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In this paper, we study the existence of strong solutions to the two-phase magnetohydrodynamic equations in a bounded domain $\Omega\subseteq \mathbb{R}^3$. The fluids are incompressible, viscous, and resistive. The surface tension is considered. The equations are reformulated using the Hanzawa transformation, which turns the free interface into a fixed one for a short time. The study of the new equations is then divided into the principal part and the nonlinear part. Due to the effect of the magnetic field and the complexity of the transformation in generic bounded domains, the Fr\'echet derivatives of nonlinearities have to be carefully estimated. The equations can then be solved using the fixed-point argument by finding a contraction mapping, which follows the estimates of the nonlinear part., Comment: 51 pages, 2 figures

In this paper, we are concerned with the two-dimensional steady supersonic combustion flows with a contact discontinuity moving through a nozzle of finite length. Mathematically, it can be formulated as a free boundary value problem governed by the two -dimensional steady combustion Euler equations with a contact discontinuity as the free boundary. The main mathematical difficulties are that the contact discontinuity is a characteristic free boundary and the equations for all states are coupled with each other due to the combustion process. We first employ the Lagrangian coordinate transformation to fix the free boundary. Then by introducing the flow slope and Bernoulli function, we further reduce the fixed boundary value problem into an initial boundary value problem for a first order hyperbolic system coupled with several ordinary differential equations. A new iteration scheme is developed near the background states by employing the intrinsic structure of the equation for the mass fraction of the non-combustion gas. We show that there is a fixed point for the iteration by deriving some novel $C^{1,\alpha}$-estimates of the solutions and applying the fixed point theorem, and then the uniqueness of the fixed point is proved by a contraction argument. On the other hand, a quasi-one-dimensional approximate system is often used to simplify the two-dimensional steady supersonic combustion model. The error between these two systems is estimated. Finally, given a piece-wise $C^{1,\alpha}$-solution containing a contact discontinuity with piece-wise constant states on the entrance of the nozzle, we can show that the solution is the piece-wise constant states with a straight contact discontinuity.

We consider the compressible Poisson-Nernst-Planck-Navier-Stokes (PNPNS) system of equations, governing the transport of charged particles under the influence of the self-consistent electrostatic potential, in a three-dimensional bounded domain. We prove the existence of global weak solutions for the initial-boundary value problem with no-slip boundary condition for the fluid's velocity, blocking boundary condition for the ionic concentrations and inhomogeneous Robin boundary condition for the electrostatic potential, without restrictions on the size of the initial data. We derive the crucial energy dissipation of the system and prove the weak sequential stability of solutions of the Poisson-Nernst-Planck subsystem with respect to the velocity field of the fluid, which enables the proof of the existence of global weak solutions for the PNPNS system. We also study the large-time behavior of the solutions and justify the incompressible limit of the compressible PNPNS system as applications of the weak sequential stability of the solutions. New techniques and estimates are developed to overcome the difficulties from the strong interaction of the fluid with the ion particles and the physical boundary conditions.

In recent years, new flexible functional materials have attracted increasing interest, but there is a lack of the designing mechanisms of flexibility design with superstructures. In traditional engineering mechanics, the maximum bending strain (MBS) was considered universal for describing the bendable properties of a given material, leading to the universal designing method of lowering the dimension such as thin membranes designed flexible functional materials.In this work, the MBS was found only applicable for materials with uniformly distributed Poisson's ratio, while the MBS increases with the thickness of the given material in case there is a variation Poisson's ratio in different areas. This means the MBS can be enhanced by certain Poisson's ratio design in the future to achieve better flexibility of thick materials. Here, the inorganic freestanding nanofiber membranes, which have a nonconstant Poisson's ratio response on stress/strain for creating nonuniformly distributed Poisson's ratio were proven applicable for designing larger MBS and lower Young's modulus for thicker samples.

