Strong solutions to a nonlinear fluid structure interaction system

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969] and considered in [V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55-82; V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173-1207]. In particular, the strong solutions here are obtained given initial datum for the Navier-Stokes equation in the space H¹, and initial data for the wave equation w₀ and w₁ in the spaces H²(Ωe) and H¹(Ωe) respectively.     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

Regularity of Traveling Free Surface Water Waves with Vorticity

We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function of class C ^2,μ admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals 1, then we obtain analyticity of the stream function. The regularity results hold not only for periodic or solitary-water waves, but also for any solution to the hydrodynamic equations of class C ^2,μ .     

On the continuation principles for the Euler equations and the quasi-geostrophic equation

We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel-Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation.     

Transport-diffusion et viscosité évanescente

In this paper, we study a convection-diffusion model with respect to a vector field having log-Lipschitz regularity. More precisely, we are interested in the viscous repartition of the mass for the solutions of the corresponding equation. We prove that mass is essentially concentrated around the image, through the flow, of the initial support. This allows us, for non-regular bidimensional vortex patches, to prove the global L^p convergence of the Navier–Stokes vorticity ω_ν to the Eulerian vorticity ω, with p>1. To cite this article: T. Hmidi, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

Pseudodifferential operators on Bochner spaces and an application

We consider pseudodifferential operators with operator valued symbols a(x,ξ) acting on a UMD Banach space X. Assuming some regularity of Hölder type in x and Mihlin type in ξ we prove L _p (? ⁿ ;X) boundedness of such operators. This result is then applied to the study of L _p -maximal regularity for nonautonomous parabolic evolution equations.     

Global Regularity for Some Oldroyd‐B Type Models

We investigate some critical models for visco‐elastic flows of Oldroyd‐B type in dimension 2. We use a transformation that exploits the Oldroyd‐B structure to prove an L^∞ bound on the vorticity which allows us to prove global regularity for our systems. © 2015 Wiley Periodicals, Inc.     

Qregularity and an extension of the Evans-Griffiths criterion to vector bundles on quadrics

Here we define the concept of Qregularity for coherent sheaves on a smooth quadric hypersurface Q_n⊂Pⁿ⁺¹. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Q_n with the Castelnuovo-Mumford regularity of their extension by zero in Pⁿ⁺¹. We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension of the Evans-Griffiths criterion to vector bundles on quadrics. In particular, we get a new and simple proof of Knörrer’s characterization of ACM bundles.     

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L₂ for small complex L_∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices.     

The Z \to c\bar c \to \gamma \gamma *, Z \to b\bar b \to \gamma \gamma * triangle diagrams and the Z → γψ, Z → γγ decays

We expounded an approach for studying the Z ?? ??? and Z ?? ???? decay based on the sum rules for the $Z \to c\bar c \to \gamma \gamma *$ and $Z \to b\bar b \to \gamma \gamma *$ amplitudes and their derivatives. We calculate the branching ratios of the Z ?? ??? and Z ?? ???? decays under different suppositions about the saturation of the sum rules. We find the lower bounds of ?? _?? BR(Z ?? ???) = 1.95 · 10 ^?7 and ?? _?? BR(Z ?? ????) = 7.23 · 10^?7 and discuss deviations from the lower bounds including the possibility of BR[Z ?? ??J/??(1S)] ?? BR[Z ?? ????(1S)] ?? 10 ^?6 , which is probably measurable at the LHC. Moreover, we calculate the angle distributions in the Z ?? ??? and Z ?? ???? decays.     

Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity

In this short note we give a link between the regularity of the solution u to the 3D Navier-Stokes equation and the behavior of the direction of the velocity u/|u|. It is shown that the control of div(u/|u|) in a suitable L ^t/p (L ^x/q ) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines. This work was supported in part by NSF Grant DMS-0607953.     

On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

We consider suitable weak solutions to an incompressible viscous Newtonian fluid governed by the Navier-Stokes equations in the half space \({\mathbb {R}^3_+}\). Our main result is a direct proof of the partial regularity up to the flat boundary based on a new decay estimate, which implies the regularity in the cylinder \({Q_\rho ^+(x_0, t_0)}\) provided

$\limsup_{R\to 0}\frac {1} {R}\int\limits_{Q_R^+(x_0, t_0)} |{\rm rot}\,\mathbf u|^2 dxdt \,\leq\, \varepsilon _0$

with ε ₀ sufficiently small. In addition, we get a new condition for the local regularity beyond Serrin’s class which involves the L ²-norm of ?u and the L ^3/2-norm of the pressure.

     

Some remarks on planar Boussinesq equations

The main purpose of this paper is to prove the well-posedness of the two-dimensional Boussinesq equations when the initial vorticity ω 0 ∈L1 (R 2 ) (or the finite Radon measure space). Using the stream function form of the equations and the Schauder fixed-point theorem to get the new proof of these results, we get that when the initial vorticity is smooth, there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.     

Haefliger’s codimension-one singular foliations, open books and twisted open books in dimension 3

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by André Haefliger as being ??₁-structures. We prove that under the obvious linear embedding condition, they are ??₁-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C ^r , r ?? 1. In particular, in dimension 3, this gives a very simple proof of Thurston??s 1976 regularization theorem without using Mather??s homology equivalence.     

Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance

We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses m₁, m₂ satisfying the resonance relation m₂=2m₁>0. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate O(|t|⁻¹) as t→±∞ in L^∞. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M. Delort, D. Fang and R. Xue [J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004) 288-323].     

Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component

We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of u₃ and ω₃, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorný (2004).     

On series Σc k f(kx) and Khinchin’s conjecture

We prove the optimality of a criterion of Koksma (1953) in Khinchin’s conjecture on strong uniform distribution. This verifies a claim of Bourgain (1988) and leads also to a near optimal a.e. convergence condition for series Σ _k=1 ^∞ c _k f(kx) with f ∈ L ². Finally, we show that under mild regularity conditions on the Fourier coefficients of f, the Khinchin conjecture is valid assuming only f ∈ L ².     

Nonlinear reflecting diffusion process,and the propagation of chaos and fluctuations associated

A nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven. Then a gaussian limit for the fluctuation field which is defined in L₀²(B) of a Wiener type space B is obtained. The covariance of the gaussian limit is computed in terms of a Hilbert-Schmidt operator on L₀²(B).