Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit

 We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4 (2000), Gauthier-Villars, Paris] and Lions & Masmoudi [Arch. Rational Mech. Anal. 158 (2001), 195–211]. (Accepted June 6, 2002) Published online December 3, 2002 Communicated by Y. BRENIER     

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.     

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).     

Smoothing Estimates for Boltzmann Equation with Full-range Interactions: Spatially Inhomogeneous Case

In this paper, we study the regularity of the solution to the Boltzmann equation with full-range interactions but for the spatially inhomogeneous case. Under the initial regularity assumption on the solution itself, we show that the solution will become immediately smooth for all the variables as long as the time is far way from zero. Our strategy relies upon the new upper and lower bounds for the collision operator established in Chen and He (Arch Ration Mech Anal 201(2):501–548, 2011), a hypo-elliptic estimate for the transport equation and the element energy method.     

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.     

L 1 and BV-type stability of the inelastic Boltzmann equation near vacuum

The L ¹ and BV-type stability to mild solutions of the inelastic Boltzmann equation is given in this paper. The result is an extension of the stability of the classical solution of the elastic Boltzmann equation proved in Ha (Arch. Ration. Mech. Anal. 173:25–42, 2004 [16]). The observation relies on the energy loss of the inelastic Boltzmann equation. This is a continuity work of Alonso (Indiana Univ. Math. J. [1]), where the author obtained the global existence of a mild solution for the inelastic Boltzmann equation. The proof is based on the mollification method and constructing some functionals as the one in Chae and Ha (Contin. Mech. Thermodyn. 17(7):511–524, 2006 [9]).     

Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.     

Sharp Observability Estimates for Heat Equations

The goal of this article is to derive new estimates for the cost of observability of heat equations. We have developed a new method allowing one to show that when the corresponding wave equation is observable, the heat equation is also observable. This method allows one to describe the explicit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain). We show that our estimate is sharp in some cases, particularly in one space dimension and in the multi-dimensional radially symmetric case. Our result extends those in Fattorini and Russell (Arch Rational Mech Anal 43:272–292, 1971) to the multi-dimensional setting and improves those available in the literature, namely those by Miller (J Differ Equ 204(1):202–226, 2004; SIAM J Control Optim 45(2):762–772, 2006; Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 17(4):351–366, 2006) and Tenenbaum and Tucsnak (J Differ Equ 243(1):70–100, 2007). Our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation, in contrast to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment on the applications of our techniques to controllability properties of heat-type equations.     

A Beale–Kato–Majda Criterion for Three Dimensional Compressible Viscous Heat-Conductive Flows

We prove a blow-up criterion in terms of the upper bound of (ρ, ρ ⁻¹, θ) for a strong solution to three dimensional compressible viscous heat-conductive flows. The main ingredient of the proof is an a priori estimate for a quantity independently introduced in Haspot (Regularity of weak solutions of the compressible isentropic Navier–Stokes equation, arXiv:1001.1581, 2010) and Sun et al. (J Math Pure Appl 95:36–47, 2011), whose divergence can be viewed as the effective viscous flux.     

On the Justification of Plate Models

In this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff–Love and Reissner–Mindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern (Arch. Ration. Mech. Anal. 4:145–152, 1959) and is based on the two-energies principle of Prager and Synge. This was half a century ago.     

Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations in the form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal. 11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lamé constants (Georgiadis et al., J. Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions (Flamant–Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the singular behavior present in the classical solutions.      

Degenerate Regularization of Forward–Backward Parabolic Equations: The Regularized Problem

We study a quasilinear parabolic equation of forward–backward type in one space dimension, under assumptions on the nonlinearity which hold for a number of important mathematical models (for example, the one-dimensional Perona–Malik equation), using a degenerate pseudoparabolic regularization proposed in Barenblatt et al. (SIAM J Math Anal 24:1414–1439, 1993), which takes time delay effects into account. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. We also study qualitative properties of such solutions, in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the Lebesgue measure. In this respect, the existence of a family of viscous entropy inequalities plays an important role.     

Interpretation of Second Law of Thermodynamics in the presence of interfaces

In 1996, Muschik and Ehrentraut (J. Non-Equilib. Thermodyn. 21:175–192, 1996) proposed an amendment to the classical Second Law of Thermodynamics, which asserts that, except in equilibria, reversible process directions in state space do not exist. As a consequence of this statement, they proved that the Second Law of Thermodynamics necessarily restricts the constitutive equations and not the thermodynamic processes. In this way, the classical Coleman–Noll approach to the exploitation of Second Law (Coleman and Noll in Arch. Rational Mech. Anal. 13:167–178, 1963) follows by a rigorous proof. In the present paper, we generalize the amendment, in order to encompass the case, not considered in Muschik and Ehrentraut (J. Non-Equilib. Thermodyn. 21:175–192, 1996), in which there are surfaces across which the unknown fields suffer jump discontinuities. Due to the generalization above, we prove that the same conclusions of Muschik and Ehrentraut (J. Non-Equilib. Thermodyn. 21:175–192, 1996) can be achieved also in the presence of non-regular processes. As an application, we study the thermodynamics of a Kortweg-type fluid interface.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

Existence of Global Weak Solution for Compressible Fluid Models of Korteweg Type

This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by Dunn and Serrin (Arch Rational Mech Anal 88(2):95–133, 1985) which can be used as a phase transition model. We improve the results of Danchin and Desjardins (Annales de l’IHP, Analyse non linéaire 18:97–133, 2001) by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient κ. To finish we show the existence of global weak solution in dimension one for a specific type of capillary coefficients with large initial data in the energy space.     

Equilibrium Problems and Limit Analysis of Masonry Beams

We consider planar straight and curved masonry beams with the constitutive equation from Orlandi (Ph.D. thesis, 1999) and Zani (Eur. J. Mech. A, Solids 23:467–484, 2004). After stating some results about the solution to the boundary value problem, the limit analysis for this kind of bodies is outlined, based on energetic considerations (Lucchesi et al. in Q. Appl. Math. 68:713–746, 2010). The static and kinematic theorems of limit analysis, which usually are justified in a heuristic way (Heyman in The Masonry Arch, 1982; Kooharian in Proc. - Am. Concr. Inst. 89:317–328, 1953), are examined from this point of view. It is seen that the kinematic theorem does not always hold but can be proved under some hypotheses that are frequently met in applications.     

Classical Solutions and the Glassey-Strauss Theorem for the 3D Vlasov-Maxwell System

R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59–90] that C ¹ solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

On Entropy Flux of Transversely Isotropic Elastic Bodies

Recently (Liu in J. Elast. 90:259–270, 2008) thermodynamic theory of elastic (and viscoelastic) material bodies has been analyzed based on the general entropy inequality. It is proved that for isotropic elastic materials, the results are identical to the classical results based on the Clausius-Duhem inequality (Coleman and Noll in Arch. Ration. Mech. Anal. 13:167–178, 1963), for which one of the basic assumptions is that the entropy flux is defined as the heat flux divided by the absolute temperature. For anisotropic elastic materials in general, this classical entropy flux relation has not been proved in the new thermodynamic theory. In this note, as a supplement of the theory presented in (Liu in J. Elast. 90:259–270, 2008), it will be proved that the classical entropy flux relation need not be valid in general, by considering a transversely isotropic elastic material body.      

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.