In this paper, we study the smooth isometric immersion of a complete, simply connected surface with a negative Gauss curvature into the three-dimensional Euclidean space. A fundamental and longstanding problem is to find a sufficient condition for a complete negatively curved surface to be isometrically embedded in R^3 [67]. It can be described as an initial and/or boundary value problem for a hyperbolic system of nonlinear partial differential equations derived from the Gauss-Codazzi equations. The mathematical theory associated with this system is largely incomplete. The global smooth isometric immersion has been proven in the literature when the Gauss curvature decays rapidly and monotonically. However, when the Gauss curvature oscillates or decays slowly, the problem becomes much more challenging and little is known. In our paper, we find a sufficient condition, consisting of a finite total Gauss curvature and appropriate oscillations of the Gauss curvature. Under this condition we prove the global existence of a smooth solution to the Gauss-Codazzi system, achieving a global smooth isometric immersion of the surface into R^3. Furthermore, we show that the finite total Gauss curvature is necessary for the existence of a solution in a special case of the Gauss-Codazzi system. New techniques are developed to overcome the difficulties posed by the slow decay and oscillations of the Gauss curvature. By observing that certain combinations of the Riemann invariants decay faster than others, we reformulate the Gauss-Codazzi equations as a symmetric hyperbolic system and uncover a crucial structure of partial dampings. These partial dampings, along with the finite total curvature and appropriate oscillations of the Gauss curvature, enable us to obtain a global smooth solution through delicate analysis, and consequently establish a global smooth isometric immersion of such surfaces., Comment: 53 pages, 1 figure; title and abstract revised, some necessary words added

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, these results also provide two different kinds of suitable boundary conditions for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the vanishing viscosity limit method. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper.

Recently, CHIME/FRB project published its first fast radio burst (FRB) catalog (hereafter, Catalog 1), which totally contains 536 unique bursts. With the help of the latest set of FRBs in this large-size catalog, we aim to investigate the dispersion measure (DM) or redshift ($z$) distribution of the FRB population, and solution of this problem could be used to clarify the question of FRB origin. In this study, we adopted the M\&E 2018 model, to fit the observed $z$ distribution of FRBs in Catalog 1. In the M\&E 2018 model, we are mostly interested in the $\Phi(z)$ function, i.e., number of bursts per proper time per comoving volume, which is represented by the star formation rate (SFR) with a power-law index $n$. Our estimated value of $n$ is $0.0_{-0.0}^{+0.6}$ ($0.0_{-0.0}^{+2.1}$) at the 68 (95) per cent confidence level, implying that the FRB population evolves with redshift consistent with, or faster than, the SFR. Specially, the consistency of the $n$ values estimated by this study and the SFR provides a potential support for the hypothesis of FRBs originating from young magnetars., Comment: 7 pages, 2 figures, accepted for publication in Astronomy Reports

The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the equation of the active particles. The existence of weak solutions to the stationary problem is established through a two-level approximation scheme, compactness estimates and weak convergence arguments. Novel techniques are developed to overcome the difficulties due to the lower regularity of stationary solutions, a Moser-type iteration is used to deal with the strong coupling of active particles and fluids, and some weighted estimates on the energy functions are achieved so that the weak solutions can be constructed for all values of the adiabatic exponent $\gamma>1$.

Convolutional neural networks (CNNs) have been successfully applied to the single target tracking task in recent years. Generally, training a deep CNN model requires numerous labeled training samples, and the number and quality of these samples directly affect the representational capability of the trained model. However, this approach is restrictive in practice, because manually labeling such a large number of training samples is time-consuming and prohibitively expensive. In this paper, we propose an active learning method for deep visual tracking, which selects and annotates the unlabeled samples to train the deep CNNs model. Under the guidance of active learning, the tracker based on the trained deep CNNs model can achieve competitive tracking performance while reducing the labeling cost. More specifically, to ensure the diversity of selected samples, we propose an active learning method based on multi-frame collaboration to select those training samples that should be and need to be annotated. Meanwhile, considering the representativeness of these selected samples, we adopt a nearest neighbor discrimination method based on the average nearest neighbor distance to screen isolated samples and low-quality samples. Therefore, the training samples subset selected based on our method requires only a given budget to maintain the diversity and representativeness of the entire sample set. Furthermore, we adopt a Tversky loss to improve the bounding box estimation of our tracker, which can ensure that the tracker achieves more accurate target states. Extensive experimental results confirm that our active learning-based tracker (ALT) achieves competitive tracking accuracy and speed compared with state-of-the-art trackers on the seven most challenging evaluation benchmarks., Comment: 12 pages

