Analysis of PDEs (math.AP)

Abstract: We study the regularity properties of a general second order H\"ormander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the H\"ormander condition, we consider the linear operator in $\R^{N}$, with $N>m+1$: \begin{equation*} \L u := \sum_{i, j= 1}^{m} a_{ij} X_{i}X_{j} u - X_0 u. \end{equation*} The vector field $X_0$ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\L u = f$ are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.

Abstract: We consider nonlinear Schr\"{o}dinger equation with strong magnetic fields in 3D. This model was derived by R L. Frank, F. M\'{e}hats, C. Sparber in 2017. We prove modified scattering for small initial data and the existence of modified wave operator for small final data. To describe asymptotic behavior of the NLS we use the time-averaged model which was derived by the same authors as "the strong magnetic confinement limit" of the NLS. We construct asymptotic solutions which satisfy both asymptotic in time evolution and convergence in the strong magnetic confinement limit. We also analyze the error between the solution to the NLS and the time-averaged model for the same initial data and obtain global estimates.

Abstract: We consider the Cauchy problem for heat equation with fractional harmonic oscillator and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global {weak-mild} solutions for small initial data and obtain decay estimates for large time in Lebesgue spaces. In particular, we show that the decay depends on the behavior of the nonlinearity near the origin. Finally, we show that for some non-negative initial data in the appropriate Orlicz space, there is no local non-negative classical solution.

Abstract: We establish one of the most important assumptions of the strong persistence theory for dynamical systems associated to cross diffusion systems of $m$ equations ($m\ge2$): the stable sets of semi-trivial steady cannot intersect the interior of the positive cone of $C(\Omega,\mathbb{R}^m)$. Many examples will be provided to show the effects of the cross diffusion.

Abstract: We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for $p\in (1,\infty)$ and $s \in (0,1)$. Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional $p-$Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov-Safanov covering arguments.

Abstract: In this paper, we study the existence of normalized ground state solutions for the following biharmonic Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \Delta^2v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad \displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $a,b>0$ are prescribed, $\lambda_1,\lambda_2\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential subcritical or critical growth in the sense of the Adams inequality. By using a minimax principle and analyzing the behavior of the ground state energy with respect to the prescribed mass, we obtain the existence of ground state solutions for the above problem.

Abstract: We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently introduced in [NS23]. The paper at hand is devoted to the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. In the course of the proof, we establish a wide variety of regularity estimates for the free boundary and for the temperature function, of interest in their own right.

Abstract: We study the following elliptic problem involving slightly subcritical non-power nonlinearity $$\left\{\begin{array}{lll} -\Delta u =\frac{|u|^{2^*-2}u}{[\ln(e+|u|)]^\epsilon}\ \ &{\rm in}\ \Omega, \\[2mm] u= 0 \ \ & {\rm on}\ \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, $n\geq 3$, $2^*=\frac{2n}{n-2}$ is the critical Sobolev exponent, $\epsilon>0$ is a small parameter. By the finite dimensional Lyapunov-Schmidt reduction method, we construct a sign changing bubble tower solution with the shape of a tower of bubbles as $\epsilon$ goes to zero.