arXiv subjects

Lili Du

Publications and source records attributed to Lili Du.

At least 19 recordsLinked to original sources

arXiv2026

Full-Density Degenerate Stagnation Points for Water Waves with General Vorticity

In this paper, we revisit the singular asymptotics of the free surface near stagnation points for two-dimensional traveling gravity water waves with vorticity. We prove the nonexistence of full-density degenerate stagnation points beyond the strict two-sided linear growth regime. Our main tools are a modified Weiss-type monotonicity formula and a modified Almgren-type frequency formula. Together, they provide a new approach that completely avoids the use of a Bessel-type differential inequality, which is an essential tool used in the previous literature to prove the nonexistence of full-density degenerate stagnation points (Ann. I. H. Poincar\'e-AN, 29, 861--885, 2012). As consequences, we obtain uniform bounds for frequency-normalized blow-ups in arbitrary dimension and strong convergence in dimension two. As an application, we extend the Stokes conjecture for rotational waves to a broader class of vorticity distributions.

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arXiv2026

3D Segment Anything Model with Visual Mamba for Diagnosing Placenta Accreta Spectrum

Placenta Accreta Spectrum (PAS) is a rare but highly dangerous obstetric disease. Early and accurate PAS diagnosis is critical for maternal health. Traditional PAS diagnosis relies on experienced doctors by analyzing the cesarean history and Magnetic Resonance Imaging (MRI) data. However, district-level hospitals often lack the expertise and resources for accurate PAS diagnosis. To address these challenges, we establish the first MRI-based PAS dataset, which includes both fine-grained segmentation and classification annotations. Meanwhile, diagnosing PAS can be significantly enhanced by segmenting lesion areas from MRI images of the uterus. To achieve automatic PAS diagnosis, we propose 3DSAMba, a novel feature learning framework for effective lesion segmentation. More specifically, we first design a 3D Segment Anything Model (SAM) and incorporate medical domain information into the model through an efficient adapter mechanism. In addition, we introduce a Multi-Level Aggregation Mamba (MLAM) to aggregate feature maps across different levels and a Fusion State Space Model (FSSM) to fuse multi-scale features from both the encoder and decoder. Finally, we apply segmentation masks to the original MRI images through element-wise multiplication, effectively isolating lesion areas for more accurate PAS diagnosis. Extensive experiments validate that our framework significantly improves the PAS diagnostic performance. To facilitate further research in PAS diagnosis, we have released the dataset and source code at https://github.com/Drchip61/PASD.

cs.CV↗

arXiv2026

Geometric structure of singular free boundary points for the logarithmic obstacle problem

In the previous work [Interfaces Free Bound., 19, 351--369, 2017], de Queiroz and Shahgholian established the optimal $C^{1,\log}_{\mathrm{loc}}$ regularity of solutions for the obstacle problem with singular logarithmic forcing term $$-\Delta u = \log u\,\chi_{\{u>0\}} \quad \text{in } \Omega,$$ where $\Omega\subset\mathbb{R}^d$ ($d\geq 2$) is a smooth bounded domain. In our earlier work [arXiv:2408.08104, 2024], we proved the $C^{1,\alpha}$ regularity of the free boundary $\Omega\cap\partial\{u>0\}$ near regular points. In this paper, we investigate the more delicate structure of the \emph{singular} free boundary. Since the nonlinearity $-\log u$ is singular near the free boundary and destroys the scaling invariance, so that neither the classical blow-up arguments nor the standard epiperimetric inequality [Weiss, Invent.\ Math., 138, 23--50, 1999] apply directly; moreover, the Weiss type monotonicity formula requires a variable-parameter correction that introduces non-integrable remainder terms into the energy estimates. Motivated by Colombo--Spolaor--Velichkov [Geom.\ Funct.\ Anal., 28, 1029--1061, 2018], we develop a new \emph{log-epiperimetric inequality} for the modified Weiss energy, also proved by the direct method. A key novelty is the introduction of an auxiliary correction term $T$ that absorbs the non-integrable errors. As consequences, we establish a logarithmic energy decay, uniqueness of blow-ups at singular points, and a $C^{1,\log}$-type geometric description of the singular strata. In dimension two, the logarithmic modulus improves to a H\"older modulus.

