue on the existence of steady flowfor incompressible fluids with arbitrary fluxes in such domains.     

Incompressible limit of all-time solutions to 3-D full Navier–Stokes equations for perfect gas with well-prepared initial condition

In this paper, we prove the incompressible limit of all-time strong solutions to the three-dimensional full compressible Navier–Stokes equations. Here the velocity field and temperature satisfy the Dirichlet boundary condition and convective boundary condition, respectively. The uniform estimates in both the Mach number \({\epsilon\in(0,\overline{\epsilon}]}\) and time \({t\in[0,\infty)}\) are established by deriving a differential inequality with decay property, where \({\overline{\epsilon}\in(0,1]}\) is a constant. Based on these uniform estimates, the global solution of full compressible Navier–Stokes equations with “well-prepared” initial conditions converges to the one of isentropic incompressible Navier–Stokes equations as the Mach number goes to zero.     

Laplacian Flow of Closed G$$_$$-Structures Inducing Nilsolitons

We study the existence of left invariant closed \(G_2\)-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these \(G_2\)-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on \({\mathbb {R}}^7\) in a similar way as in Lauret (Commun Anal Geom 19(5):831–854, 2011) we prove that the underlying metrics \(g(t)\) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as \(t\) goes to infinity.     

Ricci Flow on a Class of Noncompact Warped Product Manifolds

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.     

Global solutions to the cylindrically symmetric rotating motion of isentropic gases

We construct global solutions to the Euler equations of compressible isentropic gas dynamics with cylindrically symmetric rotating structure. A shock capturing numerical scheme is introduced to compute such a flow and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework. Earlier work of the authors, which controlled the geometrical source terms, especially as they pertain to radial flow in an unbounded region, 1, is extended here to the 3 × 3 system of cylindrically symmetric rotating flow. Arbitrary data withL bounds are allowed in these results.     

Oil displacement by water and gas in a porous medium: the Riemann problem

In this work we present the construction of the Riemann solution for a system of two conservation laws representing displacement in immiscible three-phase flow. The porousmedium is initially filled with oil and small amounts of water and gas; then a fixed proportion of water and gas is injected. We use the wave curve method to determine the wave sequences in the Riemann solution for arbitrary initial and injection data in the above mentioned class. We show the L_Loc¹-stability of the Riemann solution with variation of data. We do not verify uniqueness of the Riemann solution, but we believe that it is valid.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

Strong solutions of the Navier-Stokes equations for isentropic compressible fluids

We study strong solutions of the isentropic compressible Navier-Stokes equations in a domain . We first prove the local existence of unique strong solutions provided that the initial data ρ₀ and u₀ satisfy a natural compatibility condition. The important point in this paper is that we allow the initial vacuum: the initial density may vanish in an open subset of Ω. We then prove a new uniqueness result and stability result. Our results are valid for unbounded domains as well as bounded ones.     

Forces of a fluid in a rotating straight pipe

The fluid flow through a rotating straight pipe is considered, the axis of rotation being perpendicular to the pipe axis. The flow of the fluid is taken as fully developed, i.e. the velocity field is assumed to be the same in all transverse cross sections of the pipe. The derivation presented applies to viscous and nonviscous incompressible fluids. For constant angular pipe velocity a simple and exact (Coriolis type) relationF=2Q(t) is derived between the forceF by which the fluid acts on the (unit length of the) pipe in the direction perpendicular to the two axes, the fluid mass flow rateQ(t) through the pipe, and the angular velocity. Variable angular velocities, i.e. , introduce an additional term into the expression for the inertial forceF, which depends only on and on known (constant) parameters; this term is known for given angular velocity(t). The flow configuration investigated here is an idealization of those appearing (over short space and time intervals) in the devices measuring mass flow rateQ(t) through the (Coriolis) forceF. Therefore the exact results derived here cast some light on the degree of precision one expects in these devices, where more complicated flow configurations are present than those looked at in this paper.     

Steady compressible Navier-Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R². We show existence if a solution that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of the constant flow . In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.     

Direct problem and inverse problem for the supersonic plane flow past a curved wedge

In this paper, we consider the isentropic irrotational steady plane flow past a curved wedge. First, for a uniform supersonic oncoming flow, we study the direct problem: For a given curved wedge y = f(x), how to globally determine the corresponding shock y = g(x) and the solution behind the shock? Then, we solve the corresponding inverse problem: How to globally determine the curved wedge y = f(x) under the hypothesis that the position of the shock y = g(x) and the uniform supersonic oncoming flow are given? This kind of problems plays an important role in the aviation industry. Under suitable assumptions, we obtain the global existence and uniqueness for both problems. Copyright © 2011 John Wiley & Sons, Ltd.