Homeomorphisms of Bounded Variation

We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher dimensions we show that f ⁻¹ is of bounded variation provided that f ϵ W ^1,1(Ω; R ⁿ ) is a homeomorphism with |Df| in the Lorentz space L ^n-1,1(Ω). Dedicated to Tadeusz Iwaniec on his 60th birthday     

Navier�CStokes Equations with Nonhomogeneous Boundary Conditions in a Bounded Three-Dimensional Domain

Following the ideas developed in Girinon (Annales de l’Institut Poincaré. Analyse Non Linéaire 26:2025–2053, 2009), we prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a bounded region, W ì R³{\Omega\subset \mathbf{R}^{3}} , with nonhomogeneous Dirichlet boundary conditions on ∂Ω.     

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).     

Cauchy’s stress theorem for stresses represented by measures

A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin and Martins (Arch Ration Mech Anal 60:305–324, 1976). The main result of the paper says that unbounded Cauchy fluxes are in one-to-one correspondence with tensor valued measures with weak divergence a vector valued measure. Unavoidably, the force transmitted by a surface generally cannot be defined for all surfaces but only for almost every translation of the surface. Also conditions are given guaranteeing that the transmitted force is represented by a measure. These results are proved by using a new homotopy formula for tensor valued measure with weak divergence a vector valued measure.      

Propagation of Density-Oscillations in Solutions to the Barotropic Compressible Navier–Stokes System

Considering a bounded sequence of weak solutions to the compressible Navier–Stokes system, we introduce Young measures as in [12] in order to describe a “homogenized system” satisfied in the limit. We then study the Cauchy problem associated to this “homogenized system” when Young measures are convex combinations of Dirac measures.     

Lyapunov and Sacker–Sell Spectral Intervals

In this work, we show that for linear upper triangular systems of differential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also –under some restrictions– the Lyapunov spectral intervals. Since any bounded and continuous coefficient matrix function can be smoothly transformed to an upper triangular matrix function, our results imply that these spectral intervals may be found from scalar homogeneous problems. In line with our previous work [Dieci and Van Vleck (2003), SIAM J. Numer. Anal. 40, 516–542], we emphasize the role of integral separation. Relationships between different spectra are shown, and examples are used to illustrate the results and define types of coefficient matrix functions that lead to continuous Sacker–Sell spectrum and/or continuous Lyapunov spectrum.      

Compressible Euler Equations¶with General Pressure Law

We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found. We prove the strong compactness of any sequence that is uniformly bounded in L ^∞ and whose corresponding sequence of weak entropy dissipation measures is locally H ^-1 compact. The existence and large-time behavior of L ^∞ entropy solutions of the Cauchy problem are established. This is based on a reduction theorem for Young measures, whose proof is new even for the polytropic perfect gas. The existence result also extends to the p-system of fluid dynamics in Lagrangian coordinates. Accepted: December 16, 1999     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov’s theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228–255, 1970) can be entirely recast in L ² and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563–566, 1981; in J Diff Equ, 50, 375–390, 1983; in SIAM J Numer Anal, 21, 1013–1037; in Methods Appl Anal, 11, 515–532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505–515, 1983), and Tsai et al. (in Math Comp, 72, 159–181, 2003).     

A Simple Proof of the Generalized Cauchy’s Theorem

The Cauchy’s theorem for balance laws is proved in a general context using a simpler and more natural method in comparison to the one recently presented in Segev (Arch. Ration. Mech. Anal. 154:183–198, 2000). By “generality” we mean that the ambient space is considered to be an orientable smooth manifold, and not only the Euclidean space.     

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

On a Bound for Amplitudes of Navier�CStokes Flow with almost Periodic Initial Data

For any bounded (real) initial data it is known that there is a unique global solution to the two-dimensional Navier–Stokes equations. This paper is concerned with a bound for the sum of the modulus of amplitudes when initial velocity is spatially almost periodic in 2D. In the case of general dimension, it is bounded on local time of existence shown by Giga et al. (Methods Appl Anal 12:381–393,2005). A class of initial data is given such that the sum of the modulus of amplitudes of a solution is bounded on any finite time interval. It is shown by an explicit example that such a bound may diverge to infinity as the time goes to infinity at least for complex initial data.     

