1,544 research outputs found

    On the global regularity for anisotropic dissipative surface quasi-geostrophic equation

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    In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply, to overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic type Gronwall inequality, which may be of independent interest and potential applications.Comment: To appear in Nonlinearity. We have added some more details. The new title is "On the global regularity for anisotropic dissipative surface quasi-geostrophic equation

    Global regularity of the two-dimensional regularized MHD equations

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    In this paper, we consider the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations. The main objective of this paper is to establish the global regularity of classical solutions of the magnetohydrodynamics equations with the minimal dissipation. Consequently, our results significantly improve the previous works.Comment: This is the final version published in Dynamics of Partial Differential Equation

    A note on global regularity results for 2D Boussinesq equations with fractional dissipation

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    In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as 1α<β<min{α2,3α22α26α+5,22α4α3}1-\alpha<\beta< \min\Big\{\frac{\alpha}{2},\,\, \frac{3\alpha-2}{2\alpha^{2}-6\alpha+5}, \,\,\frac{2-2\alpha}{4\alpha-3}\Big\} with 0.77963α0<α<10.77963\thickapprox\alpha_{0}<\alpha<1. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.Comment: 15 page

    Global regularity of 2D tropical climate model with zero thermal diffusion

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    This article studies the global regularity problem of the two-dimensional zero thermal diffusion tropical climate model with fractional dissipation, given by (Δ)αu(-\Delta)^{\alpha}u in the barotropic mode equation and by (Δ)βv(-\Delta)^{\beta}v in the first baroclinic mode of the vector velocity equation. More precisely, we show that the global regularity result holds true as long as α+β2\alpha+\beta\geq2 with 1<α<21<\alpha<2. In addition, with no dissipation from both the temperature and the first baroclinic mode of the vector velocity, we also establish the global regularity result with the dissipation strength at the logarithmically supercritical level. Finally, our arguments can be extended to obtain the corresponding global regularity results of the higher dimensional cases.Comment: 21 pages, submitted for publication in 201

    Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

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    As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].Comment: This version fix several typos of the previous one. 25 page

    Shortcut Matrix Product States and its applications

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    Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in mathematics, has been proposed originally for describing an (especially one-dimensional) quantum system, and recently has found applications in various applications such as compressing high-dimensional data, supervised kernel linear classifier, and unsupervised generative modeling. However, when applied to systems which are not defined on one-dimensional lattices, a serious drawback of the MPS is the exponential decay of the correlations, which limits its power in capturing long-range dependences among variables in the system. To alleviate this problem, we propose to introduce long-range interactions, which act as shortcuts, to MPS, resulting in a new model \textit{ Shortcut Matrix Product States} (SMPS). When chosen properly, the shortcuts can decrease significantly the correlation length of the MPS, while preserving the computational efficiency. We develop efficient training methods of SMPS for various tasks, establish some of their mathematical properties, and show how to find a good location to add shortcuts. Finally, using extensive numerical experiments we evaluate its performance in a variety of applications, including function fitting, partition function calculation of 22-d Ising model, and unsupervised generative modeling of handwritten digits, to illustrate its advantages over vanilla matrix product states.Comment: 15pages, 11 figure

    Global regularity for the 2D Oldroyd-B model in the corotational case

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    This paper is dedicated to the Oldroyd-B model with fractional dissipation (Δ)ατ(-\Delta)^{\alpha}\tau for any α>0\alpha>0. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. The methods described here are quite different from the tedious iterative approach used in recent paper \cite{XY}. Moreover, in the Appendix we provide some a priori estimates to the Oldroyd-B model in the critical case which may be useful and of interest for future improvement. Finally, the global regularity to to the Oldroyd-B model in the corotational case with Δu-\Delta u replaced by (Δ)γu(-\Delta)^{\gamma}u for γ>1\gamma>1 are also collected in the Appendix. Therefore our result is more closer to the resolution of the well-known global regularity issue on the critical 2D Oldroyd-B model.Comment: 23 pages, Submitted August 201

    Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum

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    The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.Comment: 31 page

    On the Differentiability issue of the drift-diffusion equation with nonlocal L\'evy-type diffusion

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    We investigate the differentiability issue of the drift-diffusion equation with nonlocal L\'evy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial H\"older regularity, we prove that the vanishing viscosity solution is differentiable with some H\"older continuous derivatives for any positive time.Comment: 24 pages. Submitte

    Stabilizing Weighted Graphs

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    An edge-weighted graph G=(V,E)G=(V,E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of GG. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from GG yields a stable graph, for any weighted graph GG. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph GG yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NPP=NP. In this setting, we develop an O(Δ)O(\Delta)-approximation algorithm for the problem, where Δ\Delta is the maximum degree of a node in GG
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