Given an elliptic integrand of class \(\mathscr {C}^{2,\alpha }\), we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction \(\phi \). Our objective is to go beyond Einstein’s approximation \(\mu _{eff} = (1+\frac{5}{2}\phi ) \mu \). Assuming a lower bound on the minimal distance between the N particles, we are able to identify the \(O(\phi ^2)\) correction to the effective viscosity

In (Ammari et al. in SIAM J Math Anal. arXiv:1811.03905), the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal comprised of bubbles of arbitrary shape is shown. The aim of this paper is to prove that, near the Dirac points, the Bloch eigenfunctions is the sum of two eigenmodes. Each eigenmode can be decomposed into two components: one which is slowly varying and satisfies

Fluid flows around an obstacle generate vortices which, in turn, generate forces on the obstacle. This phenomenon is studied for planar viscous flows governed by the stationary Navier–Stokes equations with inhomogeneous Dirichlet boundary data in a (virtual) square containing an obstacle. In a symmetric framework the appearance of forces is strictly related to the multiplicity of solutions. Precise

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded 1-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the

We consider an isolated point defect embedded in a homogeneous crystalline solid. We show that, in the harmonic approximation, a periodic supercell approximation of the formation free energy as well as of the transition rate between two stable configurations converge as the cell size tends to infinity. We characterise the limits and establish sharp convergence rates. Both cases can be reduced to a

By a result of Ball (Proc R Soc Edinb Sect A Math 88:315–328, 1981. https://doi.org/10.1017/S030821050002014X), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball’s theorem and related results can be reached while completely avoiding the problem of

This article is devoted to studying the model of two-dimensional attractive Bose–Einstein condensates in a trap V(x) rotating at the velocity \(\Omega \). This model can be described by the complex-valued Gross–Pitaevskii energy functional. It is shown that there exists a critical rotational velocity \(0<\Omega ^*:=\Omega ^*(V)\le \infty \), depending on the general trap V(x), such that for any rotational

We establish that solving an optimal transportation problem in which the source and target densities are defined on spaces with different dimensions, is equivalent to solving a new nonlocal analog of the Monge–Ampère equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits

We present the multicomponent functionalized free energies that characterize the low-energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbit we associate a manifold of low energy membrane-type configurations parameterized by a large class of admissible interfaces. The normal coercivity of the

The second equation of Theorem 3.2

We consider a semilinear parabolic partial differential equation in \(\mathbf{R}_+\times [0,1]^d\), where \(d=1, 2\) or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic

The classical model of an isolated selfgravitating gaseous star is given by the Euler–Poisson system with a polytropic pressure law \(P(\rho )=\rho ^\gamma \), \(\gamma >1\). For any \(1<\gamma <\frac{4}{3}\), we construct an infinite-dimensional family of collapsing solutions to the Euler–Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed

We consider a class of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational

In Liu and Zhang (Arch Ration Mech Anal 235:1405–1444, 2020), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the classical Navier–Stokes system has a global unique solution. The goal of this paper is to extend this type of result to the 3-D anisotropic Navier–Stokes system (ANS) with only horizontal

We prove the existence of solutions of the reduced Hartree–Fock equations at finite temperature for a periodic crystal with a small defect, and show the total screening of the defect charge by the electrons. We also show the convergence of the damped self-consistent field iteration using Kerker preconditioning to remove charge sloshing. As a crucial step of the proof, we define and study the properties

This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order \( \mathcal {O}(\varepsilon ) \), as \(

The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results

In this paper, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards Q-tensor system modeling the hydrodynamic motion of nematic liquid crystals with either Landau–De Gennes bulk potential in \({\mathbb {R}}^3\) or Ball–Majumdar bulk potential in \(\mathbb {T}^3\), a system coupling the forced incompressible Navier–Stokes equation with a dissipative, parabolic

This paper concerns the study of some special ordered structures in turbulent flows. In particular, a systematic and relevant methodology is proposed to construct non trivial and non radial rotating vortices with non necessarily uniform densities and with different m-fold symmetries, \(m\geqq 1\). In particular, a complete study is provided for the truncated quadratic density \((A|x|^2+B){\mathbf{1}}_{{\mathbb

We study the nonlocal vectorial transport equation \(\partial _ty+ ({\mathbb {P}}y \cdot \nabla ) y=0\) on bounded domains of \({\mathbb {R}}^d\) where \({\mathbb {P}}\) denotes the Leray projector. This equation was introduced to obtain the unique optimal rearrangement of a given map \(y_0\) as the infinite time limit of the solution with initial data \(y_0\) (Angenent et al.: SIAM J Math Anal 35:61–97

Gross–Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to initial value problems, in particular the Gross–Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied

We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order \(\varepsilon \). The interaction is non-negative and scaled in such a way that its scattering length is of order \(\varepsilon /N\), while its range is proportional to \((\varepsilon /N)^{\beta

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is collinear to the velocity. The existence theory is based on multi-parameter bifurcation theory.