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arXiv2026

The singularity at degenerate points in steady axisymmetric compressible free surface flows with gravity

In this paper, we analyze the singular shape of the free boundary at degenerate points in a three dimensional axisymmetric compressible gravity flow. For all possible degenerate points on the free surface, we prove that the only nontrivial asymptotic behavior of the free surface at the stagnation points away from the axis of symmetry is the Stokes corner flow. The possible geometries for free boundaries at the non-stagnation axis points are downward pointing or upward pointing cusps. At the origin, there are only two nontrivial asymptotics possible: the Garabedian's pointed bubble or a horizontal flat surface. The problem is associated with the analysis of the degenerate points of a quasilinear free boundary problem of the Bernoulli type, and the main obstacles are the absence of a Weiss-type monotonicity formula. To achieve our goal, we establish for the first time monotonicity formulas for quasilinear Bernoulli type free boundary problems. Our formula works both when the equation becomes singular and when the free boundary condition is degenerate. Moreover, we establish a new nonlinear frequency formula at the horizontal flat points at the origin and integrate it with the compensated compactness theory for Euler equations, ensuring the strong convergence of variational solutions. Our results resolve the Stokes conjecture [Mathematical and Physical Papers, Vol. I., 1880] in a generalized compressible, three dimensional axisymmetric framework. In addition, it can also be realized as a compressible counterpart to V\v{a}rv\v{a}ruc\v{a} and Weiss [Comm. Pure Appl. Math., (67), 2014]. Our approach is completely new and gives a systematic approach for studying singularities of a singular Bernoulli type quasilinear free boundary problem.

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arXiv2026

Free boundary problem for two-dimensional ElectroHydroDynamic Equations with a gravity field

This paper studies a two-phase free boundary problem governed by the ElectroHydroDynamic equations, which describes a perfectly conducting, incompressible, irrotational fluid with gravity, surrounded by a dielectric gas. The interface separating fluid and gas is referred to as the free boundary. It is known that the free surface remains smooth away from the stagnation points, where the relative velocity of the incompressible fluid vanishes. In the presence of gravity, the Stokes conjecture, proved by Varvaruca and Weiss [Acta. Math. 206, 363-403, (2011)], implies that the corner type singularity will occur in the one-phase incompressible fluid. It is natural to ask whether this conjecture still holds in the two-phase flow problem. As a consequence, the primary objective of this work is to characterize the possible singular profiles of the free interface near the stagnation points in the presence of an electric field. Our main result is the discovery of a critical decay rate of the electric field near the stagnation points which indicates the classification of the singular profiles of the free surface. More precisely, we showed that when the decay rate of the electric field is faster than the critical decay rate, its negligible effect implies that the singular profile must be the well-known Stokes corner. When the electric field decays as the critical decay rate, the symmetry of the corner region may be broken, giving rise to either a Stokes corner or an asymmetric corner as the possible singular profile. If the decay rate is slower than the critical decay rate, the electric field dominates and completely destroys the corner structure, resulting in a cusp singularity. The analysis of these singularities relies on variational principles and geometric methods. Key technical tools include a Weiss-type monotonicity formula, a frequency formula, and a concentration-compactness argument.

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arXiv2025

Admissible solutions of the 2D Onsager's conjecture

We show that for any $\gamma < \frac{1}{3}$ there exist H\"{o}lder continuous weak solutions $v \in C^{\gamma}([0,T] \times \mathbb{T}^2)$ of the two-dimensional incompressible Euler equations that strictly dissipate the total kinetic energy, improving upon the elegant work of Giri and Radu [Invent. Math., 238 (2), 2024]. Furthermore, we prove that the initial data of these \textit{admissible} solutions are dense in $B^{\gamma}_{\infty,r<\infty}$. Our approach introduces a new class of traveling waves, refining the traditional temporal oscillation function first proposed by Cheskidov and Luo [Invent. Math., 229(3), 2022], to effectively modulate energy on any time intervals. Additionally, we propose a novel ``multiple iteration scheme'' combining Newton-Nash iteration with a Picard-type iteration to generate an energy corrector for controlling total kinetic energy during the perturbation step. This framework enables us to construct dissipative weak solutions below the Onsager critical exponent in any dimension $d \geq 2$.

math.AP↗

arXiv2025

Helically symmetric solution of 3D Euler equations with vorticity and its free boundary

