We consider the Wick ordered cubic Schrödinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e. a stable closed geodesic), which prevents any regularization in the second Picard iteration.
We consider Hölder continuous weak solutions $u \in C^{\gamma}(\Omega)$, $u\cdot n \vert_{\partial\Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega \subset \mathbb{R}^d$. If $\Omega$ is of class $C^{2,\delta}$ for some $\delta$, then the corresponding pressure satisfies $p\in C^{2\gamma}_*$ in the case $\gamma \in (0,1/2]$, chere $C^{2\gamma}_*$ is the Hölder-Zygmund space, which coincides with the usual Hölder space for $\gamma < 1/2$. This result, together with our previous one covering the case $1/2<\gamma<1$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2\gamma}_*$ estimate for the pressure on the whole space $\mathbb{R}^d$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $\gamma = 1/2$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood-Paley analysis of the modified equation in the new coordinate system.
We prove that the hydrodynamic pressure $p$ associated to the velocity $u\in C^\theta(\Omega)$, $\theta\in(0,1)$, of an inviscid incompressible fluid in a bounded and simply connected domain $\Omega\subset \mathbb{R}^d$ with $C^{2+}$ boundary satisfies $p\in C^{\theta}(\Omega)$ for $\theta \leq \frac12$ and $p\in C^{1,2\theta-1}(\Omega)$ for $\theta>\frac12$. This extends the recent result of Bardos and Titi obtained in the planar case to every dimension $d\ge2$ and it also doubles the pressure regularity for $\theta>\frac12$. Differently from Bardos and Titi, we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the $d$-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.
We consider the Euler equations on the torus in dimensions $2$ and construct invariant measures for the dynamics of these equations concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist at least locally. The proof follows a combination of a method of Kuksin and a method of Bourgain which was first implemented by Sy. We obtain in particular that these measures do not have atoms, excluding trivial invariant measures such as diracs. Then we prove that $\mu$-almost every initial data gives rise to a global solution for which the growth of the Sobolev norms are at most polynomial. We point out that up to the knowledge of the author, the only general upper-bound for the growth of the Sobolev norm to the $2d$ Euler equations is double exponential. In the other direction, some initial data are known to produce solutions exhibiting exponential growth, as in the work of Zlatos.
We consider the radial nonlinear Schrödinger equation $i\partial_tu +\Delta u = |u|^{p-1}u$ in dimension $d\geqslant 2$ for $p\in \left[1,1+\frac{4}{d}\right]$ and construct a natural Gaussian measure $\mu$ which support is almost $L^2_{\text{rad}}$ and such that $\mu$ - almost every initial data gives rise to a unique global solution. Furthermore, for $p>1+\frac{2}{d}$ the solutions constructed scatter in a space which is almost $L^2$. This paper can be viewed as the higher dimensional counterpart of the work of Burq and Thomann, in the radial case however.
We study the local well-posedness of the nonlinear Schrödinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in the Sobolev spaces $H^k$ when $k\leq\frac{3}{2}$. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in $H^k$, $k \in (1,\frac{3}{2}]$, almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.
We prove that for almost every initial data $(u_0,u_1) \in H^s \times H^{s-1}$ with $s > \frac{p-3}{p-1}$ there exists a global weak solution to the supercritical semilinear wave equation ${\partial _t^2u - \Delta u +|u|^{p-1}u=0}$ where $p>5$, in both $\mathbb{R}^3$ and $\mathbb{T}^3$. This improves in a probabilistic framework the classical result of Strauss who proved global existence of weak solutions associated to $H^1 \times L^2$ initial data. The proof relies on techniques introduced by T. Oh and O. Pocovnicu based on the pioneer work of N. Burq and N. Tzvetkov. We also improve the global well-posedness result in C. Sun and B. Xia for the subcritical regime $p<5$ to the endpoint $s=\frac{p-3}{p-1}$.
This paper establishes Carleson embeddings of Müntz spaces $M^q_{\Lambda}$ into weighted Lebesgue spaces $L^p(\mathrm{d}\mu)$, where $\mu$ is a Borel regular measure on $[0,1]$ satisfying $\mu([1-\varepsilon])\lesssim \varepsilon^{\beta}$.
In the case $\beta \geqslant 1$ we show that such measures are exactly the ones for which Carleson embeddings $L^{\frac{p}{\beta}} \hookrightarrow L^p(\mathrm{d}\mu)$ hold.
The case $\beta \in (0,1)$ is more intricate but we characterize such measures $\mu$ in terms of a summability condition on their moments.
Our proof relies on a generalization of $L^p$ estimates à la Gurariy-Macaev in the weighted $L^p$ spaces setting, which we think can be of interest in other contexts.
We continue the analysis of random series associated to the multidimensional harmonic oscillator $-\Delta + |x|^2$ on $\mathbb{R}^d$ with $d \geq 2$. More precisely we obtain a necessary and sufficient condition to get the almost sure uniform convergence on the whole space $\mathbb{R}^d$. It turns out that the same condition gives the almost sure uniform convergence on the sphere $\mathbb{S}^{d-1}$ (despite $\mathbb{S}^{d-1}$ is a zero Lebesgue measure of $\mathbb{R}^d$). From a probabilistic point of view, our proof adapts a strategy used by the first author for boundaryless Riemannian compact manifolds. However, our proof requires sharp off-diagonal estimates of the spectral function of $-\Delta + |x|^2$. Such estimates are obtained using elementary tools.
We consider the free boundary incompressible porous media equation which describes the dynamics of a density transported by a Darcy flow in the field of gravity, with a free boundary between the fluid region and the dry region above it. For any stratified density state, we identify a stability condition for the initial free boundary. Under this condition, we prove that small localized perturbations of the stratified density lead to unique local-in-time solutions in Sobolev spaces. Our proof involves analytic ingredients that are of independent interest, including tame fractional Sobolev estimates for operators that map the Dirichlet boundary function and the forcing function of Poisson's equation to its solution in domains of Sobolev regularity.
We consider the two-dimensional incompressible Euler equations and give a sufficient condition on Gaussian measures of jointly independent Fourier coefficients supported on $H^{\sigma}(\mathbb{T}^2)$ ($\sigma >3$) such that these measures are not invariant (in vorticity form). We show that this condition holds on an open and dense set in suitable topologies (and so is generic in a Baire category sense) and give some explicit examples of Gaussian measures which are not invariant. We also pose a few related conjectures which we believe to be approachable.
We prove Strichartz estimates for a class of Baouendi--Grushin operators acting either on the Euclidean space or a product of the type $\mathbb{R}^{d_1} \times M$, where $(M,g)$ is a smooth compact manifold with no boundary. We then give an application of these Strichartz estimates to the Cauchy theory for the associated Schrödinger equations.
We prove interpolation results in the spirit of the Marcinkiewicz theorem. The operators considered in this article are defined on Müntz spaces, which are not dense subspaces of $L^p$, and for which the classical interpolation theory cannot be applied directly. Our proofs crucially rely on strong decoupling of $L^p$ norms, a that was first observed by Gurariy-Macaev and later generalized.