We show that for any given solenoidal initial data in \(L^2\) and any solenoidal external force in \(L_{\text {loc}}^q \bigcap L^{3/2}\) with \(q>3\), there exist partially regular weak solutions of the Navier–Stokes equations in \(\mathbb {R}^4 \times [0,\infty [\) which satisfy certain local energy inequalities and whose singular sets have a locally finite 2-dimensional parabolic Hausdorff measure

We consider evolutionary PDE inclusions of the form $$\begin{aligned} -\lambda {\dot{u}}_\lambda + \Delta u - \mathrm {D}W_0(u) + f \ni \partial \mathscr {R}_1({\dot{u}}) \quad \text {in}\,\,{ (0,T) \times \Omega ,} \end{aligned}$$ where \(\mathscr {R}_1\) is a positively 1-homogeneous rate-independent dissipation potential and \(W_0\) is a (generally) non-convex energy density. This work constructs

For the one-dimensional nonlinear damped Klein–Gordon equation $$\begin{aligned} \partial _{t}^{2}u+2\alpha \partial _{t}u-\partial _{x}^{2}u+u-|u|^{p-1}u=0 \quad \text{ on } \mathbb {R}\times \mathbb {R}, \end{aligned}$$ with \(\alpha >0\) and \(p>2\), we prove that any global finite energy solution either converges to 0 or behaves asymptotically as \(t\rightarrow \infty \) as the sum of \(K\ge 1\)

We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional (3D) nonhomogeneous incompressible Navier–Stokes equations with density-dependent viscosity and vacuum. After establishing some key a priori exponential decay-in-time rates of the strong solutions, we obtain both the global existence and exponential stability of strong

We study a variational Ginzburg–Landau type model depending on a small parameter \(\varepsilon >0\) for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As \(\varepsilon \rightarrow 0\), these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus \({\mathfrak {g}}\) of S is different than 1. Our first main result

In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question of whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature \(\beta ^{-1}\) goes to zero. Our main results are: (1) As \(\beta \rightarrow \infty \) at fixed positive pressure \(p>0\), the Gibbs measures \(\mu _\beta

We introduce a novel variant of the JKO scheme to approximate Darcy’s law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy

In this paper we propose a notion of irreversibility for the evolution of cracks in the presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models, and we investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model

Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary

In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the

We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma–vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh–Taylor sign condition on the total pressure

By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is \(H^2\)-close to a multiply covered circle and is sufficiently rotationally symmetric.

We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin–La–Vicol; in particular, these solutions are not localizable

The structural stability for supersonic solutions of the Euler–Poisson system for hydrodynamical model in semiconductor devices and plasmas in two dimensional domain is established, under the perturbation of the flow velocity and the strength of electric field in the horizontal direction at the entrance of a channel. First, the Euler–Poisson system in the supersonic region is reformulated into a second

We prove the existence of domain walls for the Bénard–Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal forms analysis of an eight-dimensional reduced system. Domain walls are constructed as heteroclinic solutions connecting suitably chosen periodic solutions of this reduced system.

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing

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

For the three-dimensional vacuum free boundary problem with physical singularity where the sound speed is \(C^{1/2}\)-Hölder continuous across the vacuum boundary of the compressible Euler equations with damping, without any symmetry assumptions, we prove the almost global existence of smooth solutions when the initial data are small perturbations of the Barenblatt self-similar solutions to the corresponding

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

We study kinetic theories for isotropic, two-dimensional grain boundary networks which evolve by curvature flow. The number densities \(f_s(x,t)\) for s-sided grains, \(s =1,2,\ldots \), of area x at time t, are modeled by kinetic equations of the form \(\partial _t f_s + v_s \partial _x f_s =j_s\). The velocity \(v_s\) is given by the Mullins–von Neumann rule and the flux \(j_s\) is determined by

We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii–Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity

We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space \(L^1({{\mathbb {R}}}^n)\), and more generally in \(L^p({{\mathbb {R}}}^n)\). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case \(n=2\). This is motivated by the description of

We show that in 3-dimensional ideal magnetohydrodynamics there exist infinitely many bounded solutions that are compactly supported in space-time and have non-trivial velocity and magnetic fields. The solutions violate conservation of total energy and cross helicity, but preserve magnetic helicity. For the 2-dimensional case we show that, in contrast, no nontrivial compactly supported solutions exist

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of walls immersed in a plasma, and to analyze qualitative information of such a sheath layer. In the case of planar wall, Bohm proposed a criterion on the velocity of the positive ion for the formation of sheath, and several works gave its mathematical validation. It is of more interest to analyze

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