arXiv subjects

Liutang Xue

Publications and source records attributed to Liutang Xue.

At least 19 recordsLinked to original sources

arXiv2025

A revisit of patch solutions for the 2D Loglog-Euler type equation

In this paper, we revisit the patch solutions for a class of inviscid whole-space active scalar equations that interpolate between the 2D Euler equation and the $α$-SQG equation. Compared with the 2D Euler equation in vorticity form, there is an additional Fourier multiplier $m(Λ)$ ($Λ= (-Δ)^{1/2}$) in the Biot-Savart law. If the symbol $m$ satisfies the Osgood-type condition $$\int_2^{+\infty} \frac{1}{r (\log r) m(r)} dr= +\infty$$ and certain mild assumptions, the system is referred to as the 2D Loglog-Euler type equation. First, we prove a Yudovich-type theorem establishing the existence and uniqueness of a global weak solution for the Loglog-Euler type equation associated with bounded and integrable initial data. This result directly applies to patch solutions, which are weak solutions corresponding to patch initial data given by characteristic functions of disjoint, regular, bounded domains. Next, we revisit the seminal result by Elgindi ( Arch. Ration. Mech. Anal. 211(3) 965-990, 2014 ) and provide a different proof under explicit assumptions on $m$, showing that for the 2D Loglog-Euler type equation with $C^{1,μ}$ ($0<μ<1$) single-patch initial data, the evolved patch boundary globally preserves the $C^{1,μ-\varepsilon}$ regularity for any $\varepsilon \in (0,μ)$. In contrast to the frequency-space argument in Elgindi's result, we develop an entirely physical-space-based approach that avoids the Littlewood-Paley theory and offers advantages for potential extensions to the half-plane or bounded smooth domains. Furthermore, we investigate the global propagation of higher-order $C^{n,μ}$ boundary regularity for patch solutions with any $n \in \mathbb{N}^\star$, and analyze the evolution of multiple patches.

math.AP↗

arXiv2025

Doubly Connected V-States in Geophysical Models: A General Framework

In this paper, we prove the existence of doubly connected V-states (rotating patches) close to an annulus for active scalar equations with completely monotone kernels. This provides a unified framework for various results related to geophysical flows. This allows us to recover existing results on this topic while also extending to new models, such as the gSQG and QGSW equations in radial domains and 2D Euler equation in annular domains.

math.AP↗

arXiv2024

Local regularity and finite-time singularity for a class of generalized SQG patches on the half-plane

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 $m(Λ)$. When $m(Λ) = Λ^α$, where $Λ= (-Δ)^{1/2}$ and $α\in (0,2)$, the equation reduces to the well-known $α$-SQG equation. Finite-time singularity formation for patch solutions to the $α$-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlatoš [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 ($α= 0$) and the $α$-SQG equation ($α> 0$). 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 $α$-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 $r^α$ near infinity but does not have an explicit formulation.

math.AP↗

arXiv2024

Global regularity and infinite Prandtl number limit of temperature patches for the 2D Boussinesq system

We prove global regularity and study the infinite Prandtl number limit of temperature patches for the 2D non-diffusive Boussinesq system with dissipation in the full subcritical regime. The temperature satisfies a transport equation and the temperature initial data are given in the form of non-constant patches. Our first main result is a persistence of regularity of the patches globally in time. More precisely, we prove that if the boundary of the initial temperature patch lies in $C^{k+γ}$ with $k\geq 1$ and $γ\in(0,1)$ then this initial regularity is preserved for all time. Importantly, our proof is robust enough to show uniform dependence on the Prandtl number in some cases. This result solves a question in Khor and Xu \cite{KX22} concerning the global control of the curvature of the patch boundary. Besides, by studying the limit when the Prandtl number goes to infinity, we find that the patch solutions to the 2D Boussinesq-Navier-Stokes system in the torus converge to the unique patch solutions of the (fractional) Stokes-transport equation and that the $C^{k+γ}$ regularity of the patch boundary is globally preserved. This allows us to extend the $C^{k+γ}$ persistence result of Grayer II \cite{Gray23} from the range $k\in \{0,1,2\}$ to the full range $k\geq 1$.

math.AP↗

arXiv2024

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter $α\in(0,1)$. This highlights the distinct feature of the alignment operator.

math.AP↗

arXiv2023

Unified theory on V-states structures for active scalar equations

This paper revolves around the existence of V-states close to Rankine vortices for active scalar equations with completely monotone kernels. This allows to unify various results on this topic related to geophysical flows. A key ingredient is a new factorization formula for the spectrum using a universal function which is independent of the model. This function admits several interesting properties allowing to track the spectrum distribution.

math.AP↗

arXiv2023

Global well-posedness and refined regularity criterion for the uni-directional Euler-alignment system

We investigate global solutions to the Euler-alignment system in $d$ dimensions with unidirectional flows and strongly singular communication protocols $ϕ(x) = |x|^{-(d+α)}$ for $α\in (0,2)$. Our paper establishes global regularity results in both the subcritical regime $1<α<2$ and the critical regime $α=1$. Notably, when $α=1$, 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 $0<α<1$.

math.AP↗

arXiv2022

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when $α\in (0,1)$. We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

math.AP↗

arXiv2022

Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

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 $t\to \infty$.

math.AP↗

arXiv2022

On the regularity of temperature fronts for the 3D viscous Boussinesq system

We study the temperature front problem for the 3D viscous Boussinesq equation. We prove that the $C^{k,γ}$ ($k\geq 1$, $0<γ< 1$) and $W^{2,\infty}$ regularity of a temperature front is locally preserved along the evolution as well as globally preserved under a smallness condition in a critical space. In particular, beside giving another proof of the main result in \cite{GGJ20}, we also extend it to a more general class of regular patch.

math.AP↗

arXiv2021

Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough density

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 $C^{1,γ}$-regularity of the density patch boundary when the piecewise jump of density is small enough.

math.AP↗

arXiv2021

Global regularity of non-diffusive temperature fronts for the 2D viscous Boussinesq system

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 $C^{k,γ}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ and $γ\in (0,1)$ will be preserved for all the time. In particular, this naturally extends the previous work by Danchin $\&$ Zhang (2017) and Gancedo $\&$ García-Juárez (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.

math.AP↗

arXiv2020

Global regularity for a 1D Euler-alignment system with misalignment

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.

math.AP↗

arXiv2020

Boussinesq system with measure forcing

We address a question concerning the issue of existence to a Boussinesq type system with a heat source. The problem is studied in the whole two dimensional plane and the heat source is a measure transported by the flow. For arbitrary initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructing solutions and make us work in a non-standard framework of inhomogeneous Besov spaces of the $L^\infty(0,T;B^s_{p,\infty})$-type. Application of the Lagrangian coordinates yields uniqueness omitting difficulties with comparison of measures.

math.AP↗

arXiv2019

On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion

In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.

math.AP↗

arXiv2019

Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability

We construct large velocity vector solutions to the three dimensional inhomogeneous Navier-Stokes system. The result is proved via the stability of two dimensional solutions with constant density, under the assumption that initial density is point-wisely close to a constant. Key elements of our approach are estimates in the maximal regularity regime and the Lagrangian coordinates. Considerations are done in the whole $\R^3$.

math.AP↗

arXiv2018

Regularity results for a class of generalized surface quasi-geostrophic equations

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.

math.AP↗

arXiv2016

On the Differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion

We investigate the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial Hölder regularity, we prove that the vanishing viscosity solution is differentiable with some Hölder continuous derivatives for any positive time.

math.AP↗