We prove the existence of a L2-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution θ=|x|. We use a modified energy method to prove the existence time of classical solutions from 1ϵ to a time scale of 1ϵ4. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in Hps(T), with s≥0. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz

The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear

Consider a nonlocal conservation law where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation

In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bounded domain. The domain and the stationary point vortex

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity∂t2u−∂x2u+u−|u|p−1u+|u|q−1u=0,on[0,∞)×R, where 1

We construct and quantify asymptotically in the limit of large mass a variety of edge-localized stationary states of the focusing nonlinear Schrödinger equation on a quantum graph. The method is applicable to general bounded and unbounded graphs. The solutions are constructed by matching a localized large amplitude elliptic function on a single edge with an exponentially smaller remainder on the rest

The Markov and Lagrange dynamical spectra were introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be proved in the dynamical case and we do not know if they are true in classical case. They can be a good source of natural conjectures about the classical spectra: it is

We prove a stability result of constant equilibra for the three dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter ε while the incompressible part of the initial velocity is assumed to be small compared to ε. We then

We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set, proving that they are sufficiently close to critical points of a suitable nonlocal potential. We then consider the fractional perimeter in half-spaces. We prove existence of minimizers under fixed volume constraint, and we show some properties such as smoothness and rotational

In this paper we consider minimizers of the functionalmin⁡{λ1(Ω)+⋯+λk(Ω)+Λ|Ω|,:Ω⊂D open} where D⊂Rd is a bounded open set and where 0<λ1(Ω)≤⋯≤λk(Ω) are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets Ω⁎ have finite perimeter and that their free boundary ∂Ω⁎∩D is composed of a regular

We establish the regularity in 2 dimension of L2 solutions to critical elliptic systems in divergence form involving chirality operators of finite W1,2-energy.

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0∈M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0∞(M∖{q0}) is essentially self-adjoint in L2(M). A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in

We prove propagation of 1s-Gevrey regularity (s∈(0,1]) for the Vlasov-Poisson system on Td×Rd using a Fourier space method in analogy to the results proved for the 2D-Euler system in [20] and [23]. More precisely, we give quantitative estimates for the growth of the 1s-Gevrey norm and decay of the regularity radius for the solution of the system in terms of the force field and the volume of the support

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise

The family of planar linear chains are found as collision-free action minimizers of the spatial N-body problem with equal masses under DN and DN×Z2-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in [32] for the planar N-body problem. In particular, the monotone constraints required in [32] are proven to be unnecessary, as it will

We study the (hydro-)dynamics of multi-species driven by alignment. What distinguishes the different species is the protocol of their interaction with the rest of the crowd: the collective motion is described by different communication kernels, ϕαβ, between the crowds in species α and β. We show that flocking of the overall crowd emerges provided the communication array between species forms a connected

We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures with a small pure point part and global well-posedness for measures with a small total variation. Our construction is based on an L1-estimate of a solution operator

We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for C1 flows, every sectional hyperbolic set Λ is entropy expansive, and the topological entropy varies continuously with the flow. Furthermore, if Λ is Lyapunov stable, then it has positive entropy; in addition, if Λ is a chain recurrent class, then it contains a periodic orbit. As a corollary, we

In this paper we classify the nonnegative global minimizers of the functionalJF(u)=∫ΩF(|∇u|2)+λ2χ{u>0}, where F satisfies some structural conditions and χD is the characteristic function of a set D⊂Rn. We compute the second variation of the energy and study the properties of the stability operator. The free boundary ∂{u>0} can be seen as a rectifiable n−1 varifold. If the free boundary is a Lipschitz

In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly

In this paper, we investigate the spreading properties of solutions of farmer and hunter-gatherer model which is a three-component reaction-diffusion system. Ecologically, the model describes the geographical spreading of an initially localized population of farmers into a region occupied by hunter-gatherers. This model was proposed by Aoki, Shida and Shigesada in 1996. By numerical simulations and

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u=0. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied

Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations in the whole space R3. Under some additional hypotheses, stated in terms of Lebesgue and Morrey spaces, we will show that the trivial solution U→=0 is the unique solution. This type of results are known as Liouville theorems

We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state p=−A2ρα relating the pressure to the mass-energy density, in which A>0 and α∈(0,1] are constants. We express the Einstein equations in wave gauge as a system

In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain [9].

