Analysis of PDEs (math.AP)

Abstract: In this paper we prove global well-posedness and scattering for the defocusing cubic nonlinear wave equation in the hyperbolic space $\mathbb{H}^3$, under the assumption that the initial data is radial and lies in $H^{\frac{1}{2}+\delta}(\mathbb{H}^3)\times H^{-\frac{1}{2}+\delta}(\mathbb{H}^3)$

Abstract: We consider the patch problem for the $\alpha$-SQG system with the values $\alpha=0$ and $\alpha= \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint $C^{k,\beta}$ H\"older spaces, as well as in $W^{2,p},$ $1<p<\infty$ spaces. In stark contrast to the Euler case, we prove that for $0<\alpha< \frac{1}{2}$, the $\alpha$-SQG patch problem is strongly illposed in \emph{every} $C^{2,\beta} $ H\"older space with $\beta<1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for \emph{every} $W^{2,p}$ Sobolev space unless $p=2$.

Abstract: We study a class of interacting particle systems in which $n$ signed particles move on the real line. At close range particles with the same sign repel and particles with opposite sign attract each other. The repulsion and attraction are described by the same singular interaction force $f$. Particles of opposite sign may collide in finite time. Upon collision, pairs of colliding particles are removed from the system. In a recent paper by Peletier, Po\v{z}\'ar and the author, one particular particle system of this type was studied; that in which $f(x) = \frac1x$ is the Coulomb force. Global well-posedness of this particle system was shown and a discrete-to-continuum limit (i.e. $n \to \infty$) to a nonlocal PDE for the signed particle density was established. Both results rely on innovative use of techniques in ODE theory and viscosity solutions. In the present paper we extend these results to a large class of particle systems in order to cover many new applications. Motivated by these applications, we consider the presence of an external force $g$, consider interaction forces $f$ with a large range of singularities and allow $f$ to scale with $n$. To handle this class of $f$ we develop several new proof techniques in addition to those used for the Coulomb force.

Abstract: We consider here the stationary Micropolar fluid equations which are a particular generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one based mainly in the velocity field u and the other one based in the microrotation field $\omega$. We will study in this work some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establish, under some suitable decay at infinity conditions for the velocity field only, the uniqueness of the trivial solution.

Abstract: We study here the effects of a time-dependent second order perturbation to a degenerate Ornstein-Uhlenbeck type operator whose diffusive part can be either local or non-local. More precisely, we establish that some estimates, such as the Schauder and Sobolev ones, already known for the non-perturbed operator still hold, and with the same constants, when we perturb the Ornstein-Uhlenbeck operator with second order diffusions with coefficients only depending on time in a measurable way. The aim of the current work is twofold: we weaken the assumptions required on the perturbation in the local case which has been considered already in [KP17] and we extend the approach presented therein to a wider class of degenerate Kolmogorov operators with non-local diffusive part of symmetric stable type.

Abstract: The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on $\mathbb{R}^n$. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following [Doc. Math., 22, 1539--1592, 2017], using the H\"{o}rmander symbol classes $S^m_{\rho, \delta}(G)$ introduced in [Progress in Mathematics, 314. Birkh\"{a}user/Springer, 2016]. We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn-Nirenberg calculus, on $\mathbb{R}^n$ and on general graded groups [Progress in Mathematics, 314. Birkh\"{a}user/Springer, 2016]. Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\r{a}rding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups. In the particular case of the Heisenberg group $\mathbb{H}_n$, we are able to answer the fundamental questions of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on $\mathbb{H}_n$? And which are the criteria that determine it uniquely? The surprisingly simple but compelling answers raise the question whether what is true for $\mathbb{R}^n$ and $\mathbb{H}_n$ also extends to general graded groups, which we answer in the affirmative in this paper. Among other things, we discuss and investigate an analogue of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as the notion of the Poisson bracket for symbols in the setting of stratified groups, linking it to the symbolic properties of the commutators.