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W ^1,p -W ^1,q “duality pairing” with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W ^1,∞-W ^1,1 “duality pairing” which leads to optimal order error estimates in a discrete W ^1,∞-norm.     

Probabilistic Analysis of the Upwind Scheme for Transport Equations

We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129–155, 2007) and Merlet (SIAM J Numer Anal 46(1):124–150, 2007): we prove that the scheme is of order 1/2 in L^￥([0,T],L¹(\mathbb R^d)){L^{\infty}([0,T],L^1(\mathbb R^d))} for an integrable initial datum of bounded variation and of order 1/2−ε, for all ε > 0, in L^￥([0,T] ×\mathbb R^d){L^{\infty}([0,T] \times \mathbb R^d)} for an initial datum of Lipschitz regularity. Our analysis provides a new interpretation of the numerical diffusion phenomenon.     

Smoothing Estimates for Boltzmann Equation with Full-Range Interactions: Spatially Homogeneous Case

In this work, we are concerned with the regularities of the solutions to the Boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials, r ^−(p−1) with p > 2. We give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional Sobolev norms. As an application, we get the global entropy dissipation estimate which is a little stronger than that described by Alexandre et al. (Arch Rational Mech Anal 152(4):327–355, 2000). As another application, we prove the smoothing effects for the strong solutions constructed by Desvillettes and Mouhot (Arch Rational Mech Anal 193(2):227–253, 2009) of the spatially homogeneous Boltzmann equation with “true” hard potential and “true” moderately soft potential.     

Maximal L p – L q -Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal L^p – L^q-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an R{\mathcal{R}}-sectorial operator in L^p_s(W)L^{p}_{\sigma}(\Omega), 1 < p < ￥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω.     

The Benjamin–Lighthill Conjecture for Steady Water Waves (revisited)

The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin–Lighthill conjecture is proved for these waves provided Bernoulli’s constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoulli’s constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoulli’s constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433–488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.     

Group classification of one-dimensional nonisentropic equations of fluids with internal inertia

A systematic application of the group analysis method for modeling fluids with internal inertia is presented. The equations studied include models such as the nonlinear one-velocity model of a bubbly fluid (with incompressible liquid phase) at small volume concentration of gas bubbles (Iordanski Zhurnal Prikladnoj Mekhaniki i Tekhnitheskoj Fiziki 3, 102–111, 1960; Kogarko Dokl. AS USSR 137, 1331–1333, 1961; Wijngaarden J. Fluid Mech. 33, 465–474, 1968), and the dispersive shallow water model (Green and Naghdi J. Fluid Mech. 78, 237–246, 1976; Salmon 1988). These models are obtained for special types of the potential function W(r,[(r)\dot],S){W(\rho,\dot \rho,S)} (Gavrilyuk and Teshukov Continuum Mech. Thermodyn. 13, 365–382, 2001). The main feature of the present paper is the study of the potential functions with W _S  ≠ 0. The group classification separates these models into 73 different classes.     

The Rellich-Kondrachov theorem for unbounded domains

Summary The full Kondrachov compactness theorem for Sobolev imbeddings of the type W ₀ ^m,p (G) W ₀ ^j,r (G) on bounded domains G in R ⁿ is extended to a large class of unbounded domains with reasonable n- 1 dimensional boundaries. A Poincaré inequality is obtained for such domains and a compactness theorem for traces of functions in W ₀ ^m,p (G) on lower dimensional hyperplanes is also proved.     

Boundary Regularity of Shear Thickening Flows

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W^{1,(np+2-p)/(n-2)}(W){W^{1,(np+2-p)/(n-2)}(\Omega)} . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.