Div-curl Lemma Research Articles

Homogenization of a NonLinear Strongly Coupled Model of Magnetorheological Fluids

This paper concerns the rigorous periodic homogenization for a nonlinear strongly coupled system, which models a suspension of magnetizable rigid particles in a nonconducting carrier viscous Newtonian fluid. The fluid drags the particles and thus alters the magnetic field. Vice versa, the magnetic field acts on the particles, which in turn affect the fluid via the no-slip boundary condition. As the size of the particles approaches zero, it is shown that the suspension’s behavior is governed by a generalized magnetohydrodynamic system, where the fluid is modeled by a stationary Navier–Stokes system, while the magnetic field is modeled by Maxwell equations. A corrector result from the theory of two-scale convergence allows us to obtain the limit of the product of several weakly convergent sequences, where the div-curl lemma, which is a typical tool in these types of problems, is not applicable.

A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any mathsf {L}_{}^{2}-bounded sequence of vector fields with mathsf {L}_{}^{2}-bounded rotations and mathsf {L}_{}^{2}-bounded divergences as well as mathsf {L}_{}^{2}-bounded tangential traces on one part of the boundary and mathsf {L}_{}^{2}-bounded normal traces on the other part of the boundary, contains a strongly mathsf {L}_{}^{2}-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.

Existence analysis of a degenerate diffusion system for heat-conducting gases

The existence of global weak solutions to a parabolic energy-transport system in a bounded domain with no-flux boundary conditions is proved. The model can be derived in the diffusion limit from a kinetic equation with a linear collision operator involving a non-isothermal Maxwellian. The evolution of the local temperature is governed by a heat equation with a source term that depends on the energy of the distribution function. The limiting model consists of cross-diffusion equations with an entropy structure. The main difficulty is the nonstandard degeneracy, i.e., ellipticity is lost when the gas density or temperature vanishes. The existence proof is based on a priori estimates coming from the entropy inequality and the H^{-1} method and on techniques from mathematical fluid dynamics (renormalized formulation, div-curl lemma).

Heterogeneous discrete kinetic model and its diffusion limit

<p style='text-indent:20px;'>A revertible discrete velocity kinetic model is introduced when the environment is spatially heterogeneous. It is proved that the parabolic scale singular limit of the model exists and satisfies a new heterogeneous diffusion equation that depends on the diffusivity and the turning frequency together. An energy functional is introduced which takes into account spatial heterogeneity in the velocity field. The monotonicity of the energy functional is the key to obtain uniform estimates needed for the weak convergence proof. The Div-Curl lemma completes the strong convergence proof.</p>

Existence of weak solutions to a continuity equation with space time nonlocal Darcy law

In this manuscript, we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator in two space dimensions for Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality and the Div-Curl lemma.

A functional analytic perspective to the div-curl lemma

We present an abstract functional analytic formulation of the celebrated div-curl lemma found by F. Murat and L. Tartar. The viewpoint in this note relies on sequences for operators in Hilbert spaces. Hence, we draw the functional analytic relation of the div-curl lemma to differential forms and other sequences such as the Grad grad-sequence discovered recently by D. Pauly and W. Zulehner in connection with the biharmonic operator.

Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the well-known example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div–curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

Let [Formula: see text] and [Formula: see text] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [Formula: see text] and their dual spaces. More precisely, the authors prove that [Formula: see text] and, for any [Formula: see text], [Formula: see text], where [Formula: see text] denotes the local BMO space, [Formula: see text], for any [Formula: see text] and [Formula: see text], the inhomogeneous Lipschitz space and [Formula: see text] a variant of the local Orlicz–Hardy space related to the Orlicz function [Formula: see text] for any [Formula: see text] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

Global Weak Rigidity of the Gauss\u2013Codazzi\u2013Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity

We are concerned with the global weak rigidity of the Gauss–Codazzi–Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.

Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

We give an atomic decomposition of closed forms on Rn, the coefficients of which belong to some Hardy space of Musielak–Orlicz type. These spaces are natural generalizations of weighted Hardy–Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called Hlog, appears naturally when considering products of functions in the Hardy space H1 and in BMO. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in Hlog. Namely, a closed form in Hlog is the infinite sum of the wedge product between an exact form in the Hardy space H1 and an exact form in BMO. The converse result, which generalizes the classical div–curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued Hlog space can be weakly factorized.

Global existence of weak solutions for [formula omitted] system of chromatography

In the paper James et al. (1995), the authors established a compact framework for general n×n system of chromatography (1.1) by using the kinetic formulation coupled with the compensated compactness method. However, how to construct suitable approximated solutions {uil} of system (1.1) and then to prove the compactness of η(uil)t+q(uil)x in Hloc−1, for the entropy–entropy flux pairs (η,q) constructed by the kinetic formulation, with respect to the sequence {uil}, is an open problem. In this paper, we construct the approximated solutions {uiε} by using the parabolic viscosity method. By carefully calculating the Riemann invariants of system (1.1), we obtained all necessary estimates in the compact framework of James et al. (1995), and gave a complete proof of the global existence of weak solutions for the Cauchy problem (1.1) with the bounded, nonnegative initial data (1.2). As a direct by-product, when the total variation of the initial data is bounded, we obtained a simple proof of the existence of global weak solutions by applying the Div-Curl lemma in the compensated compactness theorem to some pairs of functions (c,f(ui)), where c is a constant.

