共查询到20条相似文献,搜索用时 109 毫秒
1.
The aim of this paper is to present a new system of equations describing nonlocal model of hyperbolic thermoelasticity theory. We used the Papkin and Gurtin approach based on the constitutive relations for internal energy e(x), and heat flux q(x), with integral terms. Such system of equations describes the propagation of thermal perturbation with finite velocity. Using the modified Cagniard–de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three–dimensional space. Basing on the constructed matrix of fundamental solutions in the explicit formula we represent the solution of the Cauchy problem to this system of equations in the form of some kind of convolutions. Next, applying the method of Sobolev spaces, we obtain the Lp−Lq time decay estimate to the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
2.
Jarosław Łazuka 《Mathematical Methods in the Applied Sciences》2020,43(17):10115-10137
This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the Lp−Lq time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials. 相似文献
3.
The aim of this paper is to present a new system of equations describing nonlocal model of thermoviscoelastic theory. We used the Papkin and Gurtin approach based on the constitutive relations for stress tensor σ(x), internal energy e(x) and heat flux q(x), with integral terms. Using the modified Cagniard-de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three-dimensional space. Basing on this matrix we represent in the explicit formula the solution of the Cauchy problem to this system of equations. Next, applying the method of Sobolev spaces, we proved the Lp–Lq time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
Theory of non-simple materials is different from that of simple materials because in it the first strain gradient is taken into consideration as the constitutive variable. The consequence of this fact, from mathematical point of view, is that the equation of motion consists either of higher order derivatives of displacement (four order derivatives) and some material parameters can depend not only on the temperature and the gradient of displacement but also on the second derivative of displacement. We consider the system of partial differential equations describing non-simple thermoelastic materials. This system consists of four scalar equations, three equations of motion and one of energy balance, describing the field of displacement and the temperature in an elastic body. Using the Fourier transform, we found the L p – L q time decay estimates of the solution of the Cauchy problem for the system of equation describing the non-simple thermoelastic materials, being important for proving the global-in-time solution of this problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
5.
Ta-Tsien Li Yue-Jun Peng Jérémy Ruiz 《Journal de Mathématiques Pures et Appliquées》2009,91(6):553-568
Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems. 相似文献
6.
In this work, we study the Cauchy problem of the nonlinear spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem with the initial datum in L2 enjoys an analytic regularizing effect, and the evolution of the analytic radius is the same as that of heat equations.
相似文献7.
L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained. 相似文献
8.
Zhong Tan 《Journal of Mathematical Analysis and Applications》2010,364(2):424-436
In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L2-norm. 相似文献
9.
Akbar B. Aliev Anar A. Kazimov Vusala F. Guliyeva 《Mathematical Methods in the Applied Sciences》2013,36(9):1133-1144
In this paper, we investigate the Cauchy problem for a class of the system of semilinear hyperbolic equations with damping. With the use of the Lp → Lq type estimation for the corresponding linear problem and the method of comparison of functional, the existence and nonexistence criteria of global solutions are found. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
10.
Nguyen Duc Phuong Nguyen Huy Tuan Dumitru Baleanu Nguyen Hoang Luc 《Mathematical Methods in the Applied Sciences》2019,42(18):6829-6848
The aim of this paper is to study the Cauchy problem of determining a solution of nonlinear elliptic equations with random discrete data. A study showing that this problem is severely ill posed in the sense of Hadamard, ie, the solution does not depend continuously on the initial data. It is therefore necessary to regularize the in‐stable solution of the problem. First, we use the trigonometric of nonparametric regression associated with the truncation method in order to offer the regularized solution. Then, under some presumption on the true solution, we give errors estimates and convergence rate in L2‐norm. A numerical example is also constructed to illustrate the main results. 相似文献
11.
Belkacem Said‐Houari Abdelaziz Soufyane 《Mathematical Methods in the Applied Sciences》2015,38(17):3642-3652
In this paper, we consider the Bresse system coupled with the Fourier law of heat conduction. We prove that the decay rate of the solution is very slow. In fact, we show that the L2‐norm of the solution decays with the rate of (1 + t)?1/12 similar to the one obtained for the Timoshenko system. In addition, we found that the wave speed of the first two equations still control the decay rate of the solution with respect to the regularity of the initial data. This seems to be the first result dealing with the behavior of the Cauchy problem in the Bresse–Fourier model. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
12.
We are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method converges in L1 to the weak entropy solution of the considered obstacle problem. In the case of the Cauchy problem, an L1-error bound in
is established. 相似文献
13.
Hui-hui Dai Keng-Huat Kwek Hong-jun Gao Chao-chun Qu 《Frontiers of Mathematics in China》2006,1(1):144-159
The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method
proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem
of the Camassa-Holm equation in L
p
frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the
linear dispersive coefficient contained in the equation is obtained. 相似文献
14.
Norikazu Yamaguchi 《Mathematical Methods in the Applied Sciences》2005,28(13):1507-1526
In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, Lp–Lq type estimates are obtained. By use of the Lp–Lq estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
15.
Shouming Zhou 《Mathematical Methods in the Applied Sciences》2017,40(10):3718-3732
This paper deals with the non‐uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well‐posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces Hs with s > 3/2. On the other hand, the persistence properties in weighted Lp spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
16.
We prove a theorem about global existence (in time) of the solution to the initial-value problem for a nonliear system of coupled partial differential equations of fourth order describing the thermoelasticity of non-simple materias. We consider such the case of thim system in which some nonlinear coeffcients can depend not only on the temperature and the gradient of displacement and also on the second derivative of displacement. The corresponding global existence theorem has been proved using the L p – L q time decay estmates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard continuation argument and to continue the local solution to one desired for all t ∈ (0, ∞) 相似文献
17.
Silvia Jerez 《Numerical Methods for Partial Differential Equations》2013,29(6):2133-2145
In this work, we present a monotone first‐order weighted (FORWE) method for scalar conservation laws using a variational formulation. We prove theoretical properties as consistency, monotonicity, and convergence of the proposed scheme for the one‐dimensional (1D) Cauchy problem. These convergence results are extended to multidimensional scalar conservation laws by a dimensional splitting technique. For the validation of the FORWE method, we consider some standard bench‐mark tests of bidimensional and 1D conservation law equations. Finally, we analyze the accuracy of the method with L1 and L∞ error estimates. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
18.
Starting from the generalized scheme of separation of variables, we propose a new effective method of constructing the solution of the Cauchy problem for a system of two partial differential equations, in general of infinite order with respect to the spatial variable. We consider the example of the Cauchy problem for the system of Lamé equations in the case of a two-dimensional strain.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 204–210. 相似文献
19.
N. I. Shinkarik 《Journal of Mathematical Sciences》1993,67(3):3070-3074
The method of delta sequences is used to construct hybrid Legendre-Fourier transforms on a polar axis. As a delta sequence we use the fundamental solution of the Cauchy problem for the corresponding separated system of the classical parabolic and -parabolic equations for thermal conductivity. A fundamental identity is obtained for the integral transform of a differential operator.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 51–56, 1989. 相似文献
20.
Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L
p
) orL
q
(0,T; L
p
) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic
equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach
fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed
for a class of nonlinear parabolic equations inCB([0,T);L
p
orL
q
(0,T; L
p
) with ϕ(x)∈L
r
. In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||r is sufficiently small, where (p,q,r) is an admissible triple.
Project supported by the National Natural Science Foundation of China (Grant No. 19601005). 相似文献