共查询到20条相似文献,搜索用时 15 毫秒
1.
E.Yu. Panov 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013,30(6):997-1007
We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector. 相似文献
2.
Zaihong Jiang 《Journal of Differential Equations》2009,246(11):4284-4308
The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the L1 stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A nonlinear functional for general scalar hyperbolic conservation laws, J. Differential Equations 235 (2) (2007) 658-667] and it can be viewed as a better attempt for the generalized entropy functional for general equations. 相似文献
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4.
In this paper, we study the problem of asymptotic stabilization by closed loop feedback for a scalar conservation law with a convex flux and in the context of entropy solutions. Besides the boundary data, we use an additional control which is a source term acting uniformly in space. 相似文献
5.
Małgorzata Peszyńska 《Numerical Methods for Partial Differential Equations》2014,30(1):239-264
We consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014 相似文献
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7.
Graziano Guerra 《Journal of Differential Equations》2004,206(2):438-469
We consider the Cauchy problem for the 2×2 strictly hyperbolic system
8.
Hongxia Liu 《Journal of Differential Equations》2007,235(2):658-667
A generalized entropy functional was introduced in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the scalar hyperbolic conservation laws with convex flux function. This functional was crucially used in the functional approach to the L1 stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate. However, how to construct the generalized entropy functional for scalar conservation laws with general flux, and then how to apply the functional approach to the L1 study on general systems are still open. In this paper, we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction. 相似文献
9.
We derive an algorithm for solving the initial value problem for ut = ½σ2uxx + f(u)ux. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for ut = f(u)ux. A comparison with the random choice method is illustrated by the numerical example. 相似文献
10.
We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy. 相似文献
11.
Matthias Kunik 《Mathematical Methods in the Applied Sciences》1993,16(12):895-902
In this paper we prove an explicit representation formula for the solution of a one-dimensional hyperbolic conservation law with a non-convex flux function but monotone initial data. This representation formula is similar to those of Lax [10] and Kunik [7,8] and enables us to compute the solution pointwise explicitly. This result is a generalization of a theorem given in Kunik [8] where the case of only one inflexion point for the fluxes was considered. Its proof uses the polygonal method of Dafermos [2]. The application of this method leads to a simple explicit construction of the solutions for a Kynch sedimentation process [9] and to an explicit parameter representation for the shock curves evolving during the sedimentation process. 相似文献
12.
Paolo Baiti Philippe G. LeFloch Benedetto Piccoli 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,119(2):927-945
We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well. 相似文献
13.
We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002 相似文献
14.
On exact dimensional splitting for a multidimensional scalar quasilinear hyperbolic conservation law
A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero. 相似文献
15.
J. Casti 《Journal of Optimization Theory and Applications》1978,25(4):591-595
A new conservation law relating basic linear filtering theory quantities is derived. The physical interpretation of the law is given and discussion of its possible uses is presented. 相似文献
16.
Matthias Kunik 《Mathematical Methods in the Applied Sciences》1992,15(7):495-509
We examine the existence and regularity results for a scalar conservation law with a convexity condition and solve its weak solution with shocks by using a special method of characterization combined with a representation formula for the weak solution. 相似文献
17.
Young-Sam Kwon 《Journal of Mathematical Analysis and Applications》2008,340(1):543-549
In this paper we give a simple proof of well-posedness of multidimensional scalar conservations laws with a strong boundary condition. The proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation. 相似文献
18.
Chun Shen 《Journal of Mathematical Analysis and Applications》2012,389(2):1105-1116
The aim of this paper is to study the structural stability of solutions to the Riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients. It is proved that the Riemann solution is possibly instable when one of the Riemann initial data is at the vacuum. Furthermore, we point out that the Riemann solution is also possibly instable even when the Riemann initial data stay far away from vacuum. In order to deal with it, we perturb the Riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively. It is also proved that the Riemann solutions are unstable in some certain situations under the local small perturbations of the Riemann initial data by letting the perturbed parameter ε tend to zero. In addition, the interaction of the delta standing wave and the contact vacuum state is considered which appear in the Riemann solutions. 相似文献
19.
Julien Jimenez 《Journal of Evolution Equations》2011,11(3):553-576
We deal in this paper with a scalar conservation law, set in a bounded multidimensional domain, and such that the convective
term is discontinuous with respect to the space variable. First, we introduce a weak entropy formulation for the homogeneous
Dirichlet problem associated with the first-order reaction-convection equation that we consider. Then, we establish an existence
and uniqueness property for the weak entropy solution. The method of doubling variables and a pointwise reasoning along the
curve of discontinuity are used to state uniqueness. Finally, the vanishing viscosity method allows us to prove the existence
result. Another method to obtain the existence of a solution, which relies on the regularization of the flux, is also detailled,
at least for a particular case. 相似文献
20.
We prove the existence, uniqueness, and monotonicity of the solution of an upwind conservative explicit difference scheme approximating an initial-boundary value problem for a many-dimensional nonlinear scalar conservation law with a quadratic nonlinearity under some specific conditions imposed only on the input data of the problem. We show that the resulting solution is not necessarily stable. Under some additional conditions on the input data, which provide the absence of shock waves, we prove the stability of the unique solution of the difference scheme for any finite time. 相似文献