American Institute of Mathematical Sciences

We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to \begin{document}$ r $\end{document}-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.

We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying the asymptotical behavior of the renormalization operator.

In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of fast diffusion channels along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.

On a closed \begin{document}$ 3 $\end{document}-dimensional Riemannian manifold \begin{document}$ (M,g) $\end{document} we investigate the limit of the Einstein-Lichnerowicz equation

1 \begin{document}$ \begin{equation} \triangle_g u + h u = f u^5 + \frac{\theta a}{u^7} \end{equation} $\end{document}

as the momentum parameter \begin{document}$ \theta \to 0 $\end{document}. Under a positive mass assumption on \begin{document}$ \triangle_g +h $\end{document}, we prove that sequences of positive solutions to this equation converge in \begin{document}$ C^2(M) $\end{document}, as \begin{document}$ \theta \to 0 $\end{document}, either to zero or to a positive solution of the limiting equation \begin{document}$ \triangle_g u + h u = f u^5 $\end{document}. We also prove that the minimizing solution of (1) constructed by the author in [15] converges uniformly to zero as \begin{document}$ \theta \to 0 $\end{document}.

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [10] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (ⅰ) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ⅱ) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish \begin{document}$ L^q $\end{document} time-decay for the linear homogeneous system by using a Fourier transform and finding Green's matrix. Then, we find \begin{document}$ L^q $\end{document} time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.

Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.

In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with \begin{document}$ n $\end{document} degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situations) and other classical results on formal stability of equilibria. In case of Lie stable systems we bound the solutions near the equilibrium over exponentially long times. Some examples are provided to illustrate our main contributions.

We consider the following nonlocal critical equation

\begin{document}$\begin{equation} -\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1) \end{equation}$ \end{document}

where \begin{document}$ 0<\mu_1,\mu_2<N $\end{document} if \begin{document}$ N = 3 $\end{document} or \begin{document}$ 4 $\end{document}, and \begin{document}$ N-4\leq\mu_1,\mu_2<N $\end{document} if \begin{document}$ N\geq5 $\end{document}, \begin{document}$ 2_{\mu_{i}}^\ast: = \frac{N+\mu_i}{N-2}(i = 1,2) $\end{document} is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and \begin{document}$ I_{\mu_i} $\end{document} is the Riesz potential

\begin{document}$ \begin{equation*} I_{\mu_i}(x) = \frac{\Gamma(\frac{N-\mu_i}{2})}{\Gamma(\frac{\mu_i}{2})\pi^{\frac{N}{2}}2^{\mu_i}|x|^{N-\mu_i}}, \; i = 1,2, \end{equation*} $\end{document}

with \begin{document}$ \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx $\end{document}, \begin{document}$ s>0 $\end{document}. Firstly, we prove the existence of the solutions of the equation (1). We also establish integrability and \begin{document}$ C^\infty $\end{document}-regularity of solutions and obtain the explicit forms of positive solutions via the method of moving spheres in integral forms. Finally, we show that the nondegeneracy of the linearized equation of (1) at \begin{document}$ U_0,V_0 $\end{document} when \begin{document}$ \max\{\mu_1,\mu_2\}\rightarrow0 $\end{document} and \begin{document}$ \min\{\mu_1,\mu_2\}\rightarrow N $\end{document}, respectively.

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node \begin{document}$ A $\end{document} to node \begin{document}$ B $\end{document} if, using arbitrary small perturbations, a trajectory starting from any point of \begin{document}$ A $\end{document} can be steered to any point of \begin{document}$ B $\end{document}. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node \begin{document}$ A $\end{document} to node \begin{document}$ B $\end{document}, the unstable manifold of some periodic orbit in \begin{document}$ A $\end{document} contains points that eventually map onto \begin{document}$ B $\end{document}. For special parameter values, this tower has infinitely many nodes.

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.

We give a proof by foliation that the cones over \begin{document}$ \mathbb{S}^k \times \mathbb{S}^l $\end{document} minimize parametric elliptic functionals for each \begin{document}$ k, \, l \geq 1 $\end{document}. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

We study the stochastic stability in the zero-noise limit from a quantitative point of view.

