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Jiahong Wu

Publications and source records attributed to Jiahong Wu.

At least 73 records · Page 4Linked to original sources
arXiv2024

Stability and Sharp Decay Estimates for 3D MHD Equations with Only Vertical Dissipation Near a Background Magnetic Field

This paper is concerned with the stability and large-time behavior of 3D incompressible MHD equations with only vertical dissipation near a background magnetic field. By making full use of the dissipation generated by the background magnetic field, we first establish the global stability of the solutions in $H^3$-norm. Then, the optimal decay rates of the solutions are obtained, which are consistent with the 2D classical heat equation. Moreover, some enhanced decay rates of $(u_1,b_1)$ are also achieved. In other words, the decay estimates of the second or third component of velocity/magnetic field coincide with those of 2D heat kernel, while the first component behaves like the 3D heat kernel. This is mainly due to the divergence-free condition and the anisotropic structure. The results obtained improve the previous ones due to Lin-Wu-Zhu [24,25].

math.AP↗
arXiv2023

Calmed 3D Navier-Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier-Stokes equations (NSE) in either advective or rotational form which "calms" the system in the sense that the algebraic degree of the nonlinearity is effectively reduced. This system, the calmed Navier-Stokes Equations (calmed NSE), utilizes a "calming function" in the nonlinear term to locally constrain large advective velocities. Notably, this approach avoids the direct smoothing or filtering of derivatives, thus we make no modifications to the boundary conditions. Under suitable conditions on the calming function, we are able to prove global well-posedness of calmed NSE and show the convergence of calmed NSE solutions to NSE solutions on the time interval of existence for the latter. In addition, we prove that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the existence of strong solutions to the 3D Navier-Stokes equations.

math.AP↗
arXiv2023

Multiple Threshold Schemes Under The Weak Secure Condition

In this paper, we consider the case that sharing many secrets among a set of participants using the threshold schemes. All secrets are assumed to be statistically independent and the weak secure condition is focused on. Under such circumstances we investigate the infimum of the (average) information ratio and the (average) randomness ratio for any structure pair which consists of the number of the participants and the threshold values of all secrets. For two structure pairs such that the two numbers of the participants are the same and the two arrays of threshold values have the subset relationship, two leading corollaries are proved following two directions. More specifically, the bound related to the lengths of shares, secrets and randomness for the complex structure pair can be truncated for the simple one; and the linear schemes for the simple structure pair can be combined independently to be a multiple threshold scheme for the complex one. The former corollary is useful for the converse part and the latter one is helpful for the achievability part. Three new bounds special for the case that the number of secrets corresponding to the same threshold value $ t $ is lager than $ t $ and two novel linear schemes modified from the Vandermonde matrix for two similar cases are presented. Then come the optimal results for the average information ratio, the average randomness ratio and the randomness ratio. We introduce a tiny example to show that there exists another type of bound that may be crucial for the information ratio, to which we only give optimal results in three cases.

cs.IT↗
arXiv2023

Global small solutions to the 3D compressible viscous non-resistive MHD system

Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is currently available for this 3D MHD system in the whole space $\mathbb R^3$ or the periodic box $\mathbb T^3$ even when the initial data is small or near a steady-state solution. This paper presents a global existence and stability result for smooth solutions to this 3D MHD system near any background magnetic field satisfying a Diophantine condition.

math.AP↗
arXiv2023

Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows

This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is a force exerted tangentially on the domain boundary through the Navier slip conditions. The control design has potential applications in many industrial processes such as rotating wall driven mixing, micromixers with acoustic waves, and artificial cilia mixing. The numerical algorithms have high complexity, high accuracy demand, and high computing expense, due to the multiscale nature of the mixing problem and the optimization requirements. A crucial problem is the computation of the G$â$teaux derivative of the cost functional. To this end, a hybrid approach based on variational formula and finite difference is built with high accuracy and efficiency to treat various types of control input functions. We have experimented with various optimization schemes including the steepest descent algorithm, the conjugate gradient method and two line search options (backtracking and exact line search). We are able to identify and implement the best combinations. The numerical simulations show that the mixing efficacy is limited when only one single type of control is applied, but can be enhanced when more diverse control types and more time segmentation are utilized. The mix-norm in the optimal mixings decays exponentially. The numerical study in this work demonstrates that boundary control alone could be an effective strategy for mixing in incompressible flows.