This paper investigates an incompressible steady free boundary problem of Euler equations with helical symmetry in $3$ dimensions and with nontrivial vorticity. The velocity field of the fluid arises from the spiral of its velocity within a cross-section, whose global existence, uniqueness and well-posedness with fixed boundary were established by a series of brilliant works. A perplexing issue, untouched in the literature, concerns the free boundary problem with (partial) unknown domain boundary in this helically symmetric configuration. We address this gap through the analysis of the optimal regularity property of the scalar stream function as a minimizer in a semilinear minimal problem, establishing the $C^{0,1}$-regularity of the minimizer, and the $C^{1,\alpha}$-regularity of its free boundary. More specifically, the regularity results are obtained in arbitrary cross-sections through smooth helical transformation by virtue of variational method and the rule of "flatness implies $C^{1,\alpha}$".

math.AP↗

arXiv2025

Singular and regular analysis for the free boundaries of two-phase inviscid fluids in gravity field

In this paper, we consider a free boundary problem of two-phase inviscid incompressible fluid in gravity field. The presence of the gravity field induces novel phenomena that there might be some stagnation points on free surface of the two-phase flow, where the velocity field of the fluid vanishes. From the mathematical point of view, the gradient of the stream function degenerates near the stagnation point, leading to singular behaviors on the free surface. The primary objective of this study is to investigate the singularity and regularity of the two-phase free surface, considering their mutual interaction between the two incompressible fluids in two dimensions. More precisely, if the two fluids meet locally at a single point, referred to as the possible two-phase stagnation point, we demonstrate that the singular side of the two-phase free surface exhibits a symmetric Stokes singular profile, while the regular side near this point maintains the $C^{1,\alpha}$ regularity. On the other hand, if the free surfaces of the two fluids stick together and have non-trivial overlapping common boundary at the stagnation point, then the interaction between the two fluids will break the symmetry of the Stokes corner profile, which is attached to the $C^{1,\alpha}$ regular free surface on the other side. As a byproduct of our analysis, it's shown that the velocity field for the two fluids cannot vanish simultaneously on the two-phase free boundary. Our results generalize the significant works on the Stokes conjecture in [V$\check{a}$rv$\check{a}$ruc$\check{a}$-Weiss, Acta Math., 206, (2011)] for one-phase gravity water wave, and on regular results on the free boundaries in [De Philippis-Spolaor-Velichkov, Invent. Math., 225, (2021)] for two-phase fluids without gravity.

math.AP↗

arXiv2024

Proof of the Stokes conjecture for compressible gravity water waves

In 1880, Stokes examined an incompressible irrotational periodic traveling water wave under the influence of gravity and conjectured the existence of an extreme wave with a corner of $120^{\circ}$ at the crest. The first rigorous proof of the conjecture was given by Amick, Fraenkel and Toland, as well as by Plotnikov independently via the Nekrasov integral equation. In the early 2010s, Weiss and Varvarucva revisited the conjecture by applying a new geometric method, which provided an affirmative answer to the conjecture without requiring structural assumptions such as the isolation of the stagnation points, the symmetry and the monotonicity of the free surface that were necessary in the previous works. The main purpose of this paper is to establish the validity of the Stokes conjecture in the context of compressible gravity water waves. More precisely, we prove that a sharp crest forms near each stagnation point of a compressible gravity water wave with an included angle of $120^{\circ}$, which gives a first proof to the compressible counterpart of the classical conjecture by Stokes in 1880. The central aspect of our approach is the discovery of a new monotonicity formula for quasilinear free boundary problems of the Bernoulli-type. Another observation is the introduction of a new nonlinear frequency formula, along with a compensated compactness argument for the compressible Euler system. The developed monotonicity formula enables us to do blow-up analysis at each stagnation point and helps us obtain the singular profile of the free surface near each stagnation points. The degenerate stagnation points can be further analyzed with the help of the compensated compactness argument using the frequency formula.

math.AP↗

arXiv2024

On a class of coupled obstacle systems

In this paper, we explore cooperative and competitive coupled obstacle systems, which, up to now, are new type obstacle systems and formed by coupling two equations belonging to classical obstacle problem. On one hand, applying the constrained minimizer in variational methods we establish the existence of solutions for the systems. Moreover, the optimal regularity of solutions is obtained, which is the cornerstone for further research on so-called free boundary. Furthermore, as coefficient $\lambda\to0$, there exists a sequence of solutions converging to solutions of the single classical obstacle equation. On the other hand, motivated by the heartstirring ideas of single classical obstacle problem, based on the corresponding blowup methods, Weiss type monotonicity formula and Monneau type monotonicity formula of systems to be studied, we investigate the regularity of free boundary, and on the regular and singular points in particular, as it should be, which is more challenging but exceedingly meaningful in solving free boundary problems.

