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

In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions \(\rho \) to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe \(\rho (x,t)\) as a stochastic process in x with t fixed, or as a stochastic process in t with x fixed. In this article we largely resolve this conjecture.

In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently in Lindblad and Luo (Commun Pure Appl Math 71:1273–1333, 2018) and Luo (Ann PDE 4(2):1–71, 2018), the presence of surface tension introduces severe new technical challenges, in that several

We consider the problem of finding a surface \(\Sigma \subset {\mathbb {R}}^m\) of least Willmore energy among all immersed surfaces having the same boundary, boundary Gauss map and area. Such a problem was considered by S. Germain and S.D. Poisson in the early XIX century as a model for equilibria of thin, clamped elastic plates. We present a solution in the case of boundary data of class \(C^{1,1}\)

Continuing the program initiated in Golovaty et al. (SIAM J Math Anal 51(1):276–320, 2018), we analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain \(\Omega \) we investigate the \(\varepsilon \rightarrow 0\) asymptotics

We prove the existence and regularity of minimisers for the Canham–Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the 2-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to proving the main results we establish the lower semicontinuity of the Canham–Helfrich energy under

This paper is concerned with two dual aspects of the regularity question for the Navier–Stokes equations. First, we prove a local-in-time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space \(L^3\), then the solution is locally smooth in space for some short time, which is quantified. This builds

The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These deformations are homeomorphisms \(\, h :{\mathbb {X}} \xrightarrow []{{}_{\!\!\text{ onto }\,\,\!\!}}{\mathbb {Y}}\,\) between domains \(\,{\mathbb {X}}, {\mathbb {Y}} \subset {\mathbb {R}}^n\,\) of the Sobolev class \(\,{\mathscr {W}}^{1,n}_{\text{ loc }} ({\mathbb {X}}, {\mathbb

We consider the dissipative SQG equation in bounded domains, first introduced by Constantin and Ignatova in 2016. We show global Hölder regularity up to the boundary of the solution, with a method based on the De Giorgi techniques. The boundary introduces several difficulties. In particular, the Dirichlet Laplacian is not translation invariant near the boundary, which leads to complications involving

We analyse integral representation and \(\varGamma \)-convergence properties of functionals defined on piecewise rigid functions, that is, functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally

We propose results that relate the following two contexts: (i) Given a Riemann metric G on \(\Omega ^1=\omega \times (-\frac{1}{2}, \frac{1}{2})\), we find the infimum of the averaged pointwise deficit of an immersion from attaining the orientation-preserving isometric immersion of \(G_{\mid \Omega ^h}\) on \(\Omega ^h=\omega \times (-\frac{h}{2}, \frac{h}{2})\), over all weakly regular immersions

We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific

We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on \(\mathbb {T}^3 \times \mathbb {R}^3\). Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network

We study isomerizations in quantum mechanics. We consider a neutral molecule composed of N quantum electrons and M classical nuclei and assume that the first eigenvalue of the corresponding N-particle Schrödinger operator possesses two local minima with respect to the locations of the nuclei. An isomerization is a mountain pass problem between these two local configurations, where one minimizes over

We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within an arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of \(\Gamma \)-development we recover Landau-de Gennes corrections to

We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao (Z Angew Math Phys (ZAMP) 57:419–432, 2006) or Tebou (C R Acad Sci Paris Ser I 350: 603–608, 2012) for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose to deal

Federer’s characterization states that a set \(E\subset \mathbb {R}^n\) is of finite perimeter if and only if \(\mathcal H^{n-1}(\partial ^*E)<\infty \). Here the measure-theoretic boundary \(\partial ^*E\) consists of those points where both E and its complement have positive upper density. We show that the characterization remains true if \(\partial ^*E\) is replaced by a smaller boundary consisting

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem

In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must coincide

We consider the question raised by Enciso and Peralta-Salas (Arch Ration Mech Anal 220(1):243–260, 2016): what nonconstant functions f can occur as the proportionality factor for a Beltrami field \({{\mathbf {u}}}\) on an open subset \(U \subset \mathbb {R}^3\)? We also consider the related question: for any such f, how large is the space of associated Beltrami fields? By applying Cartan’s method of

In this paper, we consider the Cauchy problem to the heat conductive compressible Navier–Stokes equations in the presence a of vacuum and with a vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions: that the scaling invariant quantity \(\Vert \rho _0\Vert _\infty (\Vert \rho _0\Vert _3+\Vert \rho _0\Vert

This paper is devoted to the study of periodic solutions of a Hamiltonian system \(\dot{z}(t)=J \nabla H(z(t))\), where H is symmetric under an action of a compact Lie group. We are looking for periodic solutions in a neighborhood of non-isolated critical points of H which form orbits of the group action. We prove a Lyapunov-type theorem for symmetric Hamiltonian systems.

We characterize the well known self-similar Blasius profiles, \([{\bar{u}}, {\bar{v}}]\), as downstream attractors to solutions [u, v] to the 2D, stationary Prandtl system. It was established in Serrin (Proc R Soc Lond A 299:491–507, 1967) using maximum principle techniques that \(\Vert u - {\bar{u}}\Vert _{L^\infty _y} \rightarrow 0\) as \(x \rightarrow \infty \). In the case of localized data near