physics.flu-dyn↗
arXiv2023

Algebraic calming for the 2D Kuramoto-Sivashinsky equations

We propose an approximate model for the 2D Kuramoto-Sivashinsky equations (KSE) of flame fronts and crystal growth. We prove that this new ``calmed'' version of the KSE is globally well-posed, and moreover, its solutions converge to solutions of the KSE on the time interval of existence and uniqueness of the KSE at an algebraic rate. In addition, we provide simulations of the calmed KSE, illuminating its dynamics. These simulations also indicate that our analytical predictions of the convergence rates are sharp. We also discuss analogies with the 3D Navier-Stokes equations of fluid dynamics.

math.AP↗
arXiv2023

The Capacity Region of Distributed Multi-User Secret Sharing under The Perfect Privacy Condition

We study the distributed multi-user secret sharing (DMUSS) problem under the perfect privacy condition. In a DMUSS problem, multiple secret messages are deployed and the shares are offloaded to the storage nodes. Moreover, the access structure is extremely incomplete, as the decoding collection of each secret message has only one set, and by the perfect privacy condition such collection is also the colluding collection of all other secret messages. The secret message rate is defined as the size of the secret message normalized by the size of a share. We characterize the capacity region of the DMUSS problem when given an access structure, defined as the set of all achievable rate tuples. In the achievable scheme, we assume all shares are mutually independent and then design the decoding function based on the fact that the decoding collection of each secret message has only one set. Then it turns out that the perfect privacy condition is equivalent to the full rank property of some matrices consisting of different indeterminates and zeros. Such a solution does exist if the field size is bigger than the number of secret messages. Finally with a matching converse saying that the size of the secret is upper bounded by the sum of sizes of non-colluding shares, we characterize the capacity region of DMUSS problem under the perfect privacy condition.

cs.IT↗
arXiv2022

Global small solutions to a special $2\frac12$-D compressible viscous non-resistive MHD system

This paper solves the global well-posedness and stability problem on a special $2\frac12$-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in $L^p$-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion, and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution's optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.

math.AP↗
arXiv2022

Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and damps electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in $\mathbb R^3$. The velocity equation in this system is the 3D Navier-Stokes equation with dissipation only in the $x_1$-direction while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field $(0,1,0)$ is globally stable in the Sobolev setting $H^3(\mathbb R^3)$. In addition, explicit decay rates in $H^2(\mathbb R^3)$ are also obtained. When there is no presence of the magnetic field, the 3D anisotropic Navier-Stokes equation in $\mathbb R^3$ is not well understood and the small data global well-posedness remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.

math.AP↗
arXiv2022

Global solutions to 3D incompressible MHD system with dissipation in only one direction

The small data global well-posedness of the 3D incompressible Navier-Stokes equations in $\mathbb R^3$ with only one-directional dissipation remains an outstanding open problem. The dissipation in just one direction, say $\partial_1^2 u$ is simply insufficient in controlling the nonlinearity in the whole space $\mathbb R^3$. The beautiful work of Paicu and Zhang \cite{ZHANG1} solved the case when the spatial domain is bounded in the $x_1$-direction by observing a crucial Poincaré type inequality. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, this paper intends to understand the global well-posedness and stability of a special 3D magnetohydrodynamic (MHD) system near a background magnetic field. The spatial domain is $\mathbb R^3$ and the velocity in this MHD system obeys the 3D Navier-Stokes with only one-directional dissipation. With no Poincaré type inequality, this problem appears to be impossible. By discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several innovative techniques to deal with the derivative loss difficulties, we are able to bound the Navier-Stokes nonlinearity and solve the desired global well-posedness and stability problem.

math.AP↗
arXiv2022

Global well-posedness for 2D non-resistive compressible MHD system in periodic domain

This paper focuses on the 2D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion in a periodic domain. We present a systematic approach to establishing the global existence of smooth solutions when the initial data is close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. When there is no magnetic diffusion, the magnetic field and the density are governed by forced transport equations and the problem considered here is difficult. This paper implements several key observations and ideas to maximize the enhanced dissipation due to hidden structures and interactions. In particular, the weak smoothing and stabilization generated by the background magnetic field and the extra regularization in the divergence part of the velocity field are fully exploited. Compared with the previous works, this paper appears to be the first to investigate such system on bounded domains and the first to solve this problem by pure energy estimates, which help reduce the complexity in other approaches. In addition, this paper combines the well-posedness with the precise large-time behavior, a strategy that can be extended to higher dimensions.