Abstract: We introduce a stochastic version of the optimal transport problem. We provide an analysis by means of the study of the associated Hamilton-Jacobi-Bellman equation, which is set on the set of probability measures. We introduce a new definition of viscosity solutions of this equation, which yields general comparison principles, in particular for cases involving terms modeling stochasticity in the optimal control problem. We are then able to establish results of existence and uniqueness of viscosity solutions of the Hamilton-Jacobi-Bellman equation. These results rely on controllability results for stochastic optimal transport that we also establish.

Abstract: This article addresses the issue of global convergence towards pushed travelling fronts for solutions of parabolic systems of the form \[ u_t = - \nabla V(u) + u_{xx} \,, \] where the potential $V$ is coercive at infinity. It is proved that, if an initial condition $x\mapsto u(x,t=0)$ approaches, rapidly enough, a critical point $e$ of $V$ to the right end of space, and if, for some speed $c_0$ greater than the linear spreading speed associated with $e$, the energy of this initial condition in a frame travelling at the speed $c_0$ is negative $\unicode{x2013}$ with symbols, \[ \int_{\mathbb{R}} e^{c_0 x}\left(\frac{1}{2} u_x(x,0)^2 + V\bigl(u(x,0)\bigr)- V(e)\right)\, dx < 0 \,, \] then the corresponding solution invades $e$ at a speed $c$ greater than $c_0$, and approaches, around the leading edge and as time goes to $+\infty$, profiles of pushed fronts (in most cases a single one) travelling at the speed $c$. A necessary and sufficient condition for the existence of pushed fronts invading a critical point at a speed greater than its linear spreading speed follows as a corollary. In the absence of maximum principle, the arguments are purely variational. The key ingredient is a Poincar\'e inequality showing that, in frames travelling at speeds exceeding the linear spreading speed, the variational landscape does not differ much from the case where the invaded equilibrium $e$ is stable. The proof is notably inspired by ideas and techniques introduced by Th. Gallay and R. Joly, and subsequently used by C. Luo, in the setting of nonlinear damped wave equations.

Abstract: In this paper, we study the asymptotic behavior of least energy nodal solutions $u_p(x)$ to the following fourth-order elliptic problem \[ \begin{cases} \Delta^2 u =|u|^{p-1}u \quad &\hbox{in}\;\Omega, \\ u=\frac{\partial u}{\partial \nu}=0 \ \ &\hbox{on}\;\partial\Omega, \end{cases} \] where $\Omega$ is a bounded $C^{4,\alpha}$ domain in $\mathbb{R}^4$ and $p>1$. Among other things, we show that up to a subsequence of $p\to+\infty$, $pu_p(x)\to 64\pi^2\sqrt{e}(G(x,x^+)-G(x,x^-))$, where $x^+\neq x^-\in \Omega$ and $G(x,y)$ is the corresponding Green function of $\Delta^2$. This generalize those results for $-\Delta u=|u|^{p-1}u$ in dimension two by (Grossi-Grumiau-Pacella, Ann.I.H.Poincar\'{e}-AN, 30 (2013), 121-140) to the biharmonic case, and also gives an alternative proof of Grossi-Grumiau-Pacella's results without assuming their comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$.

Abstract: In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a functional for certain value of the mass. We give a positive answer in dimension two while in higher dimension the situation is different. In fact, for small value of mass the ball centered at the origin is a local minimizer while for large values the ball is a maximizer among convex sets with uniform bound on the curvature.

Abstract: We obtain a nigh optimal estimate for the first eigenvalue of two natural weighted problems associated to the bilaplacian (and of a continuous family of fourth-order elliptic operators in dimension $2$) in degenerating annuli (that are central objects in bubble tree analysis) in all dimension. The estimate depends only on the conformal class of the annulus. We also show that in dimension $2$ and dimension $4$, the first eigenfunction (of the first problem) is never radial provided that the conformal class of the annulus is large enough. The other result is a weighted Poincar\'e-type inequality in annuli for those fourth-order operators. Applications to Morse theory are given.