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We establish a nonlinear Calder\'on-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^{\gamma}\Delta^N_p u=f\in L^2(\Omega)\quad {\rm in}\quad \Omega; \quad u=0 \ \mbox{on $\partial\Omega$} $$ for $n\ge2$, $ p>1$ and a large range of $\gamma>-1$, in particular, for all $p>1$ and all $ \gamma>-1$ when $n=2$. Here $\Omega\subset \mathbb{R}^n$ is a bounded convex domain, or a bounded Lipschitz domain whose boundary has small weak second fundamental form in the sense of Cianchi-Maz'ya (2018). The proof relies on an extension of an Miranda-Talenti \& Cianchi-Maz'ya type inequality, that is, for any $v\in C^\infty_0(\Omega)$ in any bounded smooth domain $\Omega$, $\|D[(|Dv|^2+\epsilon)^{\frac\gamma 2}Dv]\|_{L^2(\Omega)}$ is bounded via $\|(|Dv|^2+\epsilon)^{\frac\gamma 2} \Delta^N_{p,\epsilon}v \|_{L^2(\Omega)}$, where $\Delta^N_{p,\epsilon}v$ is the $\epsilon$-regularization of normalized $p$-Laplacian. Our results extend the well-known Calder\'on-Zygmund $L^2$-estimate for the Poisson equation, a nonlinear global second order Sobolev estimate for inhomogeneous $p$-Laplace equation by Cianchi-Maz'ya (2018), and a local $W^{2,2}$-estimate for inhomogeneous normalized $p$-Laplace equation by Attouchi-Ruosteenoja (2018).

In this paper, we investigate a class of inviscid generalized surface quasi-geostrophic (SQG) equations on the half-plane with a rigid boundary. Compared to the Biot-Savart law in the vorticity form of the 2D Euler equation, the velocity formula here includes an additional Fourier multiplier operator $m(\Lambda)$. When $m(\Lambda) = \Lambda^\alpha$, where $\Lambda = (-\Delta)^{1/2}$ and $\alpha\in (0,2)$, the equation reduces to the well-known $\alpha$-SQG equation. Finite-time singularity formation for patch solutions to the $\alpha$-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlato\v{s} [Ann. Math., 184 (2016), pp. 909-948]. We establish finite-time singularity formation for patch solutions to the generalized SQG equations under the Osgood condition \[\int_2^\infty \frac{1}{r (\log r) m(r)} dr < \infty\] along with some additional mild conditions. Notably, our result fills the gap between the globally well-posed 2D Euler equation ($\alpha = 0$) and the $\alpha$-SQG equation ($\alpha > 0$). Furthermore, in line with Elgindi's global regularity results for 2D Loglog-Euler type equations [Arch. Rat. Mech. Anal., 211 (2014), pp. 965-990], our findings suggest that the Osgood condition serves as a sharp threshold that distinguishes global regularity and finite-time singularity in these models. In addition, we generalize the local regularity and finite-time singularity results for patch solutions to the $\alpha$-SQG equation, as established by Gancedo and Patel [Ann. PDE, 7 (2021), no. 1, Art. no. 4], extending them to cases where $m(r)$ behaves like $r^\alpha$ near infinity but does not have an explicit formulation., Comment: 62 pages, 4 figures

We investigate global solutions to the Euler-alignment system in $d$ dimensions with unidirectional flows and strongly singular communication protocols $\phi(x) = |x|^{-(d+\alpha)}$ for $\alpha \in (0,2)$. Our paper establishes global regularity results in both the subcritical regime $1<\alpha<2$ and the critical regime $\alpha=1$. Notably, when $\alpha=1$, the system exhibits a critical scaling similar to the critical quasi-geostrophic equation. To achieve global well-posedness, we employ a novel method based on propagating the modulus of continuity. Our approach introduces the concept of simultaneously propagating multiple moduli of continuity, which allows us to effectively handle the system of two equations with critical scaling. Additionally, we improve the regularity criteria for solutions to this system in the supercritical regime $0<\alpha<1$., Comment: 31 pages

Let $\Omega\subset R^n$ be a bounded convex domain with $n\ge2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as $t\to \infty$., Comment: 39 pages

The paper concerns with the global well-posedness issue of the 2D incompressible inhomogeneous Navier-Stokes (INS) equations with fractional dissipation and rough density. We first establish the $L^q_t(L^p)$-maximal regularity estimate for the generalized Stokes system with fractional dissipation, and then we employ it to obtain the global existence of solution for the 2D fractional INS equations with large velocity field, provided that the $L^2\cap L^\infty$-norm of density minus constant 1 is small enough. Moreover, by additionally assuming that the density minus 1 is sufficiently small in the norm of some multiplier spaces, we prove the uniqueness of the constructed solution by using the Lagrangian coordinates approach. We also consider the density patch problem for the 2D fractional INS equations, and show the global persistence of $C^{1,\gamma}$-regularity of the density patch boundary when the piecewise jump of density is small enough., Comment: 43 pages

