Linear Differential Constraints Research Articles

Homogenization of high-contrast composites under differential constraints

Abstract We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical properties. Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in L p {L^{p}} , which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, k-th order, homogeneous differential operators with constant coefficients and constant rank.

New symmetries, group-invariant solutions, linear differential constraints of a generalized Burgers-KdV equation and its reduction

All we know that the Burgers-KdV equation is extensively used to study the liquid flow with bubbles and the liquid moving flow in the elastic pipes. In this paper, we obtain the Lie point symmetries, self-nonlinear adjointness of a generalized Burgers-KdV equation (GB-KdVE) are obtained, it follows that the conservation laws are worked out. As a reduction of the GB-KdVE, a Burgers equation with general coefficients is presented, whose new strong symmetry and new nonlocal symmetries are generated, respectively. Furthermore, the noninvariant solutions of the GB-KdVE are produced as well. Finally, we propose the double linear differential constraints for GB-KdVE-type so that some soliton solutions are singled out.

A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics

We propose a new pressure-based structure-preserving (SP) and quasi asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the unified first order hyperbolic formulation of continuum mechanics [1], which goes back to the pioneering work of Godunov [2] and further work of Godunov & Romenski [3] and Peshkov & Romenski [4]. The unified model is based on the theory of symmetric-hyperbolic and thermodynamically compatible (SHTC) systems [2,5] and includes the description of elastic and elasto-plastic solids in the nonlinear large-strain regime as well as viscous and inviscid heat-conducting fluids, which correspond to the stiff relaxation limit of the model. In the absence of relaxation source terms, the homogeneous PDE system is endowed with two stationary linear differential constraints (involutions), which require the curl of distortion field and the curl of the thermal impulse to be zero for all times. In the stiff relaxation limit, the unified model tends asymptotically to the compressible Navier-Stokes equations.The new structure-preserving scheme presented in this paper can be proven to be exactly curl-free for the homogeneous part of the PDE system, i.e. in the absence of relaxation source terms. We furthermore prove that the scheme is quasi asymptotic preserving in the stiff relaxation limit, in the sense that the numerical scheme reduces to a consistent second order accurate discretization of the compressible Navier-Stokes equations when the relaxation times tend to zero. Last but not least, the proposed scheme is suitable for the simulation of all Mach number flows thanks to its conservative formulation and the implicit discretization of the pressure terms.

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations.

On the two-state problem for general differential operators

In this note we generalize the Ball-James rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint. In particular, we prove the rigidity for approximate solutions to the two-state inclusion with incompatible states for merely L1-bounded sequences. In this way, our theorem can be seen as a result of compensated compactness in the linear-growth setting.

Periodic homogenization of integral energies under space-dependent differential constraints

A homogenization result for a family of oscillating integral energies u_{\epsilon} \mapsto \int_{\Omega} f(x,\frac{x}{\epsilon},u_{\epsilon}(x))\,dx,\quad \epsilon \to 0^+ is presented, where the fields u_{\epsilon} are subjected to first order linear differential constraints depending on the space variable x . The work is based on the theory of \mathscr A -quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of \mathscr A -quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.

On Compactness of Minimizing Sequences Subject to a Linear Differential Constraint

For \Omega\subset \mathbb R^N open, we consider integral functionals of the form F(u):=\int_\Omega f(x,u)\,dx, defined on the subspace of L^p consisting of those vector fields u which satisfy the system \mathcal{A} u=0 on \Omega in the sense of distributions. Here, \mathcal{A} may be any linear differential operator of first order with constant coefficients satisfying Murat's condition of constant rank. The main results provide sharp conditions for the compactness of minimizing sequences with respect to the strong topology in L^p . Although our results hold for bounded domains as well, our main focus is on domains with infinite measure, especially exterior domains.

Multiple integrals under differential constraints: two-scale convergence and homogenization

Two-scale techniques are developed for sequences of maps {u k } ⊂ L p (Ω; ℝ M ) satisfying a linear differential constraint Au k = 0. These, together with Γ-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type F e (u) := ∫ Ω f(x,x/e, u(x)) d x with u ∈ L p (Ω; ℝ M ), Au = 0, that generalizes current results in the case where A = curl.

Oscillations and concentrations generated by ${\mathcal A}$-free mappings and weak lower semicontinuity of integral functionals

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps {uk}k∈N ⊂ L p (Ω;R m ) satisfying a linear differential constraint Auk = 0. Applications to sequential weak lower semicontinuity of integral functionals on A-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det∇ϕk ∗ � det∇ϕ in measures on the closure of Ω ⊂ R n if ϕk �ϕ in W 1,n (Ω;R n ). This convergence holds, for example, under Dirichlet conditions. Further, we formulate a Biting-like lemma precisely stating which subsets Ωj ⊂ Ω must be removed to obtain weak lower semicontinuity of u � � Ω\Ωj v(u(x)) dx along {uk }⊂ L p (Ω;R m )∩ker A. Specifically, Ωj are arbitrarily thin boundary layers.

On integral principles for nonholonomic systems

On integral principles for nonholonomic systems

Investigation of turbulent convection under a rotational constraint

Turbulent convection for a rotating layer of fluid heated from below is studied in this paper. The boundaries of the fluid layer are taken to be free. The underlying principle, used is the Malkus hypothesis that the flow tends to transport the maximum amount of heat possible, subject to certain constraints. By taking the Prandtl number to be infinite, a linear differential constraint and an integral constraint are used. The variational problem that follows then depends on two dimensionless parameters, the Taylor number T and the Rayleigh number R.Asymptotic analysis for the turbulent regime shows that the flow arranges itself so as to tend to offset the stabilizing effect of the rotational constraint, at least in so far as the heat flux is concerned. The dimensionless heat flux, or the Nusselt number, has in general different dependence on T and R, depending on the particular region in the parameter space. For T [les ] O(R), the flow is essentially non-rotating. For O(R) [les ]T [les ] O(R4/3), the flow will always have finitely many horizontal wavenumbers, though the total number of modes increases as T increases in this region. For O(R4/3) [les ] T [les ] O (R3/2), the Nusselt number has a functional dependence proportional to R3/T2, having essentially infinitely many horizontal modes as both R and T increase indefinitely in this region. The last expression is particularly interesting, as it agrees qualitatively with results in finite-amplitude laminar convection. It is also linearly dependent on the layer thickness, as one might expect from dimensional argument. It is suggested that, in the context of the maximum principle, the result in this region of the parameter space may be applicable as well to the same fluid layer with rigid boundaries through the existence of an Ekman layer that is thinner than the thermal layer.

On “virtual displacements” of material systems with linear differential constraints of the second order

On “virtual displacements” of material systems with linear differential constraints of the second order