Analysis of PDEs (math.AP)

Abstract: We establish fractional boundary Hardy-type inequality for the critical cases. In 2004, Dyda established the inequalities for the subcritical regime, whereas we establish more general inequalities (Caffarelli-Kohn-Nirenberg type) for the critical cases.

Abstract: In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in presence of jumping nonlinearities. In the main results of the paper we prove the existence of a nontrivial solution for the problem under consideration, using variational and topological methods and applying a new linking theorems recently got by Perera and Sportelli in [10]. The existence results provided in this paper can be seen as the nonlocal counterpart of the ones obtained in [10] in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest.

Abstract: Massive and massless Dirac equations with Lorentz-covariant cubic nonlinearities are considered in spatial dimension $d=2,3$. Global well-posedness of the Cauchy problem for small initial data in scale-invariant Sobolev spaces and scattering of solutions is proved by a new approach which uses bilinear Fourier restriction estimates and atomic function spaces. Furthermore, global uniform convergence results, both in the massless and in the non-relativistic limit, are proved at optimal regularity. In both regimes, these are the first results which imply convergence of scattering states and wave operators.

Abstract: This is the first of several planned papers that study the existence and local-in-time persistence of phase fronts in the Lorentz invariant Euler equations for gases of van der Waals type, aiming at transferring earlier results of Slemrod and of Benzoni-Gavage and collaborators to the context of the theory of relativity. While the later papers will examine more general admissibility criteria, this one uses and extends the author's Lorentz invariant formulation of Korteweg's theory of capillarity, establishing a family of phase fronts that have a regular heteroclinic profile with respect to the associated Euler-Korteweg equations.

Abstract: We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in $\mathbb{R}^3\times [-T,0]$ such that the velocity is in the space $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2$ for times $t\in (-T,0)$ and is not $C^1$ at time 0.

Abstract: By definition, a wave is a $C^\infty$ solution $u(x,t)$ of the wave equation on $\mathbb R^n$, and a snapshot of the wave $u$ at time $t$ is the function $u_t$ on $\mathbb R^n$ given by $u_t(x)=u(x,t)$. We show that there are infinitely many waves with given $C^\infty$ snapshots $f_0$ and $f_1$ at times $t=0$ and $t=1$ respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave $u$ to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to shifted wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the shifted wave equation on the $n$-sphere.

Abstract: We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.

Abstract: This paper concerns the existence of multiple solutions for a Schr\"odinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\varepsilon^2\Delta u + V(x)u & =u\log u^2,\;\;\mbox{in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P_\varepsilon)} \end{equation} where $V:\mathbb{R}^N\longrightarrow \mathbb{R}$ is a continuous function that satisfies some technical conditions and $\varepsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_\varepsilon)$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.