We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a “typical” steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions \(u_0\) of the Euler equation on \({\mathbb {S}}^3\)

Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier–Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator T(t

A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin

We study the large-time behavior of a hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with the alignment which makes the large time behavior very different from the original Cucker–Smale (CS) alignment model, and far more interesting. Here we focus

The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration by using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the

We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid motions in continuum mechanics, but are also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the

We study solutions of the Newtonian n-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as \(t\rightarrow +\infty \) or as \(t\rightarrow -\infty \). In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold “at infinity”. We show that the flow near

We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in Kelvin’s-Voigt’s rheology to derive a viscoelastic plate model of von Kármán type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity considered in Friedrich and Kružík (SIAM J Math Anal 50:4426–4456, 2018) where the viscosity stress tensor complies with the principle

In this paper, we study the transition threshold problem for the 2-D Navier–Stokes equations around the Couette flow (y, 0) at high Reynolds number Re in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier–Stokes equations. In particular, three kinds of important effects—enhanced dissipation

Let \(\Omega \subset {\mathbb {R}}^3\) be a domain and let \(f\in BV_{{\text {loc}}}(\Omega ,{\mathbb {R}}^3)\) be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation \(f^{-1}\in BV_{{\text {loc}}}\). The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity

In this paper we prove sharp regularity for a differential inclusion into a set \(K\subset {{\mathbb {R}}}^{2\times 2}\) that arises in connection with the Aviles–Giga functional. The set K is not elliptic, and in that sense our main result goes beyond Šverák’s regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami

We study the general family of nonlinear evolution equations of fractional diffusive type \(\partial _t u-\text{ div }\left( |u|^{m_1}\nabla (-\Delta )^{-s} [ |u|^{m_2-1}u]\right) =f\). Such nonlocal equations are related to the porous medium equations with a fractional Laplacian pressure. Our study concerns the case in which the flow takes place in the whole space. We consider \(m_1, m_2>0\), and

We establish that the Dirichlet problem for linear growth functionals on \({\text {BD}}\), the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial \({\text {C}}^{1,\alpha }\)-regularity theory as presently available for the full gradient Dirichlet problem on \({\text {BV}}\). Functions of bounded deformation play an important role in, for example plasticity, however

A natural, “perturbative”, problem in the modelization of the fractional quantum Hall effect is to minimize a classical energy functional within a variational set based on Laughlin’s wave-function. We prove that, for small enough pair interactions, and asymptotically for large particle numbers, a minimizer can always be looked for in the particular form of uncorrelated quasi-holes superimposed to Laughlin’s

We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree \(\frac{k}{2}\) for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure

Following the publication of my paper “A Regularity Criterion for the Navier–Stokes Equation Involving Only the Middle Eigenvalue of the Strain Tensor” in the Archive for Rational Mechanics and Analysis.

We establish the short-time existence of a smooth solution to the surface diffusion equation with an elastic term and without an additional curvature regularization in three space dimensions. We also prove the asymptotic stability of strictly stable stationary sets.

An old problem since Leray (J Math Pure Appl (French) 9:1–82, 1933) asks whether homogeneous D-solutions of the 3 dimensional Navier–Stokes equation in \({\mathbb {R}}^3\) or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two cases: (1) the full 3 dimensional slab case \({\mathbb {R}}^2 \times [0, 1]\) with Dirichlet boundary condition (Theorem 1.1); (2)

We study the existence, uniqueness, regularity and asymptotic spatial behavior of a Navier–Stokes flow past a body, \({\mathscr {B}}\), moving by a time-periodic translational motion of period T, and with zero average. For example, \({\mathscr {B}}\) moves in an oscillating fashion. We then show that the flow is also time-periodic with same period T. However, sufficiently “far” from the body, the oscillatory

The Peregrine soliton \(Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})\) is an exact solution of the 1d focusing nonlinear Schrödinger equation (NLS) \(iB_t+B_{xx}=-2|B|^2B\), having the feature that it decays to \(e^{it}\) at the spatial and time infinities, and with peaks and troughs in a local region. It is considered as a prototype of the rogue wave by the ocean waves community. The 1D NLS is related

When a Gevrey smooth perturbation is applied to a quasi-convex integrable Hamiltonian, it is known that the KAM invariant tori that survive are “sticky”, that is doubly exponentially stable. We show by examples the optimality of this effective stability.

