Annals of PDE

Xishen Jin

Jiawei Liu

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

余切丛中广义拉格朗日平均曲率流的稳定性

Abstract In this paper, we give the first rigorous justification of the Benjamin-Ono equation: $$\begin{aligned} \hspace{3cm} \partial _t \zeta + (1 - \frac{\gamma }{2}\sqrt{\mu }|\textrm{D}|)\partial _x \zeta + \frac{3{\varepsilon }}{2}\zeta \partial _x\zeta =0, \hspace{2cm} \text {(BO)} \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mspace/> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> <mml:mfrac> <mml:mi>γ</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msqrt> <mml:mi>μ</mml:mi> </mml:msqrt> <mml:mrow> <mml:mo>|</mml:mo> <mml:mtext>D</mml:mtext> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>ζ</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>ζ</mml:mi> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>BO</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> as an internal water wave model on the physical time scale. Here, $${\varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> is a small parameter measuring the weak nonlinearity of the waves, $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> is the shallowness parameter, and $$\gamma \in (0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order $${\mathcal {O}}(\frac{1}{{\varepsilon }})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>ε</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for a small amount of surface tension such that $${\varepsilon }^2 \le \textrm{bo}^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>≤</mml:mo> <mml:msup> <mml:mtext>bo</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> where $$\textrm{bo}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>bo</mml:mtext> </mml:math> is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order $${\mathcal {O}}(\mu + \textrm{bo}^{-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mtext>bo</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit. The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.

Martin Oen Paulsen

Justification of the Benjamin–Ono equation as an internal water waves model

Benjamin-Ono方程作为内波模型的合理性

Jiajun Tong

Dongyi Wei

Geometric Properties of the 2-D Peskin Problem

二维Peskin问题的几何性质

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface $$\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3$$ and the outgoing null hypersurface $${{\mathcal {H}}}$$ emanating from $${\partial }\Sigma $$ , we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in $$L^2$$ . The proof uses the bounded $$L^2$$ curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

Stefan Czimek

Olivier Graf

The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

低正则性下广义相对论的类空特征柯西问题

Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $$C^\gamma $$ norm of the density $$\theta $$ with $$\gamma \approx 1/3$$ is uniformly bounded up to the singularity time.

Jiajie Chen

Thomas Y. Hou

De Huang

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

三维欧拉方程的Hou-Luo模型渐近自相似爆破

Pak-Yeung Chan

Shaochuang Huang

Man-Chun Lee

Manifolds with Small Curvature Concentration

具有小曲率集中现象的流形

Abstract We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Massimiliano Berti

Alberto Maspero

Federico Murgante

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

哈密顿-伯克霍夫正则型在具有常数涡度的重力-毛细水波中的应用：几乎全局存在性

Abstract In this work we construct global unique solutions of the dissipative Surface quasi-geostrophic equation ( $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -SQG) that lose regularity instantly when there is super-critical fractional diffusion.

Diego Córdoba

Luis Martínez-Zoroa

Global Unique Solutions with Instantaneous Loss of Regularity for SQG with Fractional Diffusion

具有分数扩散的SQG方程全局唯一解及其瞬时正则性丧失

Abbreviation	Ann. PDE
Journal Impact	2.41
Quartiles(Global)	PHYSICS, MATHEMATICAL(Q1)

ISSN	2199-2576, 2524-5317
h-index	25

Publication Information	Publisher: Springer Science + Business Media，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2015，Proportion of original research papers: 100.00%，Self Citation Rate:4.20%， Gold OA Rate: 24.66%

Annals of PDE

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Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

Justification of the Benjamin–Ono equation as an internal water waves model

Geometric Properties of the 2-D Peskin Problem

The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

Justification of the Benjamin–Ono equation as an internal water waves model

Geometric Properties of the 2-D Peskin Problem

Manifolds with Small Curvature Concentration

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

Global Unique Solutions with Instantaneous Loss of Regularity for SQG with Fractional Diffusion