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## Bounded Domain (bounded + domain)
## Selected Abstracts## Numerical evaluation of the damping-solvent extraction method in the frequency domain EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 6 2002Ushnish BasuAbstract The damping-solvent extraction method for the analysis of unbounded visco-elastic media is evaluated numerically in the frequency domain in order to investigate the influence of the computational parameters,domain size, amount of artificial damping, and mesh density,on the accuracy of results. An analytical estimate of this influence is presented, and specific questions regarding the influence of the parameters on the results are answered using the analytical estimate and numerical results for two classical problems: the rigid strip and rigid disc footings on a visco-elastic half-space with constant hysteretic material damping. As the domain size is increased, the results become more accurate only at lower frequencies, but are essentially unaffected at higher frequencies. Choosing the domain size to ensure that the static stiffness is computed accurately leads to an unnecessarily large domain for analysis at higher frequencies. The results improve by increasing artificial damping but at a slower rate as the total (material plus artificial) damping ratio ,t gets closer to 0.866. However, the results do not deteriorate significantly for the larger amounts of artificial damping, suggesting that ,t,0.6 is appropriate; a larger value is not likely to influence the accuracy of results. Presented results do not support the earlier suggestion that similar accuracy can be achieved by a large bounded domain with small damping or by a small domain with larger damping. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## RangeModel: tools for exploring and assessing geometric constraints on species richness (the mid-domain effect) along transects ECOGRAPHY, Issue 1 2008Robert K. ColwellArticle first published online: 4 FEB 200RangeModel is a computer application that offers animated demonstrations of the mechanism behind the mid-domain effect. The program also provides analytical tools for the assessment of geometric constraints in empirical datasets for one-dimensional domains (transects). The mid-domain effect (MDE) is the increasing overlap of species ranges towards the center of a shared, bounded domain due to geometric boundary constraints in relation to the distribution of range sizes, producing a peak or plateau of species richness towards the center of the domain. Domains may be spatial, temporal, or functional. RangeModel is a stand-alone, graphical-interface, freeware application for PC and Mac OS platforms. [source] ## Challenges in the application of geometric constraint models GLOBAL ECOLOGY, Issue 3 2007Craig R. McClainABSTRACT Discerning the processes influencing geographical patterns of species richness remains one of the central goals of modern ecology. Traditional approaches to exploring these patterns have focused on environmental and ecological correlates of observed species richness. Recently, some have suggested these approaches suffer from the lack of an appropriate null model that accounts for species ranges being constrained to occur within a bounded domain. Proponents of these null geometric constraint models (GCMs), and the mid-domain effect these models produce, argue their utility in identifying meaningful gradients in species richness. This idea has generated substantial debate. Here we discuss what we believe are the three major challenges in the application of GCMs. First, we argue that there are actually two equally valid null models for the random placement of species ranges within a domain, one of which actually predicts a uniform distribution of species richness. Second, we highlight the numerous decisions that must be made to implement a GCM that lead to marked differences in the predictions of the null model. Finally, we discuss challenges in evaluating the importance of GCMs once they have been implemented. [source] ## Perfectly matched layers for transient elastodynamics of unbounded domains INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004Ushnish BasuAbstract One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337,1375], the authors presented, inter alia, time-harmonic governing equations of PMLs for anti-plane and for plane-strain motion of (visco-) elastic media. This paper presents (a) corresponding time-domain, displacement-based governing equations of these PMLs and (b) displacement-based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti-plane PML is found to be symmetric, whereas that of the plane-strain PML is not. Numerical results are presented for the anti-plane motion of a semi-infinite layer on a rigid base, and for the classical soil,structure interaction problems of a rigid strip-footing on (i) a half-plane, (ii) a layer on a half-plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Analysis of parameter sensitivity and experimental design for a class of nonlinear partial differential equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2005Michael L. AndersonAbstract The purpose of this work is to analyse the parameter sensitivity problem for a class of nonlinear elliptic partial differential equations, and to show how numerical simulations can help to optimize experiments for the estimation of parameters in such equations. As a representative example we consider the Laplace,Young problem describing the free surface between two fluids in contact with the walls of a bounded domain, with the parameters being those associated with surface tension and contact. We investigate the sensitivity of the solution and associated functionals to the parameters, examining in particular under what conditions the solution is sensitive to parameter choice. From this, the important practical question of how to optimally design experiments is discussed; i.e. how