In this paper we study the validity of the so-called Oberbeck-Boussinesq approximation for compressible viscous perfect gases in the whole three-dimensional space. Both the cases of uids with positive heat conductivity and zero conductivity are considered. For small perturbations of a constant equilibrium, we establish the global existence of unique strong solutions in a critical regularity functional framework. Next, taking advantage of Strichartz estimates for the associated system of acoustic waves, and of uniform estimates with respect to the Mach number, we obtain all-time convergence to the Boussinesq system with a explicit decay rate.
There is an important literature dedicated to the limit system, that is the Oberbeck-Boussinesq equations , and , under various hypotheses over the coefficients @math and @math and the potential @math (although the most common assumption is that @math ). Loosely speaking the classical results concerning the existence issue are (see e.g. @cite_17 @cite_5 @cite_10 and the references therein): Dimension @math : Global existence of strong solutions if @math Dimension @math with @math : Global weak solutions and local strong solutions (which become global if the data are small). Dimension @math with @math : only local-in-time strong solutions are available.
@cite_5 @cite_10 @cite_17
2014137435 2094712512
This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in @math space hold true for moderately soft potential cases @math without any smallness assumption on the initial data. For very soft potential cases @math , which cover in particular the Coulomb case @math , we get local in time estimates of weak solutions in @math . In the proofs of these estimates, global ones for the special case @math and local ones for very soft potential cases @math , the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality. We seek a global minimum of @math . The solution to @math will find local minima. Using the idea of simulated annealing, we consider the diffusion process, @math , @math , where @math is the n-dimensional standard Brownian motion and @math is the annealing rate which decreases to zero as t goes to @math . Under suitable condition on @math , we prove that @math converges weakly to a probability measure @math if for large t, @math with @math , where @math has a simple expression involving the action function of the dynamical system @math , @math concentrates on the global minima of U and is the weak limit of the Gibbs densities @math .The above result can also be formulated as follows: consider the Fokker–Planck equation (forward equation) [ t V(t,y) = ... Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as @math , present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for @math as target. This monograph establishes that for @math as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations. We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math . Learning a classifier from private data collected by multiple parties is an important problem that has many potential applications. How can we build an accurate and differentially private global classifier by combining locally-trained classifiers from different parties, without access to any party's private data? We propose to transfer the knowledge' of the local classifier ensemble by first creating labeled data from auxiliary unlabeled data, and then train a global @math -differentially private classifier. We show that majority voting is too sensitive and therefore propose a new risk weighted by class probabilities estimated from the ensemble. Relative to a non-private solution, our private solution has a generalization error bounded by @math where @math is the number of parties. This allows strong privacy without performance loss when @math is large, such as in crowdsensing applications. We demonstrate the performance of our method with realistic tasks of activity recognition, network intrusion detection, and malicious URL detection. We investigate the global fluctuations of solutions to elliptic equations with random coefficients in the discrete setting. In dimension @math and for i.i.d. coefficients, we show that after a suitable scaling, these fluctuations converge to a Gaussian field that locally resembles a (generalized) Gaussian free field. The paper begins with a heuristic derivation of the result, which can be read independently and was obtained jointly with Scott Armstrong. We consider the perturbation of elliptic pseudodifferential operators @math with more than square integrable Green's functions by random, rapidly varying, sufficiently mixing, potentials of the form @math . We analyze the source and spectral problems associated with such operators and show that the rescaled difference between the perturbed and unperturbed solutions may be written asymptotically as @math as explicit Gaussian processes. Such results may be seen as central limit corrections to homogenization (law of large numbers). Similar results are derived for more general elliptic equations with random coefficients in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions. Let @math be @math -variate polynomials with rational coefficients of maximum degree @math and let @math be the set of common complex solutions of @math . We give an algorithm which, up to some regularity assumptions on @math , computes an exact representation of the global infimum @math of the restriction of the map @math to @math , i.e., a univariate polynomial vanishing at @math and an isolating interval for @math . Furthermore, it decides whether @math is reached, and if so, it returns @math such that @math . This algorithm is probabilistic. It makes use of the notion of polar varieties. Its complexity is essentially cubic in @math and linear in the complexity of evaluating the input. This fits within the best known deterministic complexity class @math . We report on some practical experiments of a first implementation that is available as a Maple package. It appears that it ca... This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear impulsive differential equation on time scale. The reasonable weak solution of nonlinear impulsive differential equation on time scale is introduced and the existence and uniqueness of the weak solution and its properties are presented. By @math strong @math weak lower semicontinuity of integral functional on time scale, we give the existence of optimal controls. Using integration by parts formula on time scale, the necessary conditions of optimality are derived. An example on mathematical programming is also presented for demonstration. The time-dependent Boltzmann equation, which describes the propagation of radiation from a point source in a random medium, is solved exactly in Fourier space. An explicit expression in real space is given in two and four dimensions. In three dimensions an accurate interpolation formula is found. The average intensity at a large distance @math from the source has two peaks, a ballistic peak at time @math and a diffusion peak at @math (with @math the velocity and @math the diffusion coefficient). We find that forward scattering adds a tail to the ballistic peak in two and three dimensions, @math and @math , respectively. Expressions in the literature do not contain this tail. In the present work, we consider the well-posedness and asymptotics of grazing collisions limit of the spatially inhomogeneous Boltzmann equation with Coulomb interaction. Under the screening hypothesis on the cross-section @math of the Boltzmann collision operator, that is, @math , where @math , we prove that there exists a common lifespan @math such that for any @math , the Boltzmann equation admits a unique nonnegative solution @math in the function space @math when the initial data belongs to the weighted Sobolev space @math with @math and @math . Moreover, it is proved that the solution @math is uniformly bounded with respect to the parameter @math in the above function spaces. As a consequence, by using the ... Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity. This paper concerns linear first-order hyperbolic systems in one space dimension of the type @math with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients @math and @math and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on @math ), which implies @math -solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity. Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding, and representable discrete polymatroids. We consider the vector linear solutions of networks over a field @math , with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. It is well known that a scalar linear solution over @math exists for a network if and only if the network is matroidal with respect to a matroid representable over @math . We define a discrete polymatroidal network and show that a linear fractional solution over a field @math exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over @math . An algorithm to construct the networks starting from certain class of discrete polymatroids is provided. Every representation over @math for the discrete polymatroid, results in a linear fractional solution over @math for the constructed network. Next, we consider the index coding problem, which involves a sender, which generates a set of messages @math , and a set of receivers @math , which demand messages. A receiver @math is specified by the tuple @math , where @math is the message demanded by @math and @math is the side information possessed by @math . We first show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid, satisfying certain conditions, which are determined by the index coding problem considered. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. The multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of , by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid. This is the second article of a series of papers on stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the @math -dimensional lattice @math with random coefficients @math of the simplest type: They are identically distributed and independent from edge to edge. On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ''homogenized'' matrix @math is characterized by @math for any direction @math , where the random field @math (the ''corrector'') is the unique solution of @math in @math such that @math , @math is stationary and @math , @math denoting the ensemble average (or expectation). In order to approximate the homogenized coefficients @math , the corrector problem is usually solved in a box @math of size @math with periodic boundary conditions, and the space averaged energy on @math defines an approximation @math of @math . Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation @math converges almost surely to @math as @math . In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size @math , but replace the elliptic operator by @math with (typically) @math , as standard in the homogenization literature. We then replace the ensemble average by a space average on @math , and estimate the overall error on the homogenized coefficients in terms of @math and @math . The @math -Means clustering problem on @math points is NP-Hard for any dimension @math , however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an @math time dynamic programming algorithm that uses @math space. It turns out that the problem has been considered under a different name more than twenty years ago. We present all the existing work that had been overlooked and compare the various solutions theoretically. Moreover, we show how to reduce the space usage for some of them, as well as generalize them to data structures that can quickly report an optimal @math -Means clustering for any @math . Finally we also generalize all the algorithms to work for the absolute distance and to work for any Bregman Divergence. We complement our theoretical contributions by experiments that compare the practical performance of the various algorithms. Given a set of objects @math , each with @math numeric attributes, a top- @math preference scores these objects using a linear combination of their attribute values, where the weight on each attribute reflects the interest in this attribute. Given a query preference @math , a top- @math query finds the @math objects in @math with highest scores with respect to @math . Given a query object @math and a set of preferences @math , a reverse top- @math query finds all preferences @math for which @math becomes one of the top @math objects with respect to @math . Previous solutions to these problems are effective only in low dimensions. In this paper, we develop a solution for much higher dimensions (up to high tens), if many preferences exhibit sparsity —i.e., each specifies non-zero weights for only a handful (say @math - @math ) of attributes (though the subsets of such attributes and their weights can vary greatly). Our idea is to select carefully a set of low-dimensional core subspaces to “cover” the sparse preferences in a workload. These subspaces allow us to index them more effectively than the full-dimensional space. Being multi-dimensional, each subspace covers many possible preferences; furthermore, multiple subspaces can jointly cover a preference, thereby expanding the coverage beyond each subspace’s dimensionality. Experimental evaluation validates our solution’s effectiveness and advantages over previous solutions. We consider @math -equivariant wave maps from @math of finite energy. We establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map @math . In particular, for each global energy solution of topological degree one, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with a companion article (Part I), where we consider the case of finite-time blow up, this gives a characterization of all @math -equivariant, degree @math wave maps in the energy regime @math . The traveling salesman problem (TSP) is a canonical NP-complete problem which is proved by Trevisan [SIAM J. Comput., 30 (2000), pp. 475--485] to be MAX-SNP hard even on high-dimensional Euclidean metrics. To circumvent this hardness, researchers have been developing approximation schemes for „simpler” instances of the problem. For instance, the algorithms of Arora and of Talwar show how to approximate TSP on low-dimensional metrics (for different notions of metric dimension). However, a feature of most current notions of metric dimension is that they are „local”: the definitions require every local neighborhood to be well-behaved. In this paper, we define a global notion of dimension that generalizes the popular notion of doubling dimension, but still allows some small dense regions; e.g., it allows some metrics that contain cliques of size @math . Given a metric with global dimension @math , we give a @math -approximation algorithm that runs in subexponential time, i.e., in $ (O(n^... This paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion. The computational model is such that each node of the graph is occupied by a processor which has its own ID. Processors are restricted to collecting data from others which are at a distance at most t away from them in t time units, but are otherwise computationally unbounded. This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data.Three results are proved within this model: • A 3-coloring of an n-cycle requires time @math . This bound is tight, by previous work of Cole and Vishkin. • Any algorithm for coloring the d-regular tree of radius r which runs for time at most @math requires at least @math colors. • In an n-vertex graph of largest degree @math , an @math -coloring may be found in time @math .