We consider the homogenization problem of the Liouville equation for non-crystalline materials namely, the coefficients are given by the composition of stationary functions with stochastic deformations. We show the asymptotic equations, which involves both macroscopic and microscopic scales.

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density \(\rho \) ahead. The averaging kernel is of exponential type: \(w_\varepsilon (s)=\varepsilon ^{-1} e^{-s/\varepsilon }\). By a transformation of coordinates, the problem can be reformulated as a \(2\times 2\) hyperbolic system with relaxation. Uniform BV bounds

We extend the results about the existence of minimizers, relaxation, and approximation proven by Bonnetier and Chambolle (SIAM J Appl Math 62:1093–1121, 2002), Chambolle and Solci (SIAM J Math Anal 39:77–102, 2007) for an energy related to epitaxially strained crystalline films, and by Braides et al. (ESAIM Control Optim Calc Var 13:717–734, 2007) for a class of energies defined on pairs of function-set

Due to typesetting mistakes, several errors have been introduced.

We show the existence of global weak solutions to the three dimensional Navier–Stokes equations with initial velocity in the weighted spaces \(L^2_{w_\gamma }\), where \(w_\gamma (x)=(1+\vert x\vert )^{-\gamma }\) and \(0<\gamma \leqq 2\), using new energy controls. As an application we give a new proof of the existence of global weak discretely self-similar solutions to the three dimensional Navier–Stokes

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier–Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the 4/5 law in 3d turbulence, predicting a constant flux of energy

We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension d of the interface. The Muskat problem is scaling invariant in the Sobolev space \(H^{s_c}({\mathbb {R}}^d)\) where \(s_c=1+\frac{d}{2}\). Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev

Given \(n\ge 3\), consider the critical elliptic equation \(\Delta u + u^{2^*-1}=0\) in \({\mathbb {R}}^n\) with \(u > 0\). This equation corresponds to the Euler–Lagrange equation induced by the Sobolev embedding \(H^1({\mathbb {R}}^n)\hookrightarrow L^{2^*}({\mathbb {R}}^n)\), and it is well-known that the solutions are uniquely characterized and are given by the so-called “Talenti bubbles”. In addition

We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. In the high frequency limit, we establish the reflection-transmission coefficients for the wave energy for the scattering off the thermostat. To our surprise, even though the thermostat fluctuations are time-dependent, the scattering does not couple wave energy at various frequencies.

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are C 1 , 1 / 2 , and that near non-degenerate points the fracture set is C 1 , α , for some α ∈ ( 0 , 1 ) .

We extend to finite elasticity the Data-Driven formulation of geometrically linear elasticity presented in Conti et al. (Arch Ration Mech Anal 229:79–123, 2018). The main focus of this paper concerns the formulation of a suitable framework in which the Data-Driven problem of finite elasticity is well-posed in the sense of existence of solutions. We confine attention to deformation gradients \(F \in

In this article we deduce necessary and sufficient conditions for the presence of “Conti-type”, highly symmetric, exactly stress-free constructions in the geometrically non-linear, planar n-well problem, generalising results of Conti et al. (Proc R Soc A 473(2203):20170235, 2017). Passing to the limit \(n\rightarrow \infty \), this allows us to treat solid crystals and nematic elastomer differential

In this article we study the low-temperature limit of a Landau–de Gennes theory. Within all \({\mathbb {S}}^2\)-valued \({\mathscr {R}}\)-axially symmetric maps (see Definition 1.1), the limiting energy functional has at least two distinct energy minimizers. One minimizer has a biaxial torus structure, while another minimizer has a split-core segment structure on the z-axis.

For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed in terms of u. The main difficulties in this context arise from the weak regularity of the domain and the possibly non-empty intersection of the graph of u and the transmission interface