Global entropy solutions of Cauchy problem for the Le Roux system

In this short paper, the Cauchy problem for the Le Roux system is studied and the global weak solutions are obtained by using a new technique from the div–curl lemma in the compensated compactness theorem, where the regular B V estimate is assumed for one characteristic field. This work extends the previous work, (LeVeque and Temple, 1985), which provided the global solutions when both the characteristic fields are B V bounded by using the Godunov’s method.

A Unified Theory for Some Non-Newtonian Fluids Under Singular Forcing

We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain nondegenerate power-law or Carreau-type fluids. Within our setting, we provide the full-range theory, namely, existence, optimal regularity, and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely, the weighted Lebesgue spaces, with weights in a respective Muckenhoupt class. The analytical highlight is derivation of existence and uniqueness theory for forcing with its regularity well below the natural duality exponent, via estimates in weighted spaces. It is a generalization of [M. Bulíček, L. Diening, and S. Schwarzacher, Anal. PDE, 9 (2016), pp. 1115--1151] to incompressible fluids. Moreover, two technical results, needed for our analysis, may be useful for further studies. They are the solenoidal, weighted, biting div-curl lemma and the solenoidal Lipschitz approximations on domains.

The Cauchy problem for multiphase first-contact miscible models with viscous fingering

In this paper, the Cauchy problem for multiphase first-contact miscible models with viscous fingering, is studied and a global weak solution is obtained by using a new technique from the Div–Curl lemma in the compensated compactness theorem. This work extends the previous works, (Juanes and Blunt, 2006; Barkve, 1989), which provided the analytical solutions and the entropy solutions respectively, of the Riemann problem, and (Lu, 2013), which provided the global solution of the Cauchy problem for the Keyfitz–Kranzer system.

Div–curl type estimates for elliptic systems of complex vector fields

In this work, we extend the classical div–curl lemma proved by Coifman, Lions, Meyer and Semmes to the setup of elliptic complexes of complex vector fields with smooth coefficients. The tools include local versions of the Gagliardo–Nirenberg estimate and Rellich–Kondrachov compactness theorem for compactly supported distributions in the local Hardy space hp(RN) that we prove to hold for 0<p≤1.

An application of weighted Hardy spaces to the Navier–Stokes equations

In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces Hp(w) with w belonging to Muckenhoupt class A∞. As a corollary, one obtains decay estimates of heat semigroup on weighted Hardy spaces. After a weighted version of the div–curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier–Stokes equations with the initial data belonging to weighted Hardy spaces.

Global Div-Curl lemma in negative Sobolev spaces

This paper studies the global version of the Div-Curl lemma in negative Sobolev spaces in bounded domains in R3. We proved that for the two sequences {uj}j=1∞ and {vj}j=1∞ converging to u and v weakly in Lr(Ω) and Lr′(Ω), respectively, with r and r′ being conjugate exponents and 1<r<∞, suppose that {curlvj}j=1∞ is compact in the dual space of W1,r(Ω), and also if {divuj}j=1∞ and the normal part of uj on boundary are compact in the dual of W01,r′(Ω) and in W−1/r,r(∂Ω) respectively or {divuj}j=1∞ and the tangential part of vj on boundary are compact in the dual of W1,r′(Ω) and in W−1/r′,r′(∂Ω) respectively, then we have ∫Ωuj⋅vjdx→∫Ωu⋅vdx provided divuj∈Lr and curlvj∈Lr′. As an application, the classical Div-Curl lemma will be given another proof.

$G$-convergence for non-divergence second order elliptic operators in the plane

The central theme of this paper is the study of $G$-convergence of elliptic operators in the plane. We consider the operator $$ \mathcal{M}[u]=\text{Tr}(A(z) D^2u)=a_{11}(z)u_{xx}+2a_{12}(z)u_{xy}+a_{22}(z)u_{yy} $$ and its formal adjoint $$ \mathcal{N}[v]=D^2(A(w)v)= (a_{11}(w)v)_{xx} + 2(a_{12}(w)v)_{xy}+ (a_{22}(w)v)_{yy}, $$ where $u\in W^{2,p}$ and $v\in L^p$, with $p>1$, and $A$ is a symmetric uniformly bounded elliptic matrix such that $\text{det}A=1$ almost everywhere. We generalize a theorem due to Sirazhudinov--Zhikov, which is a counterpart of the Div-Curl lemma for elliptic operators in non-divergence form. As an application, under suitable assumptions, we characterize the $G$-limit of a sequence of elliptic operators. In the last section we consider elliptic matrices whose coefficients are also in $VMO$; this leads us to extend our result to any exponent $p\in (1,2)$.

Existence of global entropy solutions to general system of Keyfitz–Kranzer type

In this paper, we apply the div–curl lemma in the compensated compactness theory (Tartar, 1979 [38], Murat, 1978 [34]) to the special pair of functions (c,wiε(x,t)) to obtain a very short proof of the existence of global entropy solutions for quite general system (1.1) of Keyfitz–Kranzer type (Keyfitz and Kranzer, 1980 [24]), where c is a constant and wiε(x,t) are variants in (1.1). This work extends in some sense the previous work by the author (Lu, 2011 [32]) for a special system of Keyfitz–Kranzer or Aw–Rascle type. As a by-product, a simple proof of the existence and stability of entropy solutions is also obtained for the hyperbolic system of isentropic gas dynamics in Eulerian coordinates under the compactness assumption of uxε(x,t) or ρxε(x,t) in Wloc−1,α(R×R+), α∈(1,2), where uε(x,t), ρε(x,t) are viscosity velocity and density.