We consider smooth expanding maps of the circle perturbed by additive noise. We show that in this case the zero-noise limit has a quadratic speed of convergence, as suggested by numerical experiments and heuristics published by Lin, in 2005 (see [25]). This is obtained by providing an explicit formula for the first and second term in the Taylor's expansion of the response of the stationary measure to the small noise perturbation. These terms depend on important features of the dynamics and of the noise which is perturbing it, as its average and variance.

We also consider the zero-noise limit from a quantitative point of view for piecewise expanding maps showing estimates for the speed of convergence in this case.

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a \begin{document}$ N $\end{document}-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least \begin{document}$ N $\end{document} orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility assumption does not hold, then there are at least two orthogonal Finsler geodesic chords with different values of the energy functional.

In this paper we study a \begin{document}$ 2\times2 $\end{document} semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in \begin{document}$ L^\infty $\end{document} in the space-time domain \begin{document}$ (0,1)\times [0,+\infty) $\end{document}. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space \begin{document}$ L^{\infty} $\end{document}. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in \begin{document}$ W^{1,\infty} $\end{document} for the corresponding solution to the semilinear wave equation.

Inspired by the generalized Christoffel problem, we consider a class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in \begin{document}$ \mathbb{H}^{n+1} $\end{document}. Under some sufficient conditions, we prove the a priori estimates for solutions to the Monge-Ampère type equation \begin{document}$ \det(\kappa-\mathbf{1}) = f(X, \nu(X)) $\end{document}. Moreover, we obtain an existence result for the compact horo-convex hypersurface \begin{document}$ M $\end{document} satisfying the above equation with various assumptions.

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation \begin{document}$ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $\end{document} and \begin{document}$ \alpha>0. $\end{document} Only partial results are known for the local existence in the subcritical case \begin{document}$ \alpha<(4-2b)/(N-2s) $\end{document} and much more less in the critical case \begin{document}$ \alpha = (4-2b)/(N-2s). $\end{document} In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for \begin{document}$ b = 0 $\end{document} and \begin{document}$ b>0. $\end{document}

This note treats several problems for the fractional perimeter or \begin{document}$ s $\end{document}-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as \begin{document}$ s \nearrow 1 $\end{document} is proven. It is shown that their limit as \begin{document}$ s \searrow -\infty $\end{document} can be expressed in terms of the volume.

We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach.

For \begin{document}$ n\ge 3 $\end{document}, \begin{document}$ 0<m<\frac{n-2}{n} $\end{document}, \begin{document}$ \beta<0 $\end{document} and \begin{document}$ \alpha = \frac{2\beta}{1-m} $\end{document}, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in \begin{document}$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $\end{document} of the form \begin{document}$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $\end{document} where \begin{document}$ f_{\lambda} $\end{document} is a radially symmetric function satisfying

\begin{document}$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $\end{document}

with \begin{document}$ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $\end{document} and \begin{document}$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $\end{document}, for some constant \begin{document}$ \lambda>0 $\end{document}.

As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation \begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document} in \begin{document}$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $\end{document} with initial value \begin{document}$ u_0 $\end{document} satisfying \begin{document}$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $\end{document}, \begin{document}$ \forall x\in\mathbb{R}^n\setminus\{0\} $\end{document}, such that the solution \begin{document}$ u $\end{document} satisfies \begin{document}$ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $\end{document}, \begin{document}$ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $\end{document}, for some constants \begin{document}$ \lambda_1>\lambda_2>0 $\end{document}.

We also prove the asymptotic large time behaviour of such singular solution \begin{document}$ u $\end{document} when \begin{document}$ n = 3,4 $\end{document} and \begin{document}$ \frac{n-2}{n+2}\le m<\frac{n-2}{n} $\end{document} holds. Asymptotic large time behaviour of such singular solution \begin{document}$ u $\end{document} is also obtained when \begin{document}$ 3\le n<8 $\end{document}, \begin{document}$ 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $\end{document}, and \begin{document}$ u(x,t) $\end{document} is radially symmetric in \begin{document}$ x\in\mathbb{R}^n\setminus\{0\} $\end{document} for any \begin{document}$ t>0 $\end{document} under appropriate conditions on the initial value \begin{document}$ u_0 $\end{document}.