The inviscid limit for the two-dimensional compressible viscoelastic equations on the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. However, when the deformation tensor effect is taken into account, our results show that the deformation tensor plays an important role in the vanishing viscosity process and can prevent the formation of strong boundary layers. As a result we are able to justify the inviscid limit of solutions for the compressible viscous flows under the no-slip boundary condition governed by the viscoelastic equations, based on the uniform conormal regularity estimates achieved in this paper.

We are concerned with the Cahn-Hilliard/Navier-Stokes equations for the stationary compressible flows in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations describing the compressible fluid flows and the stationary Cahn-Hilliard type diffuse equation for the mass concentration difference. We prove the existence of weak solutions when the adiabatic exponent $\gamma$ satisfies $\gamma>\frac{4}{3}$. The proof is based on the weighted total energy estimates and the new techniques developed to overcome the difficulties from the capillary stress., Comment: 22pages, 1 figure

We study the vanishing dissipation limit of the three-dimensional (3D) compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler equations. Our results are twofold. First, we prove that the 3D compressible Navier-Stokes-Fourier equations admit a family of smooth solutions that converge to the planar rarefaction wave solution of the 3D compressible Euler equations with arbitrary strength. Second, we obtain a uniform convergence rate in terms of the viscosity and heat-conductivity coefficients. For this multi-dimensional problem, we first need to introduce the hyperbolic wave to recover the physical dissipations of the inviscid rarefaction wave profile as in our previous work [29] on the two-dimensional (2D) case. However, due to the 3D setting that makes the analysis significantly more challenging than the 2D problem, the hyperbolic scaled variables for the space and time could not be used to normalize the dissipation coefficients as in the 2D case. Instead, the analysis of the 3D case is carried out in the original non-scaled variables, and consequently the dissipation terms are more singular compared with the 2D scaled case. Novel ideas and techniques are developed to establish the uniform estimates. In particular, more accurate {\it a priori} assumptions with respect to the dissipation coefficients are crucially needed for the stability analysis, and some new observations on the cancellations of the physical structures for the flux terms are essentially used to justify the 3D limit. Moreover, we find that the decay rate with respect to the dissipation coefficients is determined by the nonlinear flux terms in the original variables for the 3D limit in this paper, but fully determined by the error terms in the scaled variables for the 2D case in [29].

We consider the stability of transonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle. This is the first work on the mixed-type problem of transonic flows across a contact discontinuity as a free boundary in nozzles. We start with the Euler-Lagrangian transformation to straighten the contact discontinuity in the new coordinates. However, the upper nozzle wall in the subsonic region depending on the mass flux becomes a free boundary after the transformation. Then we develop new ideas and techniques to solve the free-boundary problem in three steps: (1) we fix the free boundary and generate a new iteration scheme to solve the corresponding fixed boundary value problem of the hyperbolic-elliptic mixed type by building some powerful estimates for both the first-order hyperbolic equation and a second-order nonlinear elliptic equation in a Lipschitz domain; (2) we update the new free boundary by constructing a mapping that has a fixed point; (3) we establish via the inverse Lagrangian coordinates transformation that the original free interface problem admits a unique piecewise smooth transonic solution near the background state, which consists of a smooth subsonic flow and a smooth supersonic flow with a contact discontinuity.

We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero., Comment: 47 pages, Accepted by Comm. Math. Sci. 2020

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations for compressible fluids and a stationary Cahn-Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent., Comment: 39 pages, To appear in Mathematical Models and Methods in Applied Sciences

We consider a vanishing viscosity sequence of weak solutions of the three-dimensional Navier--Stokes equations on a bounded domain. In a seminal paper [25] Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato's criterion holds for H\"older continuous solutions with the regularity index arbitrarily close to the Onsager's critical exponent through a new boundary layer foliation and a global mollification., Comment: 20 pages, 2 figures

In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point., Comment: 31 pages