This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the long-time behavior of solutions of quasi-stationary phase field models and prove the existence of a global attractor.
A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle $\K$ is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite.
This article covers the construction of exponential attractors in two different functional space settings; one is in Hilbert's space, and the other is in the Banach space. The former relies on the squeezing properties of solution trajectories, but the latter does not. We present these different methods for constructing exponential attractors using the three-dimensional Rotating Navier-Stokes-$α$ equations.
We establish the effective {\em finite dimensionality} of the dynamics corresponding to a flow-plate interaction PDE model arising in aeroelasticity: a nonlinear panel, in the absence of rotational inertia, immersed in an inviscid potential flow. An intrinsic component of the analysis is the study of a plate equation with a {\it delay} term---a fundamentally non-gradient dynamics. First, we construct a compact global attractor and observe that the attractor is smooth, with finite fractal dimension in the state space. Secondly, by fattening the attractor, we obtain an exponential attractor, tho…
This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation whose potential is divided into a convex part and a non-…
Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.
We study the global attractors for the damped 3D Euler--Bardina equations with the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ endowed with periodic boundary conditions as well as their damped Euler limit $α\to0$. We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even the non-existence of such solutions in the distributional sense, the limit dynamics of the corresponding dissipative solutions introduced by P.\,Lions can be described in terms of attractors of the properly constructed trajectory dynamical system. Moreover, the con…
An application of dimensional reduction results for gradient constrained problems is provided for 3D-2D dimension reduction for supremal functionals, in the case when the domain is convex.
We give a survey of results regarding existence and regularity for autonomous functionals of linear growth that depend on the symmetric rather than the full gradients.
We consider variational integrals of the form $\int F(D^2u)$ where $F$ is convex and smooth on the Hessian space. We show that a critical point $u\in W^{2,\infty}$ of such a functional under compactly supported variations is smooth if the Hessian of $u$ has a small oscillation.
We study the solvability of second boundary value problems of fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and monopolist's problem in economics. The right hand sides of our Abreu type equations are quasilinear expressions of second order; they are highly singular and a priori just measures. However, our analysis in particular shows that minimizers of t…
We investigate regularity of minimizers in two dimensions for certain classes of non-smooth convex functionals. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov and others
We study Sobolev regularity results for minimisers of autonomous, convex variational of linear growth which depend on the symmetric gradient rather than the full gradient. This extends the results available in the literature for the BV-setting to the case of functionals whose full gradients are a priori not known to exist as matrix-valued Radon measures.
We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous works of Carlier-Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations. 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of b…
We study the convex lift of Mumford-Shah type functionals in the space of rectifiable currents and we prove a generalized coarea formula in dimension one, for finite linear combinations of SBV graphs. We use this result to prove the equivalence between the minimum problems for the Mumford-Shah functional and the lifted one and, as a consequence, we obtain a weak existence result for calibrations in one dimension.
We revisit entropy methods to prove new sharp trace logarithmic Sobolev and sharp Gagliardo-Nirenberg-Sobolev inequalities on the half space, with a focus on the entropy inequality itself and not the actual flow, allowing for somewhat robust and self-contained proofs.
For displacement convex functionals in the probability space equip\-ped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Łoja\-sie\-wicz inequalities. \chg{We also discuss the more general case of $λ$-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo…
We prove global $W^{1,q}(Ω,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_ΩF(x,Du)\mathrm{d} x$. $W^{1,q}(Ω,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $α$-Hölder continuity assumption in $x$ and controlled $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$.
We establish partial regularity of BD-minima for variational integrals of linear growth which depend on the symmetric gradients and satisfy a weak ellipticity condition. Since there is no Korn Inequality in the $L^{1}$-Setup, the result does not follow from the respective result on BV due to Anzellotti and Giaquinta.
Motivated by the direct method in the calculus of variations in $L^{\infty}$, our main result identifies the notion of convexity characterizing the weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria f…
This article is devoted to obtain the $Γ$-limit, as $ε$ tends to zero, of the family of functionals $$F_ε(u)=\int_Ωf\Bigl(x,\frac{x}ε,..., \frac{x}{ε^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.
Sharp mean Hadamard inequalities and polyconvex integrands that give rise to convex functionals
University of Surrey（萨里大学）；Czech Academy of Sciences, Institute of Information Theory and Automation（捷克科学院信息与自动化研究所）；Czech Technical University in Prague（捷克理工大学）
Subdifferentials of a singular convex functional representing the surface free energy of a crystal under the roughening temperature are characterized. The energy functional is defined on Sobolev spaces of order -1, so the subdifferential mathematically formulates the energy's gradient which formally involves 4th order spacial derivatives of the surface's height. The subdifferentials are analyzed in the negative Sobolev spaces of arbitrary spacial dimension on which both a periodic boundary condition and a Dirichlet boundary condition are separately imposed. Based on the characterization theore…
We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: Ω_1\timesΩ_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(Ω_1\timesΩ_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication "$\mathcal{E}(u) = 0 \Rightarrow u \in S(Ω_1\timesΩ_2)$" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by provin…
We present quantitative results for the homogenization of uniformly convex integral functionals with random coefficients under independence assumptions. The main result is an error estimate for the Dirichlet problem which is algebraic (but sub-optimal) in the size of the error, but optimal in stochastic integrability. As an application, we obtain quenched $C^{0,1}$ estimates for local minimizers of such energy functionals.
We show weak lower semi-continuity of functionals assuming the new notion of a "convexly constrained" $\mathcal A$-quasiconvex integrand. We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$ which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent "convexly constrained" $\mathcal A$-free sequences. In a motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convex constraint is positive semi-definiteness of a matrix field.
The $Γ$-limit of a family of functionals $u\mapsto \int_{Ω}f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ is obtained for $s=1,2$ and when the integrand $f=f\left( y,z,v\right) $ is a continous function, periodic in $y$ and $z$ and convex with respect to $v$ with nonstandard growth. The reiterated two-scale limits of second order derivative are characterized in this setting.
We prove partial regularity of stationary solutions and minimizers $u$ from a set $Ω\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_ΩF(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an $ε$-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.
We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has $p$-growth from below (with $p>d$, the dim…
