We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.
This work investigates a mass-critical focusing inhomogeneous NLS with an inverse square potential. The paper proves the finite time blow-up of solutions for datum with negative energy without any radial or finite variance assumption. Moreover, the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document} concentration of nonglobal mass-critical solutions is established. This note complements (arXiv:2306.15210v1 [math.AP]) to the mass-critical regime and (Nonlinearity 35(8), 2022) to the case with inverse square potential. The proof is based on a localized Virial type identity via the decaying factor |x|−b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|x|^{-b}$\end{document}, which gives a control, away from the origin, for the terms arising from the nonlinearity. Moreover, Hardy estimate is used to get the norm equivalence ∥⋅∥H˙1(RN)∼∥−Δ+a|x|2⋅∥L2(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|\cdot \|_{\dot{H}^{1}(\mathbb{R}^{N})}\sim \|\sqrt{-\Delta + \frac {a}{|x|^{2}}}\cdot \|_{L^{2}(\mathbb{R}^{N})}$\end{document}, which enables to handle the inverse square potential. This needs in particular the restrictions N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document} and a>−(N−2)24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>-\frac{(N-2)^{2}}{4}$\end{document}. Furthermore, the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document} concentration is based on a compactness result in the spirit of (Int. Math. Res. Not. 46:2815-2828, (2005))
We establish nonexistence conditions for nonnegative nontrivial solutions to a class of semilinear parabolic equations with a positive potential on weighted graphs, extending results in arXiv:2404.12058 [math.AP] to a broader setting that includes both the Porous Medium Equation and the Fast Diffusion Equation. We identify conditions related to the graph's geometry, the potential's behaviour at infinity, and bounds on the Laplacian of the distance function under which nonexistence holds. Using a test function argument, we derive explicit parameter ranges for nonexistence.
In this paper we study the $5$th Order Kadomstev-Petviashvili (KP) equations posed on the real line. In particular we adapt the energy estimate argument from Guo-Molinet (arXiv:2404.12364v1 [math.AP]) to conclude unconditional uniqueness of the solution to data map for $5$th order KP type equations. Applying short-time $X^{s,b}$ methods to improve classical energy estimates provides more than sufficient decay when considering estimates on the interior of the time interval $[0,T]$. The issue is how we deal with the boundary. By abusing symmetry we can apply multilinear interpolation to gain access to $L^4$ Strichartz estimates, which provide improved derivative gain. When taken together, the regularity of our resultant function space can be arbitrarily close to $L^2$, which in the context of unconditional uniqueness results is almost sharp.
We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schr\"odinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schr\"odinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments.
Wuyan Wang, Timothy Wessler, Nancy Rodr'iguez
et al.
Protest activity, a constitutionally protected right in the United States under the First Amendment, serves as a key tool for individuals with limited individual influence to unite collectively and amplify their impact. Despite its legal recognition, the historical interaction between the government and protesters has been complex. In this study, we extend a model proposed in arXiv:1502.04725 [math.AP] to incorporate the influence of protest management strategies. Our research establishes the existence and stability of traveling wave solutions, supported by both theoretical analyses and numerical simulations. We delve into the impact of two distinct protest management approaches on the qualitative and semi-quantitative characteristics of these traveling waves. These metrics can aid in categorizing different types of protests.
In the seminal work of Benjamin (1974 Nonlinear Wave Motion (American Mathematical Society)), in the late 70s, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin–Ono equation, During the 90s, there was renewed interest in it. Benjamin (1992 J. Fluid Mech. 245 401–11; 1996 Phil. Trans. R. Soc. A 354 1775–806) studied the problem for existence of solitary waves, followed by works of Bona–Chen (1998 Adv. Differ. Equ. 3 51–84), Albert–Bona–Restrepo (1999 SIAM J. Appl. Math. 59 2139–61), Pava (1999 J. Differ. Equ. 152 136–59), who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in 2024 (arXiv:2404.04711 [math.AP]), Abdallah et al, proved existence, orbital stability and uniqueness results for these waves, but only for large values of cγ2≫1. In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to all L2 subcritical cases (i.e. power nonlinearities (|u|p−2u)x, 2<p⩽6). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers 2<p<∞, but produces some unstable waves, for large p. Some open questions and a conjecture regarding this last result are proposed for further investigation.
We consider the singular limit of a chemotaxis model of bacterial collective motion recently introduced in arXiv:2009.11048 [math.AP]. The equation models aggregation-diffusion phenomena with advection that is discontinuous and depends sharply on the gradient of the density itself. The quasi-linearity of the problem poses major challenges in the construction of the solution and complications arise in the proof of regularity. Our method overcomes these obstacle by relying solely on entropy inequalities and the theory of monotone operators. We provide existence, uniqueness and smoothing estimates in any dimensional space.
We establish global-in-time well-posedness of the one-dimensional hydrodynamic Gross-Pitaevskii equations in the absence of vacuum in $(1 + H^s) \times H^{s-1}$ with $s \geq 1$. We achieve this by a reduction via the Madelung transform to the previous global-in-time well-posedness result for the Gross-Pitaevskii equation in arXiv:1801.08386v2 [math.AP] and arXiv:2204.06293v1 [math.AP]. Our core result is a local bilipschitz equivalence between the relevant function spaces.
