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## Parabolic Systems (parabolic + system)
## Selected Abstracts## Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2010Qionglei ChenIn this paper, we prove global well-posedness for compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium. This result allows us to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms. © 2010 Wiley Periodicals, Inc. [source] ## Boundedness and exponential stabilization in a signal transduction model with diffusion MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2008Michael WinklerAbstract The influence of diffusion in a model arising in the description of signal transduction pathways in living cells is investigated. It is proved that all solutions of the corresponding semilinear parabolic system, consisting of four equations, are global in time and bounded. Under the additional assumption that certain two of the diffusion coefficients are equal, it is furthermore demonstrated that all solutions approach a spatially homogeneous steady state as t,,,,. This equilibrium is uniquely determined by the initial data, and the rate of convergence is shown to be at least exponential. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Weak solutions of a phase-field model for phase change of an alloy with thermal properties MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2002José Luiz BoldriniThe phase-field method provides a mathematical description for free-boundary problems associated to physical processes with phase transitions. It postulates the existence of a function, called the phase-field, whose value identifies the phase at a particular point in space and time. The method is particularly suitable for cases with complex growth structures occurring during phase transitions. The mathematical model studied in this work describes the solidification process occurring in a binary alloy with temperature-dependent properties. It is based on a highly non-linear degenerate parabolic system of partial differential equations with three independent variables: phase-field, solute concentration and temperature. Existence of weak solutions for this system is obtained via the introduction of a regularized problem, followed by the derivation of suitable estimates and the application of compactness arguments. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## A second-order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Neumann boundary value conditions NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2004Ling-yun ZhangAbstract A linearized three-level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Neumann boundary value problem of a nonlinear parabolic system. It is proved that the difference scheme is uniquely solvable and second-order convergent in L, -norm. A numerical example is given. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 230,247, 2004 [source] ## A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2001Danping YangAbstract A miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection-diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L2 -norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229,249, 2001 [source] ## The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2004Mu-Tao WangLet , be a bounded C2 domain in ,n and , ,, , ,m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : , , ,m with f|,, = , and with the graph of f a minimal submanifold in ,n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if , : ¯, , ,m satisfies 8n, sup, |D2,| + ,2 sup,, |D,| < 1, then the Dirichlet problem for ,|,, is solvable in smooth maps. Here , is the diameter of ,. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with , as initial data. © 2003 Wiley Periodicals, Inc. [source] ## On-line robust parameter identification for parabolic systems INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 6 2001M. A. DemetriouAbstract In this paper a robust scheme for the adaptive parameter identification of parabolic distributed parameter systems is developed. Results from the finite-dimensional treatment of the parameter projection method and , (sigma) modifications to the standard adaptation rules are extended to infinite-dimensional systems. For the class of systems under study, modifications to these standard parameter adaptation rules were deemed necessary in order to account for the additional mathematical subtleties that arise when dealing with infinite-dimensional systems. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Resolvent estimates in W,1,p related to strongly coupled-linear parabolic systems with coupled nonsmooth capacities MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2007Annegret GlitzkyAbstract We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W,1, p spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain Lp estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Global and blow-up solutions for non-linear degenerate parabolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003Zhi-wen DuanAbstract In this paper the degenerate parabolic system ut=u(uxx+av). vt=v(vxx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (,1=,2) of solution are analysed. For , the blow-up time, blow-up rate and blow-up set of blow-up solution are estimated and the asymptotic behaviour of solution near the blow-up time is discussed by using the ,energy' method. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## A second-order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Neumann boundary value conditions NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2004Ling-yun ZhangAbstract A linearized three-level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Neumann boundary value problem of a nonlinear parabolic system. It is proved that the difference scheme is uniquely solvable and second-order convergent in L, -norm. A numerical example is given. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 230,247, 2004 [source] ## Stability of one-dimensional boundary layers by using Green's functions COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2001Emmanuel GrenierThe aim of this paper is to investigate the stability of one-dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and, more precisely, to prove that spectral stability implies linear and nonlinear stability of approximate solutions. In particular, we replace the smallness condition obtained by the energy method [10, 13] by a weaker spectral condition. © 2001 John Wiley & Sons, Inc. [source] ## The attractor for a nonlinear reaction-diffusion system in an unbounded domain COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2001Messoud A. EfendievIn this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc. [source] |