Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints

Guerra, André; Raiţă, Bogdan

doi:10.48550/arxiv.1909.03923

Cited by 10 publications

(21 citation statements)

References 76 publications

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“…Another angle from which we would like to investigate linear systems of PDE comes from the analysis of continuum mechanics problems, where one often studies a nonlinear relation without derivatives, coupled with a linear PDE [26]; this is the so called theory of compensated compactness [7,10,18,25]. Since the natural mode of convergence for such problems is a convergence of measurements (averages), spaces of smooth functions are not spaces where one can expect existence of solutions for the nonlinear problems.…”

Section: We Start From the Following Vague Questionmentioning

confidence: 99%

“…The question we are interested in is that of lower semicontinuity of integral functionals with respect to weakly convergent sequences v j that satisfy Av j = 0. This problem was solved completely under the real constant rank assumption in [2,7,10]. However, in the absence of the rank condition, very little is known [15,17].…”

Section: We Start From the Following Vague Questionmentioning

confidence: 99%

“…Another important property is that of constant rank, which is particularly relevant in the study of compensated compactness [7,10,18]. Definition 2.8.…”

Section: Classes Of Operatorsmentioning

confidence: 99%

“…Remark 5.13. We will see that for the conclusion of Proposition 5.9 to hold, it suffices to require that z ∈ span W S in (10).…”

Section: Applications To the Calculus Of Variationsmentioning

confidence: 99%

“…|F (zP + Sw j ) − F (z p + Sw j,δ )| = P \P δ |F (z P + Sw j ) − F (z p + Sw j,δ )| |P \ P δ |c 6 c 6 ε|P |.We thus obtain that lim infj→∞ III P ∈Fm −c 6 ε|P | + P F (z P + w j,δ ) − F (z P ) −c 6 ε|Ω|,where we use the quasiconvexity inequality(10) in P . cf.…”

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Syzygies, constant rank, and beyond

Härkönen¹,

Nicklasson²,

Raiță³

2021

Preprint

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We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.

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Section: We Start From the Following Vague Questionmentioning

confidence: 99%

Section: We Start From the Following Vague Questionmentioning

confidence: 99%

“…Another important property is that of constant rank, which is particularly relevant in the study of compensated compactness [7,10,18]. Definition 2.8.…”

Section: Classes Of Operatorsmentioning

confidence: 99%

“…Remark 5.13. We will see that for the conclusion of Proposition 5.9 to hold, it suffices to require that z ∈ span W S in (10).…”

Section: Applications To the Calculus Of Variationsmentioning

confidence: 99%

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Syzygies, constant rank, and beyond

Härkönen¹,

Nicklasson²,

Raiță³

2021

Preprint

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A Data-Driven Approach to Viscous Fluid Mechanics: The Stationary Case

Lienstromberg

Schiffer

Schubert

2023

Arch Rational Mech Anal

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We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81–101, 2016; Conti et al. in Arch Ration Mech Anal 229:79–123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and $${\mathscr {A}}$$ A -quasiconvexity, we show a $$\Gamma $$ Γ -convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.

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Endpoint $$L^1$$ estimates for Hodge systems

2022

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Let $$d\ge 2$$ d ≥ 2 . In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$\begin{aligned} \Vert I_\alpha F\Vert _{{\dot{B}}^{0,1}_{d/(d-\alpha ),1}(\mathbb {R}^d;\mathbb {R}^k)} \le C \Vert F \Vert _{L^1(\mathbb {R}^d;\mathbb {R}^k)} \end{aligned}$$ ‖ I α F ‖ B ˙ d / ( d - α ) , 1 0 , 1 ( R d ; R k ) ≤ C ‖ F ‖ L 1 ( R d ; R k ) for all $$F \in L^1(\mathbb {R}^d;\mathbb {R}^k)$$ F ∈ L 1 ( R d ; R k ) which satisfy a first order cocancelling differential constraint, where $$\alpha \in (0,d)$$ α ∈ ( 0 , d ) and $$I_\alpha $$ I α is a Riesz potential. We show how this implies endpoint Besov–Lorentz estimates for Hodge systems with $$L^1$$ L 1 data via fractional integration for exterior derivatives.

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Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints

Cited by 10 publications

References 76 publications

Syzygies, constant rank, and beyond

Syzygies, constant rank, and beyond

A Data-Driven Approach to Viscous Fluid Mechanics: The Stationary Case

Endpoint $$L^1$$ estimates for Hodge systems

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