We consider a class of non-doubling manifolds $\mathcal{M}$ defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$ and the Euclidean dimension $n_i$ are not necessarily all the same. In arXiv:1805.00132v3 [math.AP], Hassell and Sikora proved that the Riesz transform on $\mathcal{M}$ is weak type $(1,1)$, bounded on $L^{p}(\mathcal{M})$ for all $1<p<n^*$ where $n^* = \min_k n_k$ and is unbounded for $p \ge n^*$. In this note we show that the Riesz transform is bounded from Lorentz space $L^{n^* ,1}(\mathcal{M})$ to $L^{n^*,1}(\mathcal{M})$. This complete the picture by obtaining the end point results for $p=n^*$. Our approach is based on parametrix construction described in arXiv:1805.00132v3 [math.AP] and a generalisation of Hardy-Hilbert type inequalities first studied by Hardy, Littlewood and Pólya.
In this paper, we consider the following nonlinear Schrödinger equations with the critical Sobolev exponent and mixed nonlinearities: \[\left\{\begin{aligned} & -\Delta u+\lambda u=t|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\ & u\in H^{1}(\mathbb{R}^{N}), \end{aligned}\right.\] where $N\geq 3$, $t>0$, $\lambda >0$ and $20$ sufficiently large, which gives a rigorous proof of the numerical conjecture in [J. Dávila, M. del Pino and I. Guerra. Non-uniqueness of positive ground states of non-linear Schrödinger equations. Proc. Lond. Math. Soc. 106 (2013), 318–344.]; (2) there exists $t_q^{*}>0$ for $2t_4^{*}$ in the case of $q=4$, while the above equation has no ground-states for $0\overline {t}_{a,q}$ with $\int _{\mathbb {R}^{N}}|u|^{2}{\rm d}x=a^{2}$, which, together with our recent study in [J. Wei and Y. Wu, Normalized solutions for Schrodinger equations with critical Sobolev exponent and mixed nonlinearities, e-print arXiv:2102.04030[Math.AP].], gives a completed answer to the open question proposed by Soave in [N. Soave. Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case. J. Funct. Anal. 279 (2020) 108610.]. Finally, as applications of our new argument, we also study the following Schrödinger equation with a partial confinement: \[\left\{ \begin{aligned} & -\Delta u+\lambda u+(x_1^{2}+x_2^{2})u=|u|^{p-2}u\quad\text{in }\mathbb{R}^{3},\\ & u\in H^{1}(\mathbb{R}^{3}),\quad \int_{\mathbb{R}^{3}}|u|^{2}{\rm d}x=r^{2}, \end{aligned}\right.\] where $x=(x_1,x_2,x_3)\in \mathbb {R}^{3}$, $\frac {10}{3}0$ is a constant and $(u, \lambda )$ is a pair of unknowns with $\lambda$ being a Lagrange multiplier. We prove that the above equation has a second positive solution, which is also a mountain-pass solution, for $r>0$ sufficiently small. This gives a positive answer to the open question proposed by Bellazzini et al. in [J. Bellazzini, N. Boussaid, L. Jeanjean and N. Visciglia. Existence and Stability of Standing Waves for Supercritical NLS with a Partial Confinement. Commun. Math. Phys. 353 (2017), 229–251].
Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 107\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^7$$\end{document}. We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of LtqLxp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_t^q L_x^p$$\end{document} norm of the velocity with 3/p+2/q≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3/p + 2/q \le 1$$\end{document}. Our numerical results for the cases of (p,q)=(4,8),(6,4),(9,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q) = (4,8),\; (6,4),\; (9,3)$$\end{document} and (p,q)=(∞,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q)=(\infty ,2)$$\end{document} provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of (p,q)=(3,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q)=(3,\infty )$$\end{document} is more difficult to verify numerically due to the extremely slow growth rate in the L3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^3$$\end{document} norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global L3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^3$$\end{document} norm of the velocity grows very slowly, the localized version of the L3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^3$$\end{document} norm of the velocity experiences rapid dynamic growth relative to the localized L3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^3$$\end{document} norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.
Let $X$ be a topological space and $\Omega \subset X$. Suppose $f:\Omega\rightarrow X$ is a function defined in a complete space $ \Omega $ and $ \tau $ is a vector in $ \mathbb{R} $ taking values in $X$. Suppose $ f $ is ap-Sequential Henstock integrable with respect to $\tau$, is $ f $ ap-Sequential Topological Henstock integrable with respect to $\tau$? It is the purpose of this paper to proffer affirmative answer to this question.
We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^3$$\end{document}. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^3$$\end{document} by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^4_3$$\end{document}-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.
This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, \begin{align*} u_t=\Delta (u+1)^m - \chi \nabla \cdot (u(u+1)^{\alpha - 1} \nabla v) + \lambda(|x|) u - \mu(|x|) u^\kappa, \quad 0=\Delta v - v + u, \quad x\in\Omega,\ t>0, \end{align*} where $\Omega:=B_{R}(0)\subset\mathbb{R}^n\ (n\ge3)$ is a ball with some $R>0$; $m>0$, $\chi>0$, $\alpha>0$ and $\kappa\ge1$; $\lambda$ and $\mu$ are spatially radial nonnegative functions. About this problem, Winkler (Z. Angew. Math. Phys.; 2018; 69; Art. 69, 40) found the condition for $\kappa$ such that solutions blow up in finite time when $m=\alpha=1$. In the case that $m=1$ and $\alpha\in(0,1)$ as well as $\lambda$ and $\mu$ are constant, some conditions for $\alpha$ and $\kappa$ such that blow-up occurs were obtained in a previous paper (Math. Methods Appl. Sci.; 2020; 43; 7372-7396). Moreover, in the case that $m\ge1$ and $\alpha=1$ Black, Fuest and Lankeit (arXiv:2005.12089[math.AP]) showed that there exists initial data such that solutions blow up in finite time under some conditions for $m$ and $\kappa$. The purpose of the present paper is to give conditions for $m\ge1$, $\alpha>0$ and $\kappa\ge1$ such that solutions blow up in finite time.