to choose the shape of the domain and the type of measurements to be performed, such that a subsequent inversion of the measured data for the model parameters yields maximal accuracy in the parameters. We investigate this through numerical studies of the behaviour of the eigenvalues of the sensitivity matrix and their relation to experimental design. These studies show that the accuracy with which parameters can be identified from given measurements can be improved significantly by numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A new stable space,time formulation for two-dimensional and three-dimensional incompressible viscous flow INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001Donatien N'driAbstract A space,time finite element method for the incompressible Navier,Stokes equations in a bounded domain in ,d (with d=2 or 3) is presented. The method is based on the time-discontinuous Galerkin method with the use of simplex-type meshes together with the requirement that the space,time finite element discretization for the velocity and the pressure satisfy the inf,sup stability condition of Brezzi and Babu,ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two-dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 16 2006Issam S. StrubAbstract This article proves the existence and uniqueness of a weak solution to a scalar conservation law on a bounded domain. A weak formulation of the boundary conditions is needed for the problem to be well posed. The existence of the solution results from the convergence of the Godunov scheme. This weak formulation is written explicitly in the context of a strictly concave flux function (relevant for highway traffic). The numerical scheme is then applied to a highway scenario with data from highway Interstate-80 obtained from the Berkeley Highway Laboratory. Finally, the existence of a minimiser of travel time is obtained, with the corresponding optimal boundary control. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## New variance expressions for systematic sampling: the filtering approach JOURNAL OF MICROSCOPY, Issue 3 2006XIMO GUAL-ARNAUSummary We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n -boxes in general, on a bounded domain in n -dimensional Euclidean space ,n, n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted , termed the filtering approach , is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation. [source] ## Non-homogeneous Navier,Stokes systems with order-parameter-dependent stresses MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010Helmut AbelsAbstract We consider the Navier,Stokes system with variable density and variable viscosity coupled to a transport equation for an order-parameter c. Moreover, an extra stress depending on c and ,c, which describes surface tension like effects, is included in the Navier,Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two-phase flow of viscous incompressible fluids. The so-called density-dependent Navier,Stokes system is also a special case of our system. We prove short-time existence of strong solution in Lq -Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Does a ,volume-filling effect' always prevent chemotactic collapse? MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2010Michael WinklerAbstract The parabolic,parabolic Keller,Segel system for chemotaxis phenomena, is considered under homogeneous Neumann boundary conditions in a smooth bounded domain ,,,n with n,2. It is proved that if ,(u)/,(u) grows faster than u2/n as u,, and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass ,,u(x, t)dx may attain arbitrarily small positive values. In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4):501,543), the results indicate how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2009Gianluca MolaAbstract We consider a conserved phase-field system on a tri-dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ,, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ,, which is coupled with a viscous Cahn,Hilliard type equation governing the order parameter ,. The latter equation contains a nonmonotone nonlinearity , and the viscosity effects are taken into account by a term ,,,,t,, for some ,,0. Rescaling the kernel k with a relaxation time ,>0, we formulate a Cauchy,Neumann problem depending on , and ,. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {,,,,} for our problem, whose basin of attraction can be extended to the whole phase,space in the viscous case (i.e. when ,>0). Moreover, we prove that the symmetric Hausdorff distance of ,,,, from a proper lifting of ,,,0 tends to 0 in an explicitly controlled way, for any fixed ,,0. In addition, the upper semicontinuity of the family of global attractors {,,,,,} as ,,0 is achieved for any fixed ,>0. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Global weak solution to the flow of liquid crystals system MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2009Fei JiangAbstract In this paper, we study a simplified system for the flow of nematic liquid crystals in a bounded domain in the three-dimensional space. We derive the basic energy law which enables us to prove the global existence of the weak solutions under the condition that the initial density belongs to L,(,) for any . Especially, we also obtain that the weak solutions satisfy the energy inequality in integral or differential form. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009M. EfendievAbstract We consider the following doubly nonlinear parabolic equation in a bounded domain ,,,3: where the nonlinearity f is allowed to have a degeneracy with respect to ,tu of the form ,tu|,tu|p at some points x,,. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite-dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Artificial boundary conditions for viscoelastic flows MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2008Sergueï A. NazarovAbstract The steady three-dimensional exterior flow of a viscoelastic non-Newtonian fluid is approximated by reducing the corresponding nonlinear elliptic,hyperbolic system to a bounded domain. On the truncation surface with a large radius R, nonlinear, local second-order artificial boundary conditions are constructed and a new concept of an artificial transport equation is introduced. Although the asymptotic structure of solutions at infinity is known, certain attributes cannot be found explicitly so that the artificial boundary conditions must be constructed with incomplete information on asymptotics. To show the existence of a solution to the approximation problem and to estimate the asymptotic precision, a general abstract scheme, adapted to the analysis of coupled systems of elliptic,hyperbolic type, is proposed. The error estimates, obtained in weighted Sobolev norms with arbitrarily large smoothness indices, prove an approximation of order O(R,2+,), with any ,>0. Our approach, in contrast to other papers on artificial boundary conditions, does not use the standard assumptions on compactly supported right-hand side f, leads, in particular, to pointwise estimates and provides error bounds with constants independent of both R and f. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## The transient equations of viscous quantum hydrodynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2008Michael DreherAbstract We study the viscous model of quantum hydrodynamics in a bounded domain of space dimension 1, 2, or 3, and in the full one-dimensional space. This model is a mixed-order partial differential system with nonlocal and nonlinear terms for the particle density, current density, and electric potential. By a viscous regularization approach, we show existence and uniqueness of local in time solutions. We propose a reformulation as an equation of Schrödinger type, and we prove the inviscid limit. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On the Lp,Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2007Rieko ShimadaAbstract We obtain the Lp,Lq maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ,n (n,2). The Robin condition consists of two conditions: v , u=0 and ,u+,(T(u, p)v , ,T(u, p)v, v,v)=h on the boundary of the domain with ,, ,,0 and ,+,=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when ,=1 and non-slip one when ,=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## On non-stationary viscous incompressible flow through a cascade of profiles MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2006Miloslav FeistauerAbstract The paper deals with theoretical analysis of non-stationary incompressible flow through a cascade of profiles. The initial-boundary value problem for the Navier,Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004V. A. GalaktionovAbstract As a basic example, we consider the porous medium equation (m > 1) (1) where , , ,N is a bounded domain with the smooth boundary ,,, and initial data . It is well-known from the 1970s that the PME admits separable solutions , where each ,k , 0 satisfies a non-linear elliptic equation . Existence of at least a countable subset , = {,k} of such non-linear eigenfunctions follows from the Lusternik,Schnirel'man variational theory from the 1930s. The first similarity pattern t,1/(m,1),0(x), where ,0 > 0 in ,, is known to be asymptotically stable as t , , and attracts all nontrivial solutions with u0 , 0 (Aronson and Peletier, 1981). We show that if , is discrete, then it is evolutionary complete, i.e. describes the asymptotics of arbitrary global solutions of the PME. For m = 1 (the heat equation), the evolution completeness follows from the completeness-closure of the orthonormal subset , = {,k} of eigenfunctions of the Laplacian , in L2. The analysis applies to the perturbed PME and to the p -Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Interaction of elementary waves for scalar conservation laws on a bounded domain MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003Hongxia LiuAbstract This paper is concerned with the interaction of elementary waves on a bounded domain for scalar conservation laws. The structure and large time asymptotic behaviours of weak entropy solution in the sense of Bardos et al. (Comm. Partial Differential Equations 1979; 4: 1017) are clarified to the initial,boundary problem for scalar conservation laws ut+,(u)x=0 on (0,1) × (0,,), with the initial data u(x,0)=u0(x):=um and the boundary data u(0,t)=u -,u(1,t)=u+, where u±,um are constants, which are not equivalent, and ,,C2 satisfies ,,,>0, ,(0)=f,(0)=0. We also give some global estimates on derivatives of the weak entropy solution. These estimates play important roles in studying the rate of convergence for various approximation methods to scalar conservation laws. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Partial differential equations of chemotaxis and angiogenesis MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2001B. D. SleemanThe topic of this paper is concerned with an investigation of the qualitative properties of solutions to the following problem. Let ,,Rn be a bounded domain with boundary ,,. We seek solutions P,,,Rm+1 of the system (1) subject to the ,no-flux' boundary condition (2) where n denotes the inward pointing normal to ,,. To close the system we prescribe the initial conditions (3) In this system D is a constant diffusion coefficient, P is a population density and , is a vector of nutrients or chemicals whose dynamics influences the movement of P. Notice here that the substances , do not diffuse. If they do then the second equation of (1) is modified to (4) where d is a positive semi-definite diagonal matrix. This more general system includes the so-called Keller,Segel model of Biology ([1] Keller EF, Segel LA. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology 1970; 26: 339,415). To motivate our study of system (1),(3) we begin by outlining two themes. One basic to developmental biology and the other from angiogenesis. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients MATHEMATISCHE NACHRICHTEN, Issue 9 2009Vladimir KozlovAbstract We consider the Dirichlet problem for non-divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi-linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,,, , , [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the decay of the solutions of second order parabolic equations with Dirichlet conditions MATHEMATISCHE NACHRICHTEN, Issue 8 2007Brice FrankeArticle first published online: 8 MAY 200Abstract We use rearrangement techniques to investigate the decay of the parabolic Dirichlet problem in a bounded domain. The coefficients of the second order term are used to introduce an isoperimetric problem. The resulting isoperimetric function together with the divergence of the first order coefficients and the value distribution of the zero order part are then used to construct a symmetric comparison equation having a slower heat-flow than the original equation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Interior regularity criterion via pressure on weak solutions to the Navier,Stokes equations MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007Tomoyuki SuzukiAbstract Consider the nonstationary Navier,Stokes equations in , × (0, T), where , is a bounded domain in ,3. We prove interior regularity for suitable weak solutions under some condition on the pressure in the class of scaling invariance. The notion of suitable weak solutions makes it possible to obtain better information around the singularities. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Equivalence of weak Dirichlet's principle, the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functions MATHEMATISCHE NACHRICHTEN, Issue 4 2006Christian G. SimaderAbstract For boundary data , , W1,2(G ) (where G , ,N is a bounded domain) it is an easy exercise to prove the existence of weak L2 -solutions to the Dirichlet problem ",u = 0 in G, u |,G = , |,G". By means of Weyl's Lemma it is readily seen that there is , , C,(G ), ,, = 0 and , = u a.e. in G . On the contrary it seems to be a complicated task even for this simple equation to prove continuity of , up to the boundary in a suitable domain if , , W1,2(G ) , C0(). The purpose of this paper is to present an elementary proof of that fact in (classical) Dirichlet domains. Here the method of weak solutions (resp. Dirichlet's principle) is equivalent to the classical approaches (Poincaré's "sweeping-out method", Perron's method). (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·) MATHEMATISCHE NACHRICHTEN, Issue 1 2004Lars DieningAbstract We study the Riesz potentials I,f on the generalized Lebesgue spaces Lp(·)(,d), where 0 < , < d and I,f(x) , , |f(y)| |x , y|, , ddy. Under the assumptions that p locally satisfies |p(x) , p(x)| , C/(, ln |x , y|) and is constant outside some large ball, we prove that I, : Lp(·)(,d) , Lp,(·)(,d), where . If p is given only on a bounded domain , with Lipschitz boundary we show how to extend p to on ,d such that there exists a bounded linear extension operator , : W1,p(·)(,) , (,d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings Wk,p(·)(,d) ,Lp*(·)(Rd) with and W1,p(·)(,) , Lp*(·)(,) for k = 1. We show compactness of the embeddings W1,p(·)(,) , Lq(·)(,), whenever q(x) , p*(x) , , for some , > 0. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Solving a fourth-order fractional diffusion-wave equation in a bounded domain by decomposition method NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2008Hossein JafariAbstract In this article, the Adomian decomposition method has been used to obtain solutions of fourth-order fractional diffusion-wave equation defined in a bounded space domain. The fractional derivative is described in the Caputo sense. Convergence of the method has been discussed with some illustrative examples. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] ## Semilinear parabolic problem with nonstandard boundary conditions: Error estimates NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Marián SlodiAbstract We study a semilinear parabolic partial differential equation of second order in a bounded domain , , ,N, with nonstandard boundary conditions (BCs) on a part ,non of the boundary ,,. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ,non is given, and the solution along ,non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ,(t). We prove the well-posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167,191, 2003 [source] ## Ginzburg-Landau vortex dynamics driven by an applied boundary current COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 12 2010Ian TiceIn this paper we study the time-dependent Ginzburg-Landau equations on a smooth, bounded domain , , ,2, subject to an electrical current applied on the boundary. The dynamics with an applied current are nondissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit , , 0. We first consider the original time scale in which the vortices do not move and the solutions undergo a "phase relaxation." Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current. © 2010 Wiley Periodicals, Inc. [source] ## Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2009Yuri TrakhininWe study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5, 12]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity, and we briefly discuss its extension to general relativity. © 2009 Wiley Periodicals, Inc. [source] ## About smoothness of solutions of the heat equations in closed, smooth space-time domains COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2005Hongjie DongWe consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ,2 = {(x1, x2)} of class C2k. We give sufficient conditions for u to have kth -order continuous derivatives with respect to (x1, x2) in D, for integers k , 2. The equation is supplemented with C2k boundary data, and we assume that f , C2(k,1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc. [source] |