We establish that for any non-empty, compact set K\subset\mathbb{R}_{\operatorname{sym}}^{3\times 3} the 1 - and \infty -symmetric div-quasiconvex hulls \smash{K^{(1)}} and \smash{K^{(\infty)}} coincide. This settles a conjecture in a recent work of Conti,Müller & Ortiz [Arch. Ration. Mech. Anal. 235 (2020)] in the affirmative. As a key novelty, we construct an \operatorname{L}^{\infty} -truncation that preserves both symmetry and solenoidality of matrix-valued maps in \operatorname{L}^{1} . For comparison, we moreover give a construction of \mathcal{A} -free truncations in the regime 1<p<\infty which, however, does not apply to the case p=1 .

In this paper, we prove that for each closed differential form $$\displaystyle u \in L^1(\mathbb {R}^N;(\mathbb {R}^N)^{*} \wedge ... \wedge (\mathbb {R}^N)^{*})$$ u ∈ L 1 ( R N ; ( R N ) ∗ ∧ . . . ∧ ( R N ) ∗ ) , which is almost in $$\displaystyle L^{\infty }$$ L ∞ in the sense that $$\begin{aligned} \int _{\{y \in \mathbb {R}^N :\vert u(y) \vert \ge L \}} \vert u(y) \vert dy< \varepsilon \end{aligned}$$ ∫ { y ∈ R N : | u ( y ) | ≥ L } | u ( y ) | d y < ε for some $$\displaystyle L>0$$ L > 0 and a small $$\displaystyle \varepsilon >0$$ ε > 0 , we may find a closed differential form v, such that $$\displaystyle \Vert u - v \Vert _{L^1}$$ ‖ u - v ‖ L 1 is again small, and v is, in addition, in $$\displaystyle L^{\infty }$$ L ∞ with a bound on its $$\displaystyle L^{\infty }$$ L ∞ norm depending only on N and L. In particular, the set $$\displaystyle \{ v \ne u\}$$ { v ≠ u } has measure at most $$\displaystyle CL^{-1} \varepsilon .$$ C L - 1 ε . As an application of this theorem, we are able to prove that the $$\displaystyle \mathcal {A}$$ A -p-quasiconvex hull of a set does not depend on p. Furthermore, we can prove a classification theorem for $$\displaystyle \mathcal {A}$$ A -$$\displaystyle \infty $$ ∞ -Young measures.

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.

We consider a homogeneous differential operator $${\mathcal {A}}$$ A and show an improved version of Murat’s condition for $${\mathcal {A}}$$ A -quasiaffine functions, provided that the operator $${\mathcal {A}}$$ A satisfies the constant rank condition. As a consequence, we obtain that affinity along the characteristic cone of $${\mathcal {A}}$$ A implies $${\mathcal {A}}$$ A -quasiaffinity if $${\mathcal {A}}$$ A admits a first order potential.