In this paper we address the temperature patch problem of the 2D viscous Boussinesq system without heat diffusion term. The temperature satisfies the transport equation and the initial data of temperature is given in the form of non-constant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that our partially viscous Boussinesq system admits a unique global regular solution and the initial $C^{k,\gamma}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ and $\gamma\in (0,1)$ will be preserved for all the time. In particular, this naturally extends the previous work by Danchin $\&$ Zhang (2017) and Gancedo $\&$ Garc\'ia-Ju\'arez (2017). In the proof of the persistence result of higher boundary regularity, we introduce the striated type Besov space $\mathcal{B}^{s,\ell}_{p,r,W}(\mathbb{R}^d)$ and establish a series of refined striated estimates in such a function space, which may have its own interest., Comment: 50 pages. The striated estimates are simplified

We study one-dimensional Eulerian dynamics with nonlocal alignment interactions, featuring strong short-range alignment, and long-range misalignment. Compared with the well-studied Euler-alignment system, the presence of the misalignment brings different behaviors of the solutions, including the possible creation of vacuum at infinite time, which destabilizes the solutions. We show that with a strongly singular short-range alignment interaction, the solution is globally regular, despite the effect of misalignment., Comment: 38 pages, 1 figure

In this paper, we study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{tt}-\Delta u+u+(|x|^{-4}\ast|u|^2)u=0$ in the spatial dimension $d \geq 5$. We utilize the strategy in [S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation. Analysis and PDE., 4 (2011), 405-460.] derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the soliton-like solution. Employing technique from [B. Pausader, Scattering for the Beam Equation in Low Dimensions. Indiana Univ. Math. J., 59 (2010), 791-822.], we consider a virial-type identity in the direction orthogonal to the momentum vector so as to exclude such solution., Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:math/0612028

Suppose that $n\ge3$ and $H(p)\in C^0(\mathbb{R}^n)$ is a locally strongly convex and concave Hamiltonian. We obtain the everywhere differentiability of all absolute minimizers for $H$ in any domain of $\mathbb{R}^n$., Comment: 21 pages

Suppose that $H \in C^0 (\mathbb{R}^2)$ satisfies \begin{enumerate} \item[(H1)] $H$ is locally strongly convex and locally strongly concave in $\rr^2$, \item[(H2)] $H(0)=\min_{p\in\rr^2}H(p)=0$. \end{enumerate} Let $\Omega\subset \rr^2$ be any domain. For any $u$ absolute minimizer for $H$ in $\Omega$, or if $H\in C^1(\rr^2)$ additionally, for any viscosity solution to the Aronsson equation $$\mathscr A_H[u]=\sum_{i,j=1}^2 H_{p_i}(Du) H_{p_j}(Du)u_{x_ix_j}=0 \quad \mbox{ in $\Omega$,}$$ the following are proven in this paper: \begin{enumerate} \item[(i)] We have $[H(Du)]^\alpha\in W^{1,2}_\loc(\Omega)$ whenever $\alpha>1/2-\tau_H(0)$; some quantative upper bounds are also given. Here $\tau_H(0)=1/2$ when $H\in C^2(\rr^2)$, and $0< \tau_H(0)\le 1/2$ in general. \item[(ii)] If $H\in C^1(\rr^2)$, then the distributional determinant $-{\rm det}D^2u\,dx$ is a nonnegative Radon measure in $\Omega$ and enjoys some quantative lower/upper bounds. \item[(iii)] If $H\in C^1(\rr^2)$, then for all $\alpha>\frac12-\tau_H(0)$, we have $$\mbox{$\langle D [H(Du )]^\alpha ,D_p H(Du )\rangle=0 $ almost everywhere in $\Omega$}.$$ \end{enumerate}, Comment: 54 pages

For a bounded domain $U\subset\rn$, consider the $L^\fz$-functional involving a nonnegative Hamilton function $H:\overline U\times\rn\to [0,\fz)$. In this paper, we will establish the uniqueness of absolute minimizers $u\in W^{1,\fz}_\loc(U)\cap C(\overline U)$ for $H$, under the Dirichlet boundary value $g\in C(\partial U)$, provided \noindent (A1) $H$ is lower semicontinuous in $\overline U\times\rn$, and $H(x,\cdot)$ is convex for any $x\in\overline U$. \noindent (A2) $\displaystyle H(x,0)=\min_{p\in \rn}H(x,p)=0$ for any $ x\in \overline U$, and $\displaystyle\bigcup_{x\in \overline U}\big\{p: H(x,p)=0\big\}$ is contained in a hyperplane of $\rn$. \noindent (A3) For any $\lz>0$, there exist $\displaystyle 0 0\ \mbox{and}\ x\in \overline U.$$ This generalizes the uniqueness theorem by \cite{j93, jwy, acjs} and \cite{ksz} to a large class of Hamiltonian functions $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by {\cite{jwy}}. The proofs rely on geometric structure of the action function $\mathcal L_t(x,y)$ induced by $H$, and the identification of the absolute subminimality of $u$ with convexity of the Hamilton-Jacobi flow $t\mapsto T^tu(x)$, Comment: 53 pages