Görtler Vortices with System Rotation

The steady primary instability of Görtler vortices developing along a curved Blasius boundary layer subject to spanwise system rotation is analysed through linear and nonlinear approaches, to clarify issues of vortex growth and wavelength selection, and to pave the way to further secondary instability studies.A linear marching stability analysis is carried out for a range of rotation numbers, to yield the (predictable) result that positive rotation, that is rotation in the sense of the basic flow, enhances the vortex development, while negative rotation dampens the vortices. Comparisons are also made with local, nonparallel linear stability results (Zebib and Bottaro, 1993) to demonstrate how the local theory overestimates vortex growth. The linear marching code is then used as a tool to predict wavelength selection of vortices, based on a criterion of maximum linear amplification.Nonlinear finite volume numerical simulations are performed for a series of spanwise wave numbers and rotation numbers. It is shown that energy growths of linear marching solutions coincide with those of nonlinear spatially developing flows up to fairly large disturbance amplitudes. The perturbation energy saturates at some downstream position at a level which seems to be independent of rotation, but that increases with the spanwise wavelength. Nonlinear simulations performed in a long (along the span) cross section, under conditions of random inflow disturbances, demonstrate that: (i) vortices are randomly spaced and at different stages of growth in each cross section; (ii) upright vortices are the exception in a universe of irregular structures; (iii) the average nonlinear wavelengths for different inlet random noises are close to those of maximum growth from the linear theory; (iv) perturbation energies decrease initially in a linear filtering phase (which does not depend on rotation, but is a function of the inlet noise distribution) until coherent patches of vorticity near the wall emerge and can be amplified by the instability mechanism; (v) the linear filter represents the receptivity of the flow: any random noise, no matter how strong, organizes itself linearly before subsequent growth can take place; (vi) the Görtler number, by itself, is not sufficient to define the state of development of a vortical flow, but should be coupled to a receptivity parameter; (vii) randomly excited Görtler vortices resemble and scale like coherent structures of turbulent boundary layers.A.Z. has been supported, during his stay at EPFL, by an ERCOFTAC Visitor Grant. A.B. acknowledges the Swiss National Fund, Grant No. 21-36035.92, for travel support associated with this research. This work was also supported by the Swedish Board of Technical Development (NUTEK), the Swedish Technical-Scientific Council (TFR), and an ERCOFTAC Visitor Grant, through which the stay of B.G.B.K. at the EPFL was made possible. Cray-2 computing time for this research was generously provided by the Service Informatique Centrale of EPFL.     

KdV Limit of the Euler–Poisson System

Consider the scaling ${\varepsilon^{1/2}(x-Vt) \to x, \varepsilon^{3/2}t \to t}$ in the Euler–Poisson system for ion-acoustic waves (1). We establish that as ${\varepsilon \to 0}$ , the solutions to such Euler–Poisson systems converge globally in time to the solutions of the Korteweg–de Vries equation.     

Subsonic Flow for the Multidimensional Euler–Poisson System

We establish the existence and stability of subsonic potential flow for the steady Euler–Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on a non-insulated boundary from a fixed point at the exit, and prescribing the pressure at the exit of the nozzle. The Euler–Poisson system for subsonic potential flow can be reduced to a nonlinear elliptic system of second order. In this paper, we develop a technique to achieve a priori \({C^{1,\alpha}}\) estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain enclosed by a Lipschitz continuous boundary. In particular, we discovered a special structure of the Euler–Poisson system which enables us to obtain \({C^{1,\alpha}}\) estimates of the velocity potential and the electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler–Poisson system under perturbations of various data.     

Isolated Singularity for the Stationary Navier—Stokes System

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in B_R \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|^{-(n - 1)/2}) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L^{2n/(n - 1)}(B_R) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? L^b(B_R) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.     

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:

Free Boundary Problems for a Lotka–Volterra Competition System

In this paper we investigate two free boundary problems for a Lotka–Volterra type competition model in one space dimension. The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the right-half-space as time \(t\) goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of the solutions and criteria for spreading and vanishing are also obtained. This paper is an improvement and extension of Guo and Wu (J Dyn Differ Equ 24:873–895, 2012).     

Asymptotic Stability of Landau Solutions to Navier–Stokes System

It is known that the three-dimensional Navier–Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L ²-perturbation.     

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.     

