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## Weak Solutions (weak + solution)
Kinds of Weak Solutions
## Selected Abstracts## Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in Lp (p,1) MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2009Pierre-Étienne DruetAbstract Accurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L1. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and Lp right-hand side, with p,1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the Lp norm of the heat sources and to prove the existence of weak solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Weak solutions for time-dependent boundary integral equations associated with the bending of elastic plates under combined boundary data MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2004Igor ChudinovichAbstract The existence, uniqueness, stability, and integral representation of distributional solutions are investigated for the equations of motion of a thin elastic plate with a combination of displacement and moment-stress components prescribed on the boundary. Copyright © 2004 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] ## Bounds on outputs of the exact weak solution of the three-dimensional Stokes problem INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2009Zhong ChengAbstract A method for obtaining rigorous upper and lower bounds on an output of the exact weak solution of the three-dimensional Stokes problem is described. Recently bounds for the exact outputs of interest have been obtained for both the Poisson equation and the advection-diffusion-reaction equation. In this work, we extend this approach to the Stokes problem where a novel formulation of the method also leads to a simpler flux calculation based on the directly equilibrated flux method. To illustrate this technique, bounds on the flowrate are calculated for an incompressible creeping flow driven by a pressure gradient in an endless square channel with an array of rectangular obstacles in the center. Copyright © 2009 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] ## Solutions to a nonlinear Poisson,Nernst,Planck system in an ionic channel MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2010L. HadjadjAbstract A limiting one-dimensional Poisson,Nernst,Planck (PNP) equations is considered, when the three-dimensional domain shrinks to a line segment, to describe the flows of positively and negatively charged ions through open ion channel. The new model comprises the usual drift diffusion terms and takes into account for each phase, the bulk velocity defined by (4) including the water bath for ions. The existence of global weak solution to this problem is shown. The proof relies on the use of certain embedding theorem of weighted sobolev spaces together with Hardy inequality. Copyright © 2010 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] ## A weak solution approach to a reaction,diffusion system modeling pattern formation on seashells MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2009Jan KelkelAbstract We investigate a reaction,diffusion system proposed by H. Meinhardt as a model for pattern formation on seashells. We give a new proof for the existence of a local weak solution for general initial conditions and parameters upon using an iterative approach. Furthermore, the solution is shown to exist globally for suitable initial data. The behavior of the solution in time and space is illustrated through numerical simulations. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature-dependent viscosity MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009Bernard DucometAbstract We consider an initial-boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature-dependent viscosity µ(,) and conductivity ,(,). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(,) and ,(,) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Existence of a weak solution to the Navier,Stokes equation in a general time-varying domain by the Rothe method MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2009í NeustupaAbstract We assume that ,t is a domain in ,3, arbitrarily (but continuously) varying for 0,t,T. We impose no conditions on smoothness or shape of ,t. We prove the global in time existence of a weak solution of the Navier,Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q[0,,T) := {(x,,t);0,t,T, x,,t}. The solution satisfies the energy-type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## The play operator on the rectifiable curves in a Hilbert space MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2008Vincenzo RecuperoAbstract The vector play operator is the solution operator of a class of evolution variational inequalities arising in continuum mechanics. For regular data, the existence of solutions is easily obtained from general results on maximal monotone operators. If the datum is a continuous function of bounded variation, then the existence of a weak solution is usually proved by means of a time discretization procedure. In this paper we give a short proof of the existence of the play operator on rectifiable curves making use of basic facts of measure theory. We also drop the separability assumptions usually made by other authors. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2007Mark LichtnerAbstract We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a function space of continuous functions the weak solutions generate a smooth skew product semiflow. Using slow fast structure and dissipativity we prove the existence of smooth exponentially attracting invariant centre manifolds for the non-autonomous model. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Convergence rates toward the travelling waves for a model system of the radiating gas MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007Masataka NishikawaAbstract The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t),1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## A rain water infiltration model with unilateral boundary condition: qualitative analysis and numerical simulations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2006I. BorsiAbstract We present a rigorous mathematical treatment of a model describing rain water infiltration through the vadose zone in case of runoff of the excess water. The main feature of the mathematical problem emerging from the model lies on the boundary condition on the ground surface which is in the form of a unilateral constraint. Existence and uniqueness of a weak solution is proved under general assumptions. We present also the results of a numerical study comparing the proposed model with other models which approach in a different way the rain water infiltration problem. 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] ## Boundary value problem for the N -dimensional time periodic Vlasov,Poisson system MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2006M. BostanAbstract In this work, we study the existence of time periodic weak solution for the N -dimensional Vlasov,Poisson system with boundary conditions. We start by constructing time periodic solutions with compact support in momentum and bounded electric field for a regularized system. Then, the a priori estimates follow by computations involving the conservation laws of mass, momentum and energy. One of the key point is to impose a geometric hypothesis on the domain: we suppose that its boundary is strictly star-shaped with respect to some point of the domain. These results apply for both classical or relativistic case and for systems with several species of particles. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Existence and uniqueness of solutions of elastic string with moving ends MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2004M. A. RinconAbstract A mathematical model for the small vibration of an elastic string is considered. The model takes into account the change of tension due to the movement of the end points of the string. Under the assumptions that the speed of the moving ends be less than the characteristic speed of the equation, the existence and the uniqueness of local weak solution and global strong solution are proved. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2002Petr PlecháAbstract Evolution of fine structure in martensite undergoing an isothermal process is modelled on a microscopic level by using a positive homogeneous dissipation potential which can reflect a specific energy needed for a phase transformation between different variants of martensite. The model thus naturally incorporates an activation phenomenon. Existence of a weak solution is proved together with convergence of finite-element approximations. Numerical experiments showing the expected rate-independent hysteresis response are also presented. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2002Yang ZhijianWe consider the blowup of solutions of the initial boundary value problem for a class of non-linear evolution equations with non-linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, ,}, where m, , and p are non-negative real numbers and m+1, ,+1, p+1 are, respectively, the growth orders of the non-linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above-mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non-linear terms, the states of the initial energy and the existence and non-existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Single-phase flow in composite poroelastic media MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2002R. E. ShowalterThe mathematical formulation and analysis of the Barenblatt,Biot model of elastic deformation and laminar flow in a heterogeneous porous medium is discussed. This describes consolidation processes in a fluid-saturated double-diffusion model of fractured rock. The model includes various degenerate cases, such as incompressible constituents or totally fissured components, and it is extended to include boundary conditions arising from partially exposed pores. The quasi-static initial,boundary problem is shown to have a unique weak solution, and this solution is strong when the data are smoother. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## An analytical and numerical study of the Stefan problem with convection by means of an enthalpy method MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001E. Casella1Abstract In this paper, we consider a theoretical and numerical study of the Stefan problem with convection, described by the Navier,Stokes equations with no-slip boundary conditions. The mathematical formulation adopted is based on the enthalpy method. The existence of a weak solution is proved in the bidimensional case. The numerical effectiveness of the model considered is confirmed by some numerical results. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2001D. HömbergAbstract We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy-balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Finite element analysis of thermally coupled nonlinear Darcy flows NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010Jiang ZhuAbstract We consider a coupled system describing nonlinear Darcy flows with temperature dependent viscosity and with viscous heating. We first establish existence, uniqueness, and regularity of the weak solution of the system of equations. Next, we decouple the coupled system by a fixed point algorithm and propose its finite element approximation. Finally, we present convergence analysis with an error estimate between continuous solution and its iterative finite element approximation.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] ## Distributions for which div v = F has a continuous solution COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2008Thierry De PauwThe equation div v = F has a continuous weak solution in an open set U , ,m if and only if the distribution F satisfies the following condition: the F(,i) converge to 0 for every sequence {,i} of test functions such that the support of each ,i is contained in a fixed compact subset of U, and in the L1 norm, {,i} converges to 0 and {,,i} is bounded. © 2007 Wiley Periodicals, Inc. [source] ## On the number of singular points of weak solutions to the Navier-Stokes equations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2001Gregory A. SereginWe consider a suitable weak solution to the three-dimensional Navier-Stokes equations in the space-time cylinder , × ]0, T[. Let , be the set of singular points for this solution and , (t) , {(x, t) , ,}. For a given open subset , , , and for a given moment of time t ,]0, T[, we obtain an upper bound for the number of points of the set ,(t) , ,. © 2001 John Wiley & Sons, Inc. [source] ## Physicochemical factors controlling the release of dissolved organic carbon from columns of forest subsoils EUROPEAN JOURNAL OF SOIL SCIENCE, Issue 2 2002J.-M. MünchSummary Retention of dissolved organic carbon in soil depends on the chemical and physical environment. We studied the release of organic carbon from three carbonate-free forest subsoil materials (Bs1, Bs2, Bg) in unsaturated column experiments as influenced by (i) variations of the flow regime and (ii) varied chemical properties of the irrigation solution. We investigated the effect of flow initiation, constant irrigation, interruptions to flow, and variation in the effective pore water velocity on the release of organic C. The influence of ionic strength and cation valence in the irrigation solution was studied by stepped pulses of NaCl and CaCl2. The release of C from all materials was characterized by an initial large output and a decline to constant concentrations under long-term irrigation. Interrupting the flow increased its release when flow was resumed. The release from the Bs1 material was not related to the duration of the interruption. The Bs2 material, in contrast, released organic carbon in a way that was successfully described by a kinetic first-order model. Increased pore water velocity decreased the concentrations of C in the effluent from it. The pH of the irrigation solution had negligible effects on the mobilization of C. Increased ionic strength reduced the release, whereas rinsing with distilled water increased the concentrations of C in the effluent. The response of dissolved C to pulses of weak solutions, however, was sensitive to the type of cation in the previous step with strong solutions. The results suggest that the release of organic matter in the soils depends on its colloidal properties. [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] ## Existence of solutions to a phase transition model with microscopic movements MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2009Eduard FeireislAbstract We prove the existence of weak solutions for a 3D phase change model introduced by Michel Frémond in (Non-smooth Thermomechanics. Springer: Berlin, 2002) showing, via a priori estimates, the weak sequential stability property in the sense already used by the first author in (Comput. Math. Appl. 2007; 53:461,490). The result follows by passing to the limit in an approximate problem obtained adding a superlinear part (in terms of the gradient of the temperature) in the heat flux law. We first prove well posedness for this last problem and then,using proper a priori estimates,we pass to the limit showing that the total energy is conserved during the evolution process and proving the non-negativity of the entropy production rate in a suitable sense. Finally, these weak solutions turn out to be the classical solution to the original Frémond's model provided all quantities in question are smooth enough. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in Lp (p,1) MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2009Pierre-Étienne DruetAbstract Accurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L1. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and Lp right-hand side, with p,1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the Lp norm of the heat sources and to prove the existence of weak solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three-dimensional domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2009Michael LukaschewitschAbstract The forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point-wise asymptotic behaviour of weak solutions to this problem in the three-dimensional case. Copyright © 2008 John Wiley & Sons, Ltd. [source] |