math.AP↗
arXiv2021

Zero-Shot Instance Segmentation

Deep learning has significantly improved the precision of instance segmentation with abundant labeled data. However, in many areas like medical and manufacturing, collecting sufficient data is extremely hard and labeling this data requires high professional skills. We follow this motivation and propose a new task set named zero-shot instance segmentation (ZSI). In the training phase of ZSI, the model is trained with seen data, while in the testing phase, it is used to segment all seen and unseen instances. We first formulate the ZSI task and propose a method to tackle the challenge, which consists of Zero-shot Detector, Semantic Mask Head, Background Aware RPN and Synchronized Background Strategy. We present a new benchmark for zero-shot instance segmentation based on the MS-COCO dataset. The extensive empirical results in this benchmark show that our method not only surpasses the state-of-the-art results in zero-shot object detection task but also achieves promising performance on ZSI. Our approach will serve as a solid baseline and facilitate future research in zero-shot instance segmentation.

cs.CV↗
arXiv2021

Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is $\mathbb R^2$, the stability problem in a Sobolev setting remains open. When the spatial domain is $\mathbb T\times \mathbb R$, this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $θ$ into the horizontal average $(\bar u, \barθ)$ and the corresponding oscillation $(\widetilde u, \widetilde θ)$, and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space $H^2$. In addition, we prove that the oscillation $(\widetilde u, \widetilde θ)$ decays exponentially to zero in $H^1$ and $(u, θ)$ converges to $(\bar u, \barθ)$. This result reflects the stratification phenomenon of buoyancy-driven fluids.

math.AP↗
arXiv2020

AinnoSeg: Panoramic Segmentation with High Perfomance

Panoramic segmentation is a scene where image segmentation tasks is more difficult. With the development of CNN networks, panoramic segmentation tasks have been sufficiently developed.However, the current panoramic segmentation algorithms are more concerned with context semantics, but the details of image are not processed enough. Moreover, they cannot solve the problems which contains the accuracy of occluded object segmentation,little object segmentation,boundary pixel in object segmentation etc. Aiming to address these issues, this paper presents some useful tricks. (a) By changing the basic segmentation model, the model can take into account the large objects and the boundary pixel classification of image details. (b) Modify the loss function so that it can take into account the boundary pixels of multiple objects in the image. (c) Use a semi-supervised approach to regain control of the training process. (d) Using multi-scale training and reasoning. All these operations named AinnoSeg, AinnoSeg can achieve state-of-art performance on the well-known dataset ADE20K.

cs.CV↗
arXiv2020

The stabilizing effect of the temperature on buoyancy-driven fluids

The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for the linearized system. The results presented in this paper reveal a remarkable phenomenon for buoyancy driven fluids. That is, the temperature actually smooths and stabilizes the fluids. If the temperature were not present, the fluid is governed by the 2D Navier-Stokes with only vertical dissipation and its stability remains open. It is the coupling and interaction between the temperature and the velocity in the Boussinesq system that makes the stability problem studied here possible. Mathematically the system can be reduced to degenerate and damped wave equations that fuel the stabilization.

math.AP↗
arXiv2020

Stability of Couette flow for 2D Boussinesq system with vertical dissipation

This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator $y\partial_x -ν\partial_{yy}$ in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow $(y, 0)$ is no more than the viscosity to a suitable power (in the Sobolev space $H^b$ with $b>\frac43$), we prove that the solution of the 2D Boussnesq system with only vertical dissipation on $\mathbb T\times \mathbb R$ remains close to the Couette at the same order. A special consequence of this result is the stability of the Couette for the 2D Navier-Stokes equations with only vertical dissipation.

math.AP↗
arXiv2020

Global regularity of the three-dimensional fractional micropolar equations

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Δ)^αu$ and $(-Δ)^βw$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $α\geq \frac{5}{4}$, $β\geq 0$ and $α+β\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.

math.AP↗