arXiv subjects
Publications and source records attributed to Zhifei Zhang.
In this paper, we prove the global well-posedness of the incompressible MHD equations near a homogeneous equilibrium in the domain $R^k\times T^{d-k}, d\geq2,k\geq1$ by using the comparison principle and constructing the comparison function.
We consider the compressible Oldroyd-B model derived in \cite{Barrett-Lu-Suli}, where the existence of global-in-time finite energy weak solutions was shown in two dimensional setting. In this paper, we first state a local well-posedness result for this compressible Oldroyd-B model. In two dimensional setting, we give a (refined) blow-up criterion involving only the upper bound of the fluid density. We then show that, if the initial fluid density and polymer number density admit a positive lower bound, the weak solution coincides with the strong one as long as the latter exists. Moreover, if the fluid density of a weak solution issued from regular initial data admits a finite upper bound, this weak solution is indeed a strong one; this can be seen as a corollary of the refined blow-up criterion and the weak-strong uniqueness.
In this paper, we prove the local well-posedness of plasma-vacuum interface problem for ideal incompressible magnetohydrodynamics under the stability condition: the magnetic field $\mathbf{h}$ and the vacuum magnetic field $\hat{\mathbf{h}}$ are non-collinear on the interface(i.e., $|\mathbf{h}\times \hat{\mathbf{h}}|>0$), which was introduced by Trakhinin as a stability condition for the compressible plasma-vacuum interface problem.
In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.
"If I provide you a face image of mine (without telling you the actual age when I took the picture) and a large amount of face images that I crawled (containing labeled faces of different ages but not necessarily paired), can you show me what I would look like when I am 80 or what I was like when I was 5?" The answer is probably a "No." Most existing face aging works attempt to learn the transformation between age groups and thus would require the paired samples as well as the labeled query image. In this paper, we look at the problem from a generative modeling perspective such that no paired samples is required. In addition, given an unlabeled image, the generative model can directly produce the image with desired age attribute. We propose a conditional adversarial autoencoder (CAAE) that learns a face manifold, traversing on which smooth age progression and regression can be realized simultaneously. In CAAE, the face is first mapped to a latent vector through a convolutional encoder, and then the vector is projected to the face manifold conditional on age through a deconvolutional generator. The latent vector preserves personalized face features (i.e., personality) and the age condition controls progression vs. regression. Two adversarial networks are imposed on the encoder and generator, respectively, forcing to generate more photo-realistic faces. Experimental results demonstrate the appealing performance and flexibility of the proposed framework by comparing with the state-of-the-art and ground truth.
In this paper, we solve Gallay's conjecture on the spectral lower bound and pseudospecrtal bound for the linearized operator of the Navier-Stokes equation in $R^2$ around rapidly rotating Oseen vortices.
In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class $2-θ$ for any $θ>0$. This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.
The staggering amount of streaming time series coming from the real world calls for more efficient and effective online modeling solution. For time series modeling, most existing works make some unrealistic assumptions such as the input data is of fixed length or well aligned, which requires extra effort on segmentation or normalization of the raw streaming data. Although some literature claim their approaches to be invariant to data length and misalignment, they are too time-consuming to model a streaming time series in an online manner. We propose a novel and more practical online modeling and classification scheme, DDE-MGM, which does not make any assumptions on the time series while maintaining high efficiency and state-of-the-art performance. The derivative delay embedding (DDE) is developed to incrementally transform time series to the embedding space, where the intrinsic characteristics of data is preserved as recursive patterns regardless of the stream length and misalignment. Then, a non-parametric Markov geographic model (MGM) is proposed to both model and classify the pattern in an online manner. Experimental results demonstrate the effectiveness and superior classification accuracy of the proposed DDE-MGM in an online setting as compared to the state-of-the-art.
We consider the zero viscosity limit of the incompressible Navier-Stokes equations with non-slip boundary condition in the half-space for the initial vorticity located away from the boundary. By using the vorticity formulation and Cauchy-Kowaleskaya theorem, Maekawa proved the local in time convergence of the Navier-Stokes equations in the half- plane to the Euler equations outside a boundary layer and to the Prandtl equations in the boundary layer. In this paper, we develop the direct energy method to generalize Maekawa's result to the half-space.
In this paper, we study the MHD equations with small viscosity and resistivity coefficients, which may be different. This is a typical setting in high temperature plasmas. It was proved that the MHD equations are globally well-posed if the initial velocity is close to 0 and the initial magnetic field is close to a homogeneous magnetic field in the weighted Hölder space, where the closeness is independent of the dissipation coefficients.
This paper focuses on the stability analysis of WKB approximate solutions in geometric optics with the absence of strong transparency conditions. We introduce a compatible condition and a singular localization method which allows us to prove the stability of WKB solutions over long time intervals. This compatible condition is weaker than the strong transparency condition. The singular localization method allows us to do delicate analysis near resonances. As an application, we show the long time approximation of Klein-Gordon equations by Schrödinger equations in the non-relativistic limit regime.
This paper applies the recently developed framework for integral control on nonlinear spaces to two non-standard cases. First, we show that the property of perfect target stabilization in presence of actuation bias holds also if this bias is state dependent. This might not be surprising, but for practical purposes it provides an easy way to robustly cancel nonlinear dynamics of the uncontrolled plant. We specifically illustrate this for robust stabilization of a pendulum at arbitrary angle, a problem posed as non-trivial by some colleagues. Second, as previous work has been restricted to systems with as many control inputs as configuration dimensions, we here provide results for integral control of a non-holonomic system. More precisely, we design robust steering control of a rigid body under velocity bias.
Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^{p}(M)}\big)\leq Λ\quad \text{for some}\,\,p>1, \eeno where $τ(u_n)$ is the tension field of $u_n$, then there hold the so called energy identity and neckless property during blowing up. This result is sharp by Parker's example, where the tension fields of the mappings from Riemannian surface are bounded in $L^1(M)$ but the energy identity fails.
In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.
Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3<p,q<\infty$ and $u_h\in L^{\infty}(0,T; BMO^{-1}(\R^3))$ with $u_h(T)\in VMO^{-1}(\R^3)$, then $u$ can be extended beyond $T$. This result generalizes the recent result proved by Gallagher, Koch and Planchon, which requires $u\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$. Our proof is based on a new interior regularity criterion in terms of one velocity component.
In this paper, we solve a long-standing open problem: nonlinear stability of current-vortex sheet in the ideal incompressible Magneto-Hydrodynamics under the linear stability condition. This result gives a first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instability.
In this paper, we prove the decay estimates of the velocity and $H^1$ scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960.
Let us consider an initial data $v_0$ for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any $p$ in $]4,\infty[$, and any unit vector $e$ of $\R^3$, the $L^p$ norm in time with value in $\dot{H}^{\frac 12+\frac 2 p }$ of $(v|e)_{\R^3}$ blows up at $T^\star$