math.AP↗

arXiv2024

The free boundary for the singular obstacle problem with logarithmic forcing term

In the previous work [Interfaces Free Bound., 19, 351-369, 2017], de Queiroz and Shahgholian investigated the regularity of the solution to the obstacle problem with singular logarithmic forcing term \begin{equation*} -Δu = \log u \, χ_{\{u>0\}} \quad \text{in} \quad Ω, \end{equation*} where $χ_{\{u>0\}}$ denotes the characteristic function of the set $\{u>0\}$ and $Ω\subset \mathbb{R}^n$ ($n \geq 2$) is a smooth bounded domain. The solution solves the minimum problem for the following functional, \begin{equation*} \mathscr{J}(u):=\int_Ω\left(\frac{|\nabla u|^2}{2}-u^+ (\log u-1)\right) \, dx, \end{equation*} where $u^+=\max{\{0,u\}}$. In this paper, based on the regularity of the solution, we establish the $C^{1,α}$ regularity of the free boundary $Ω\cap \partial\{u>0\}$ near the regular points for some $α\in (0,1)$. The logarithmic forcing term becomes singular near the free boundary $Ω\cap\partial\{u>0\}$ and lacks the scaling properties, which are very crucial in studying the regularity of the free boundary. Despite these challenges, we draw inspiration for our overall strategy from the "epiperimetric inequality" method introduced by Weiss in 1999 [Invent. Math., 138, 23-50, 1999]. Central to our approach is the introduction of a new type of energy contraction. This allows us to achieve energy decay, which in turn ensures the uniqueness of the blow-up limit, and subsequently leads to the regularity of the free boundary.

math.AP↗

arXiv2024

Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations

In this article, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^{\gamma}W_{x}^{s,p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Lady\v{z}enskaya-Prodi-Serrin condition with the triplet $(s,\gamma,p)=(s,\infty, \frac{2}{2\alpha-1+s})$ and $(s, \frac{2\alpha}{2\alpha-1+s}, \infty)$. As a good observation, we use the intermittency of the temporal concentrated function in an almost optimal way. The research results extend the recent elegant works on 2D Navier-Stokes equations in [Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987--1054; Cheskidov and Luo, Ann. PDE, 9:13 (2023)] to the hyper-dissipative case $\alpha \in(1,\frac{3}{2})$, and are also applicable in Lebesgue and Besov spaces. It is proved that even in the case of high viscosity, the behavior of the solution remains unpredictable and stochastic due to the lack of integrability and regularity.

math.AP↗

arXiv2024

The free boundary for a semilinear non-homogeneous Bernoulli problem

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a "regular" part and a "singular" part, as Alt and Caffarelli have shown in their pioneer work (J. Reine Angew. Math., 325, 105-144, 1981) that regular points are $C^{1,γ}$ in two-dimensions. Later, Weiss (J. Geom. Anal., 9, 317-326, 1999) first realized that in higher dimensions a critical dimension $d^{*}$ exists so that the singularities of the free boundary can only occur when $d\geqslant d^{*}$. In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a $C^{1,γ}$ function for some $γ\in(0,1)$. In addition, there exists a critical dimension $d^{*}$ so that the singular set is empty if $d d^{*}$. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.

math.AP↗

arXiv2024

Singular profile of free boundary of incompressible inviscid fluid with external force

This article is devoted to investigate the singular profile of the free boundary of two-dimensional incompressible inviscid fluid with external force near the stagnation point. More precisely, given an external force with some polynomial type decay close to the stagnation point, the singular profile of the free boundary at stagnation point possible are corner wave, flat and cusp singularity. Through excluding the cusp and flat singularity, we know the only singular profile is corner wave singularity, and the corner depends on the decay rate of the solution near the stagnation point. The analysis depends on the geometric method to a class of Bernoulli-type free boundary problem with given degenerate gradient function on free boundary. This work is motivated by the significant work [E. V$\breve{a}$rv$\breve{a}$ruc$\breve{a}$ and G. Weiss, Acta Math, 206, 363-403, (2011)] on Stokes conjecture to the incompressible inviscid fluid acted on by gravity.