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This

We consider the third order Benjamin-Ono equation on the torus∂tu=∂x(−∂xxu−32uH∂xu−32H(u∂xu)+u3). We prove that for any t∈R, the flow map continuously extends to Hr,0s(T) if s≥0, but does not admit a continuous extension to Hr,0−s(T) if 00, but is not weakly sequentially continuous in Lr,02(T). We then classify the traveling wave solutions for the third order Benjamin-Ono equation in Lr,02(T) and study

We consider the logarithmic Schrödinger equation (logNLS) in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In this paper, we construct multi-solitons (or multi-Gaussons) for logNLS, with estimates in H1∩F(H1). We also construct solutions to logNLS behaving (in L2) like

We establish local C1,α estimates for one source near field refractors under structural assumptions on the target, and with no assumptions on the smoothness of the densities.

This paper is concerned with the Cauchy problem of the 2D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space Hs(R2) for s>−1/4, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of S1-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic vector spaces, we give a

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in Rn is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning

We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon

This paper is concerned with the classical two-species Lotka-Volterra diffusion system with strong competition. The sharp dynamical behavior of the solution is established in two different situations: either one species is an invasive one and the other is a native one or both are invasive species. Our results seem to be the first that provide a precise spreading speed and profile for such a strong

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures

In this paper we obtain comparison results for the quasilinear equation −Δp,xu−uyy=f with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable x, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in y, and the proof of a comparison principle for the discrete version of the

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations (−Δ)σu=up with an isolated singularity, where σ∈(0,1) and nn−2σ

The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that solutions of the linear problem present this kind of singularities. Then we show that the corresponding solutions of the nonlinear problem present dispersive blow-up

In this paper we prove local in time well-posedness for the incompressible Euler equations in Rn for the initial data in L1(1)1(Rn), which corresponds to a critical case of the generalized Campanato spaces Lq(N)s(Rn). The space is studied extensively in our companion paper [9], and in the critical case we have embeddings B∞,11(Rn)↪L1(1)1(Rn)↪C0,1(Rn), where B∞,11(Rn) and C0,1(Rn) are the Besov space

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when

We propose a duality theory for multi-marginal repulsive cost that appears in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the N-marginals relaxed cost in terms

In this paper we develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces Ep,qs with exponentially decaying weights (s<0,1

We study the three-dimensional Navier–Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray–Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and

In dynamical systems, understanding statistical properties shared by most orbits and how these properties depend on the system are basic and important questions. Statistical properties may persist as one perturbs the system (statistical stability is said to hold), or may vary wildly. The latter case is our subject of interest, and we ask at what timescale does statistical stability break down. This

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps

In this paper, we establish the first rigorous mathematical result on the validation of the hypersonic similarity globally, which is also called the Mach-number independence principle, for the two dimensional steady potential flow. The hypersonic similarity is equivalent to the Van Dyke's similarity theory, that is, if the hypersonic similarity parameter K is fixed, the shock solution structures (after

This paper studies hamiltonization of nonholonomic systems using geometric tools, building on [1], [5]. The main novelty in this paper is the use of symmetries and suitable first integrals of the system to explicitly define a new bracket on the reduced space that codifies the nonholonomic dynamics and carries, additionally, an almost symplectic foliation (determined by the common level sets of the

We construct nontrivial bounded, real analytic domains Ω⊆Rn of the form Ω0∖Ω‾1, bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem{−Δu=1,u>0 in Ω,u=0,∂νu=const on ∂Ω0,u=const,∂νu=const on ∂Ω1, where ν stands for the inner unit normal to ∂Ω. From results by Reichel [1] and later by Sirakov [2], it was known that the condition ∂νu≤0 on ∂Ω1 is sufficient

We study how physical measures vary with the underlying dynamics in the open class of Cr, r>1, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics. A main ingredient in the proof is a new Pliss-like Lemma which

This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top of a dry region. An important feature considered here is that the layer of fluid is under an impervious wall. This physical situation has been widely study by mean

This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix the sets from which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for: the mean field game of impulse

We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of