Multi—Function Fluidization Data Acquisition System （FDAS）

Fluidization data acquired, processed and printed out inone integral instrument: pressure drop versus gas velocity fluctuating height versus gas velocity minimum fluidization velocity quality of fluidization expressed in terms of bed collapsing curves: rate of bubble escape rate of particulate sedimentation in dense phase rate of consolidation of packed solids printout of dimensionless subsidence time     

Algebraic-Complex Scheme for Dirichlet–Neumann Data for Parabolic System

In this paper, we use the Laplace–Laplace transformation and complex analysis to give a systematical scheme to determine the proper boundary conditions for initial-boundary value problems in the half space and to construct exponentially sharp pointwise structures of the boundary data. Here, we have used the boundary value problems with the Robin boundary conditions for the convection heat equations and the linearized compressible Navier–Stokes equation with a constant convection velocity to demonstrate this scheme.     

Decay Rates for the Three‐Dimensional Linear System of Thermoelasticity

. We consider the two and three‐dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as . First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lamé system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component. We show that when the domain is convex, the decay rate is never uniform. In fact, the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents. We also give an example, due to D. Hulin, in which these geometric properties hold. In three space dimensions we indicate (without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite‐dimensional subspace of solutions except when the domain is a ball or an annulus. Finally we discuss some closely related controllability and spectral issues. (Accepted May 14, 1998)     

Existence and Uniqueness Theorems for the Two-Dimensional Ericksen–Leslie System

In this paper we study the two dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals if the moment of inertia of the molecules does not vanish. We prove short time existence and uniqueness of strong solutions for the initial value problem in two situations: the space-periodic problem and the case of a bounded domain with spatial Dirichlet boundary conditions on the Eulerian velocity and the cross product of the director field with its time derivative. We also show that the speed of propagation of the director field is finite and give an upper bound for it.     

Time-Periodic Solutions to the Full Navier–Stokes–Fourier System

We consider the full Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting fluid driven by a time-periodic external force. We show the existence of at least one weak time periodic solution to the problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the boundary. Such a condition is in fact necessary, as energetically closed fluid systems do not possess non-trivial (changing in time) periodic solutions as a direct consequence of the Second law of thermodynamics.     

Measure Valued Solutions of the 2D Keller–Segel System

We develop a theory of global measure-valued solutions for the classical Keller–Segel model. These solutions are obtained considering the limit of solutions of a regularized problem. We also prove that different regularizations yield different limit measures in the case in which classical solutions of the Keller–Segel system are not globally defined in time.     

Global Existence for the Vlasov–Poisson System in Bounded Domains

In this paper we prove global existence for solutions of the Vlasov–Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary.     

The Toda System and Clustering Interfaces in the Allen–Cahn equation

We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ₀ be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ₀, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ₀ as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.     

The Vadasz–Olek Model Regarded as a System of Coupled Oscillators

The Vadasz–Olek model of chaotic convection in a porous layer is revisited in this article. The first-order differential equations of this Lorenz-type model are transformed in the governing equations of a damped nonlinear oscillator, modulated by a linear degenerated overdamped oscillator (relaxator) which in turn is coupled to former one by a nonlinear cross force. The benefit of this mechanical analogy is an intuitive picture of the regular and chaotic dynamics described by the Vadasz–Olek model. Thus, there turns out that the “eyes” of the chaotic attractor correspond to the minima of the potential energy of the modulated nonlinear oscillator having a double-well shape. Several new aspects of the subcritical and supercritical dynamic regimes are discussed in some detail.     

Existence of Global Strong Solutions to a Beam–Fluid Interaction System

We study an unsteady nonlinear fluid–structure interaction problem which is a simplified model to describe blood flow through viscoelastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action–reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain, in particular that contact between the viscoelastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, and of the existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.     

Weak�CStrong Uniqueness for the Isentropic Compressible Navier�CStokes System

We prove weak–strong uniqueness results for the isentropic compressible Navier–Stokes system on the torus. In other words, we give conditions on a weak solution, such as the ones built up by Lions (Compressible Models, Oxford Science, Oxford, 1998), so that it is unique. It is of fundamental importance since uniqueness of these solutions is not known in general. We present two different methods, one using relative entropy, the other one using an improved Gronwall inequality due to the author; these two approaches yield complementary results. Known weak–strong uniqueness results are improved and classical uniqueness results for this equation follow naturally.     

Classical Limit for a System of Non-Linear Random Schrödinger Equations

This work is concerned with the semi-classical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.