On Subsonic and Subsonic-Sonic Flows with General Conservatives Force in Exterior Domains

In this paper, we study the irrotational subsonic and subsonic-sonic fows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general ca...     

Examples of steady subsonic flows in a convergent-divergent approximate nozzle

We construct special solutions of the full Euler system for steady compressible flows in a convergent-divergent approximate nozzle and study the stability of the purely subsonic flows. For a given pressure p₀ prescribed at the entry of the nozzle, as the pressure p₁ at the exit decreases, the flow patterns in the nozzle change continuously: there appear subsonic flow, subsonic-sonic flow, transonic flow and transonic shocks. Our results indicate that, to determine a subsonic flow in a two-dimensional nozzle, if the Bernoulli constant is uniform in the flow field, then this constant should not be prescribed if the pressure, density at the entry and the pressure at the exit of the nozzle are given; if the Bernoulli constant and both the pressures at the entrance and the exit are given, the average of the density at the entrance is then totally determined.     

Asymptotic behavior for the Navier-Stokes equations with nonzero external forces

We estimate the asymptotic behavior for the Stokes solutions, with external forces first. We found that if there are external forces, then the energy decays slowly even if the forces decay quickly. Then, we also obtain the asymptotic behavior in the temporal-spatial direction for weak solutions of the Navier-Stokes equations. We also provide a simple example of external forces which shows that the Stokes solution does not decay quickly.     

Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method. More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π/2; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.     

Smooth or singular solutions to the Navier-Stokes system ?

The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including stationary ones), and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type.     

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.     

Asymptotic behavior of solutions to Euler–Poisson equations for bipolar hydrodynamic model of semiconductors

In this paper we study the Cauchy problem for 1-D Euler–Poisson system, which represents a physically relevant hydrodynamic model but also a challenging case for a bipolar semiconductor device by considering two different pressure functions and a non-flat doping profile. Different from the previous studies (Gasser et al., 2003 [7], Huang et al., 2011 [12], Huang et al., 2012 [13]) for the case with two identical pressure functions and zero doping profile, we realize that the asymptotic profiles of this more physical model are their corresponding stationary waves (steady-state solutions) rather than the diffusion waves. Furthermore, we prove that, when the flow is fully subsonic, by means of a technical energy method with some new development, the smooth solutions of the system are unique, exist globally and time-algebraically converge to the corresponding stationary solutions. The optimal algebraic convergence rates are obtained.     

Optimal Hölder Continuity for a Class of Degenerate Elliptic Problems with an Application to Subsonic-Sonic Flows

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.     

On subsonic compressible flows in a two-dimensional duct

This paper concerns subsonic flows passing a two-dimensional duct for the steady compressible Euler system. If the Bernoulli constant is uniform in the flow field, the density at the entry and both the pressures at the entrance and the exit are given, we show that the problem is generally ill-posed; but if we give the pressure at the exit with a constant difference, then under the same other conditions as above we establish the existence of subsonic flows.     

A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation

We present the existence of the subsonic solution to a two-dimensional Riemann problem governed by a self-similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle. Thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle. By utilizing the barrier methods and iterative methods, we show the well-posedness of the transonic shock in the entire subsonic region and thus establish the global solution. This result does not rely on any smallness of Riemann data.     

Non-negative global weak solutions for a degenerated parabolic system approximating the two-phase Stokes problem

We establish the existence of non-negative global weak solutions for a strongly coupled degenerated parabolic system which was obtained as an approximation of the two-phase Stokes problem driven solely by capillary forces. Moreover, the system under consideration may be viewed as a two-phase generalization of the classical Thin Film equation.     

Stability of two‐dimensional magnetohydrodynamic motions in the periodic case

We investigate the stability of two‐dimensional periodic solutions to magnetohydrodynamics equations in the class of three‐dimensional periodic solutions. We show the existence of global, strong three‐dimensional solutions to magnetohydrodynamics equations, which are close to two‐dimensional solutions. The advantage of our approach is that neither these solutions nor the external forces have to vanish at infinity. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and uniqueness analysis of a detached shock problem for the potential flow

We study a problem for two-dimensional steady potential and isentropic Euler equations in a bounded domain, where an artificial detached shock interacts with a wedge. Using the stream function, we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted Hölder spaces. The proof is based on various Hölder and Schauder estimates for second-order elliptic equations and fixed point theorems. Moreover, we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional.     

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.     

Estimates on enstrophy, palinstrophy, and invariant measures for 2-D turbulence

We construct semi-integral curves which bound the projection of the global attractor of the 2-D Navier-Stokes equations in the plane spanned by enstrophy and palinstrophy. Of particular interest are certain regions of the plane where palinstrophy dominates enstrophy. Previous work shows that if solutions on the global attractor spend a significant amount of time in such a region, then there is a cascade of enstrophy to smaller length scales, one of the main features of 2-D turbulence theory. The semi-integral curves divide the plane into regions having limited ranges for the direction of the flow. This allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy. We also derive a sharp, universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero.     

Blow up and grazing collision in viscous fluid solid interaction systems

We investigate qualitative properties of strong solutions to a classical system describing the fall of a rigid ball under the action of gravity inside a bounded cavity filled with a viscous incompressible fluid. We prove contact between the ball and the boundary of the cavity implies blow up of strong solutions and such a contact has to occur in finite time under symmetry assumptions on the initial data.     

Two-dimensional variational vibroequilibria and Faraday’s drops

When contacting with acoustically-vibrated structures a fluid volume can take a [time-averaged] geometric shape differing from capillary equilibrium. In accordance with theorems by Beyer et al. (2001) this shape (vibroequilibrium) furnishes a local minimum of a [quasi-potential energy] functional. The variational problem contains five dimensionless parameters evaluating the fluid volume, the wave number of acoustic field in the fluid domain, the contact angle and two newly-introduced numbers (1, 2) giving relationships between (surface tension, gravitation) and Kapitsas vibrational forces/energy. The paper focuses on negligible small wave numbers (incompressible fluid) and two-dimensional flows. Although the variational problem may in some isolated cases have analytical solutions, it requires in general numerical approaches. Numerical examples simulate experiments by Wolf (1969) and Ganiyev et al. (1977) on vibroequilibria in horizontally vibrating tanks. These show that there appear at least two types of stable vibroequilibria associated with symmetric (possible non-connected) and asymmetric surface shapes. The paper represents also numerical results on flattening and vibrostabilisation of a drop hanging beneath a vibrating plate (experiments by Faraday (1831)).Received: October 17, 2002; revised: June 30, 2003     

On the incompressible limits for the full magnetohydrodynamics flows

In this article the incompressible limits of weak solutions to the governing equations for magnetohydrodynamics flows on both bounded and unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number and Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains and the derivation of uniform bounds.     

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H ¹. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.     

On the regularity criteria for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations. Some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations, which generalize the results in [C. He, Z. Xin, On the regularity of solutions to the magneto-hydrodynamic equations, J. Differential Equations 213 (2) (2005) 235-254]. Our results reveal that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations.