One-dimensional Cauchy Problem Research Articles

Some properties of the one-dimensional potentials

The main aim of this paper is to study the properties of the one-dimensional potentials. In this paper, we have studied the connection between the one-dimensional potentials and the self-adjoint part of the operator LK−1, which LK−1 is the solution to the one-dimensional Cauchy problem. Moreover, a new method is used that allows us to reduce the spectral problem for the Helmholtz potential to the equivalent problem.

Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model

<abstract><p>In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow.</p></abstract>

Existence theorem for global in time solutions to Burgers equation with a time delay

In this paper, we consider the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay and analyze a delay effect. We prove the existence of the global in time solution when the product of the size of the delay parameter and the one of the initial history is suitably small. Moreover we show theorem concerning with the regularity of the global in time solution.

Stability of rarefaction wave for viscous vasculogenesis model

<p style='text-indent:20px;'>In this paper, we are concerned with the large time behavior of solutions to the one-dimensional Cauchy problem on a hyperbolic-parabolic-elliptic model for vasculogenesis in the case when far field states of initial data are distinct. It turns out that the solutions exist for all time and tend to a weak rarefaction wave whose strength is not necessarily small under small perturbation. All the results are based on the assumption <inline-formula><tex-math id="M1">\begin{document}$ 2A-\frac{{\mu}a}{b}>0 $\end{document}</tex-math></inline-formula> which guarantees the dissipation of this model.</p>

ITERATIVE METHOD APPLIED TO THE FRACTIONAL NONLINEAR SYSTEMS ARISING IN THERMOELASTICITY WITH MITTAG-LEFFLER KERNEL

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with [Formula: see text]-homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the future technique, we consider three different cases and analyzed the projected model in terms of fractional order. Moreover, the physical behavior of the obtained solution has been captured in terms of plots for diverse fractional order, and the numerical simulation is demonstrated to ensure the exactness. The obtained results elucidate that the proposed scheme is easy to implement, highly methodical as well as accurate to analyze the behavior of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law

In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a $2 × 2$ system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence

We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. I t corresponds to the compressible Navier-Stokes system with large Mach number equal to 1 √ e for e going to 0. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation– ρe converges to the unique solution to the porous medium equation [13,14]. For viscosity coefficientµ(ρe) = ρ α with α > 1, we obtain a rate of convergence of ρe in L ∞ (0,T;H −1 (R)); for 1 < α ≤ 3 the solution ρe converges in L ∞ (0,T;L 2 (R)). For compactly supported initial data, we prove that most of the mass corresponding to solution ρe is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of e.

Decay Rates of The Solution to the Cauchy Problem of the Type III Timoshenko Model Without Any Mechanical Damping

AbstractIn this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow and, in the case of nonequal wave speeds, are of regularity-loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term.

Leader in a diffusion competition model

A one-dimensional Cauchy problem is considered for a system of reaction-diffusion equations that, in the point version, generalizes the Volterra competition model. It is proved that the number of the leader in the propagation velocity of nonvanishing solution values at the periphery is independent of nonnegative finite initial distributions.

A STUDY OF WEAKLY DISCONTINUOUS SOLUTIONS FOR HYPERBOLIC DIFFERENTIAL EQUATIONS BASED ON WAVELET TRANSFORM METHODS

A new approach to prove the one-dimensional Cauchy problem's weakly discontinuous solutions for hyperbolic PDEs are on the characteristics is discussed in this paper. To do so, I use wavelet singularity detection methods or WTMM [1] based on two-dimensional wavelet transform and combine it with the Lipschitz index to strengthen the detection.

Global existence and decay of solutions of the Cauchy problem in thermoelasticity with second sound

We consider the one-dimensional Cauchy problem in non-linear thermoelasticity with second sound, where the heat conduction is modelled by Cattaneo’s law. After presenting decay estimates for solutions to the linearized problem, including refined estimates for data in weighted Lebesgue-spaces, we prove a global existence theorem for small data together with improved decay estimates, in particular for derivatives of the solutions.

CONSTRUCTION OF THE 2D RIEMANN SOLUTIONS FOR A NONSTRICTLY HYPERBOLIC CONSERVATION LAW

In this note, we consider the Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws. Without the restriction that each jump of the initial data projects one planar elementary wave, six topologically distinct solutions are constructed by applying the generalized characteristic analysis method, in which the delta shock waves and the vacuum states appear. Moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct global solutions.

Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws

The Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws is considered. Without the restriction that each jump of the initial data projects one planar elementary wave, ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis. Some of these solutions involve the nonclassical waves, i.e., the delta shock wave and the delta contact discontinuity, for which we explicitly give the expressions of their strengths, locations and propagation speeds. Moreover, we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct unique global solutions.

Nonclassical interactions portrait in a macroscopic pedestrian flow model

In this paper we describe the main characteristics of the macroscopic model for pedestrian flows introduced in [R.M. Colombo, M.D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci. 28 (13) (2005) 1553–1567] and recently sperimentally verified in [D. Helbing, A. Johansson, H.Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 75 (4) (2007) 046109]. After a detailed study of all the possible wave interactions, we prove the existence of a weighted total variation that does not increase after any interaction. This is the main ingredient used in [R.M. Colombo, M.D. Rosini, Existence of nonclassical Cauchy problem modeling pedestrian flows, technical report, Brescia Department of Mathematics, 2008] to tackle the Cauchy problem through wave front tracking, see [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl., vol. 20, Oxford Univ. Press, Oxford, 2000, The one-dimensional Cauchy problem; A. Bressan, The front tracking method for systems of conservation laws, in: C.M. Dafermos, E. Feireisl (Eds.), Handbook of Differential Equations; Evolutionary Equations, vol. 1, Elsevier, 2004, pp. 87–168; R.M. Colombo, Wave front tracking in systems of conservation laws, Appl. Math. 49 (6) (2004) 501–537]. From the mathematical point of view, this model is one of the few examples of conservation laws in which nonclassical solutions have a physical motivation, see [P.G. Lefloch, Hyperbolic Systems of Conservation Laws, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2002, The theory of classical and nonclassical shock waves], and an existence result is available.

Nonlinear Hyperbolic Heat Conduction Problem: Closed-Form Solutions

A nonlinear rigid heat conductor obeying the first and second laws of thermodynamics, Cattaneo's law, and a generalized energy-entropy relation in which both the energy and entropy are parabolic functions of the heat flux, is revisited. For a one-dimensional Cauchy problem in which both the temperature and heat flux are time-dependent only, a solution in terms of elementary functions is obtained. Also, for a one-dimensional traveling wave problem, a solution in terms of elementary functions is presented. Graphs illustrating the solutions are included.

Traffic Flow on a Road Network

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions. Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.

Two-sided linear numerical methods for the Cauchy problem for normal equations

Two-sided approximating collocation polynomials of the same order are constructed. These polynomials are used to develop two-sided extrapolation and interpolation difference methods and Krylov-type methods for the one-dimensional Cauchy problem for normal equations. The required integrals are evaluated by matrix numerical integration algorithms.

Some properties of the Upper Convected Maxwell model for viscoelastic fluid flow

We have studied in two dimensions the upper convected Maxwell constitutive equation for a viscoelastic fluid together with the usual equations for conservation of mass and momentum. A small degree of compressibility is allowed. This can be considered as an artificial compressibility. We show convergence from the compressible to the incompressible solution for the periodic one-dimensional Cauchy problem. Given smooth initial data, an initially positive definite modified stress tensor \\ ̃ gt = τ + (η/λ) I is shown to be a sufficient condition for short time well-posedness, i.e. existence of a unique solution, smooth both in time and space, for the periodic two-dimensional Cauchy problem. Hyperbolicity will be lost if the eigenvalues of \\ ̃ gt go to zero but, in finite time, this is only possible if the velocity gradients go to infinity in finite time.

Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity

FOR THE equations of one-dimensional nonlinear thermoelasticity Slemrod [17] established in 1981 the global existence of smooth solutions for small initial data in the case of a bounded reference configuration for a homogeneous medium. The boundary conditions (traction free, constant temperature or rigidly clamped, insulated) he considered yield, through the differential equation, an additional boundary condition which was crucial for obtaining the necessary energy estimates. Subsequently, Jiang [7] discussed the same boundary conditions as in [17] in the case of an unbounded reference configuration (the half line). Kawashima [lo], Zheng and Shen [ 181, and Hrusa and Tarabek [6] dealt with the one-dimensional Cauchy problem. Recently, Racke and Shibata [13] proved a global existence theorem for the Dirichlet initial boundary value problem (rigidly clamped, constant temperature) in a bounded interval by using the decay estimates for the linearized equations and the L*-energy method, and the author [8] obtained a similar result for the problem in an unbounded interval (the half line) by only applying the L*-energy method. More recently, using a method similar to that in [13], Shibata [16] showed that global small solutions exist for the Neumann problem (traction free, insulated) in the case of a bounded reference configuration. For an unbounded reference configuration (the half line), the case of the Neumann boundary conditions remained open. Our purpose in this paper is to prove a global existence theorem for the Neumann problem in the half line and in a bounded interval for small data by applying the L2-energy method. The proof given here closely follows the pattern of the proof in [8], subject to the necessary modifications. We note that a globally defined classical solution should not be expected for large data. Indeed, Dafermos and Hsiao [4], and Hrusa and Messaoudi [5] showed that for specialized constitutive equations if the initial data are large, then the solution to the Cauchy problem will blow up in finite time. For three-dimensional nonlinear models, we refer the reader to the papers [3, 9, 10-12, 151 (and the references cited therein). This paper is organized as follows. In Section 1 the equations of one-dimensional nonlinear thermoelasticity are presented and the main theorem is stated. In Section 2 we give a local