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arXiv2024

Regularity of the free boundaries for the two-phase axisymmetric inviscid fluid

In the seminal paper (Alt, Caffarelli and Friedman, Trans. Amer. Math. Soc., 282, (1984).), the regularity of the free boundary of two-phase fluid in two dimensions via the so-called ACF energy functional was investigated. It was shown the $C^1$ regularity of the free boundaries and asserted that the two free boundaries coincide under some additional assumptions. Later on the standard technique of Harnack inequality could be applied to improve the regularity to $C^{1,\eta}$. A recent significant breakthrough in the regularity of two-phase fluid is due to De Philippis, Spolaor and Velichkov, who investigated the free boundary of the two-phase fluid with the two-phase functional (De Philippis, Spolaor and Velichkov, Invent. Math., 225, (2021).), and the $C^{1,\eta}$ regularity of the whole free boundaries was given in dimension two. Moreover, the free boundaries of the two-phase fluids do not coincide and the zero level set may process positive Lebesgue measure. In this paper, we consider the free boundaries for the two-phase axisymmetric fluid and show the free boundary is $C^{1,\eta}$ smooth. The Lebesgue measure of the zero level set of may also be positive, and the main difference lies in the degenerate elliptic operator and the free boundary conditions. More precisely, we use partial boundary Harnack inequalities and establish a linearized problem, whose regularity of the solutions implies the flatness decay of the two-phase free boundaries. Then the iteration argument gives the smoothness of the free boundaries.

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arXiv2023

The non-existence of horizontally flat singularity for steady axisymmetric free surface flows near stagnation points

In a recent research on degenerate points of steady axisymmetric gravity flows with general vorticity, it has been shown that the possible asymptotics near any stagnation point must be the "Stokes corner", the "horizontal cusp", or the "horizontal flatness" (Theorem 1.1, Du, Huang, Pu, Commun. Math. Phys., 400, 2137-2179, 2023). In this paper, we focus on the horizontally flat singularity and show that it is not possible, and therefore the "Stokes corner" and the "cusp" are the only possible asymptotics at the stagnation points. The basic idea of our proof relies on a perturbation of the frequency formula for the two-dimensional problem (Varvaruca, Weiss, Acta Math., 206, 363-403, 2011). Our analysis also suggests that, for steady axisymmetric rotational gravity flows, the singular asymptotic profiles at stagnation points are similar to the scenario observed in two-dimensional waves with vorticity (Varvaruca, Weiss, Ann. I. H. Poincare-AN, 29, 861-885, 2012).

math.AP↗

arXiv2023

The free boundary of steady axisymmetric inviscid flow with vorticity II: near the non-degenerate points

This is the sequel of the recent work (Du, Huang, Pu, Commun. Math. Phys, 2023, doi: 10.1007/s00220-023-04651-7) on axially symmetric gravity water waves with general vorticities, which has investigated the singular wave profile of the free boundary near the degenerate points. In this companion paper, we are interested in the regularity of the free surface of the water wave near the non-degenerate point. Precisely, we showed that the free boundary is $C^{1,\gamma}$ smooth for some $\gamma\in(0,1)$ near all non-degenerate points. The problem is intrinsically intertwined with the regularity theory of the semilinear Bernoulli-type free boundary problem. Our approach is closely related to the monotonicity formula developed by Weiss in his celebrated work (Weiss, J. Geom. Anal. 9: 317-326, 1999), and to a partial boundary Harnack inequality for the one-phase free boundary problem, which is dedicated to De Silva (De Silva, Interfaces Free Bound. 13, 223-238, 2011). Mathematically, we associate the existence of viscosity solutions with the Weiss boundary-adjusted energy. Compared to the classical approach of Caffarelli (Caffarelli, Ann. Scuola Norm. Sup. Pisa, 15, 583-602, 1988), we provide an alternative proof of the existence of viscosity solutions for a large class of semilinear free boundary problems.

math.AP↗

arXiv2023

Uniqueness of blowup at singular points for superconductivity problem

In this paper, we prove that the uniqueness of blowup at the maximum point of coincidence set of the superconductivity problem, mainly based on the Weiss-type and Monneau-type monotonicity formulas, and the proof of the main results in this paper is inspired the recent paper \cite{CFL22} by Chen-Feng-Li.

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