In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive a sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with

This paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for \(n\times n\) hyperbolic conservation laws in one space dimension. These estimates are achieved by a “post-processing algorithm”, checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number \(\varepsilon >0\) is vanishing. Precisely, whenever \(\varepsilon >0\) is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter \(a\in [\frac{2}{3},1]\), we construct

We construct forward self-similar solutions (expanders) for the compressible Navier–Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity due to cavitation at the origin.

The relativistic quantum Boltzmann equation (or the relativistic Uehling–Uhlenbeck equation) describes the dynamics of single-species fast-moving quantum particles. With the recent development of relativistic quantum mechanics, the relativistic quantum Boltzmann equation has been widely used in physics and engineering, for example in the quantum collision experiments and the simulations of electrons

In this paper we prove a conjecture by P.-L. Lions on maximal regularity of \(L^q\)-type for periodic solutions to \(-\Delta u + |Du|^\gamma = f\) in \(\mathbb {R}^d\), under the (sharp) assumption that \(q > d \frac{\gamma -1}{\gamma }\).

We present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schrödinger type. For Hamiltonian dispersive equations with a non-singular symplectic form and d conserved quantities (in addition to the Hamiltonian), one expects that generically \({{\mathcal {L}}}\), the linearization around a periodic traveling wave, will have a particular Jordan

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour

We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary

We study the exterior problem for stationary Navier–Stokes equations in two dimensions describing a viscous incompressible fluid flowing past an obstacle. It is shown that, at small Reynolds numbers, the classical solutions constructed by Finn and Smith are unique in the class of D-solutions (that is, solutions with finite Dirichlet integral). No additional symmetry or decay assumptions are required

A Correction to this paper has been published: https://doi.org/10.1007/s00205-021-01608-9

In this paper we consider the regularity problem of the Navier–Stokes equations in \( {\mathbb {R}}^{3} \). We show that the Serrin-type condition imposed on one component of the velocity \( u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))\) with \( \frac{2}{p}+ \frac{3}{q} <1\), \( 3

We consider an analogue of the Lieb–Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb–Thirring constant converges to the optimal constant of the one-body Gagliardo–Nirenberg interpolation inequality without interaction.

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation

In this paper we prove the convergence of a suitable particle system towards the BGK model. More precisely, we consider an interacting stochastic particle system in which each particle can instantaneously thermalize locally. We show that, under a suitable scaling limit, propagation of chaos does hold and the one-particle distribution function converges to the solution of the BGK equation.

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form $$\begin{aligned} \mathcal {E}[V](X):=\sum _{1\leqq i\sqrt{2}\), in which case \({\overline{\mathcal E}_{\mathrm {sq}}[V]}=-4\). To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.

This paper considers the Alt–Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction \(e \in S^{d-1}\). In \(d=2\) we characterize the regularity properties

We consider ionic electrodiffusion in fluids, described by the Nernst–Planck–Navier–Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data in bounded domains with a smooth boundary in three space dimensions, in the following situations. We consider: a arbitrary positive Dirichlet boundary conditions for the ionic concentrations, arbitrary Dirichlet boundary conditions

We study central configurations lying on a common circle in the Newtonian four-body problem. Using a topological argument we prove that there is at most one co-circular central configuration for each cyclic ordering of the masses on the circle.

The seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of

For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting

In this paper, we develop dispersive PDE techniques for the Fermi–Pasta–Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg–de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium

We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters

In this paper we prove the uniform-in-time \(L^p\) convergence in the inviscid limit of a family \(\omega ^\nu \) of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution \(\omega \) of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of \(\omega ^{\nu }\) to \(\omega \) in

We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected

We prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic

In the paper [5, Section 6], we quoted a

We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric

We study the existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with \(C^{1,\alpha }\) interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

In the Newtonian n-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for homothetic solutions with arbitrary energy, we give a simple and precise formula that relates the Morse indices of these homothetic solutions to the spectra of the normalized

We consider the motion of a finite though large number N of hard spheres in the whole space \({\mathbb R}^n\). Particles move freely until they experience elastic collisions. We use our recent theory of Compensated Integrability in order to estimate how much the particles are deviated by collisions. Our result, which is expressed in terms of hodographs, tells us that only \(O(N^2)\) collisions are

We provide a new analysis of the Boltzmann equation with a constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is \(L^2_{x,v}\); we prove the global well-posedness and a version of scattering, assuming that the data \(f_0\) is sufficiently smooth and localized, and the \(L^2_{x,v}\) norm of \(f_0\) is sufficiently small. The proof relies upon a new scaling-critical

Bernoulli’s free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are “stable” solutions and tractable by the super and subsolution method, variational methods and the implicit function theorem

We study the large-time behavior of systems driven by radial potentials, which react to anticipated positions, \(\mathbf{x}^\tau (t)=\mathbf{x}(t)+\tau \mathbf{v}(t)\), with anticipation increment \(\tau >0\). As a special case, such systems yield the celebrated Cucker–Smale model for alignment, coupled with pairwise interactions. Viewed from this perspective, such anticipation-driven systems are expected

We investigate the regularity of the free boundary for the Signorini problem in \({\mathbb {R}}^{n+1}\). It is known that regular points are \((n-1)\)-dimensional and \(C^\infty \). However, even for \(C^\infty \) obstacles \(\varphi \), the set of non-regular (or degenerate) points could be very large—e.g. with infinite \({\mathcal {H}}^{n-1}\) measure. The only two assumptions under which a nice

A singularly perturbed phase field model used to model lithium-ion batteries including chemical and elastic effects is considered. The underlying energy is given by $$\begin{aligned} I_\varepsilon [u,c ] := \int _{\varOmega }\left( \frac{1}{\varepsilon }f(c)+\varepsilon \Vert \nabla c\Vert ^2+\frac{1}{\varepsilon }{\mathbb {C}}(e(u)-ce_0):(e(u)-ce_0)\right) \, \mathrm{d}x, \end{aligned}$$ where f is

Charged domain walls are a type of domain wall in thin ferromagnetic films which appear due to global topological constraints. The non-dimensionalized micromagnetic energy for a uniaxial thin ferromagnetic film with in-plane magnetization \(m \in {\mathbb {S}}^1\) is given by $$\begin{aligned} E_\varepsilon [m] \ = \ \varepsilon \Vert \nabla m\Vert _{L^2}^2 + \frac{1}{\varepsilon } \Vert m \cdot e_2\Vert

We provide new direct methods to establish symmetrization results in the form of a mass concentration (that is, integral) comparison for fractional elliptic equations of the type \((-\Delta )^{s}u=f \, (0

How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via \(\Gamma \)-convergence for answering this question to leading order in the shell’s thickness and other small parameters. The observed patterns involve “ordered” regions of well-defined wrinkles alongside “disordered” regions

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.

We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic