Explicit generalized solutions to a system of conservation laws

This paper studies a special 3 by 3 system of conservation laws which cannot be solved in the classical distributional sense. By adding a viscosity term and writing the system in the form of a matrix Burgers equation an explicit formula is found for the solution of the pure initial value problem. These regularized solutions are used to construct solutions for the conservation laws with initial conditions, in the algebra of generalized functions of Colombeau. Special cases of this system were studied previously by many authors.     

Riemann solutions without an intermediate constant state for a system of two conservation laws

We present a class of systems consisting of two conservation laws in one spatial dimension that share an intriguing property: they admit structurally stable Riemann solutions without the standard constant state. This striking phenomenon emerges in sharp contrast to what is known for strictly hyperbolic systems of conservation laws, in which the existence of constant states is necessary for the structural stability of Riemann solutions. We prove that, together, coincidence of characteristic speeds and a certain amount of genuine nonlinearity are sufficient to trigger the aforementioned phenomenon. The proof revolves about the presence of a singular point in the coincidence set that organizes the construction of our Riemann solutions.     

Construction of solutions for mixed hyperbolic elliptic Riemann initial value system of conservation laws

In this paper, we apply Adomian decomposition method (ADM) to develop a fast and accurate algorithm for systems of conservation laws of mixed hyperbolic elliptic type. The solutions of our model equations are calculated in the form of convergent power series with easily computable components. The results obtained are compared with our Modification of Adomian decomposition method (MADM) Az-Zo’bi and Al-Khaled (2010) [1]. The study outlines the significant features of the two methods. With application to van der Waals system, we obtain the stability of the approximate solutions graphically when the system changes type with more efficiency of the MADM.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

Regularity of solutions to the perturbed conservation laws

A kind of regularity for the mild solution of perturbed conservation laws is proposed. This regularity is described in term of variations measured in the L¹-norm. A dissipativity condition from the semigroup approach is used to show that the mild solution stays within a class of bounded variation in this sense of regularity. This shows that this class of functions is an invariant of the semigroup. The same analysis carries over to the periodic problem. The class of boundedL¹-variation functions used here can be normed to give a Banach space structure. It also has an analogue with the space of Lipschitz functions     

On the conservation laws and invariant solutions of the mKdV equation

In this paper, we consider modified Korteweg-de Vries (mKdV) equation. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws for the mKdV equation are presented. It is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. In addition, invariant solution is obtained by utilizing the relationship between conservation laws and Lie-point symmetries of the equation.     

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

This paper considers the two‐dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non‐selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non‐selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

A two-dimensional hyperbolic system of nonlinear conservation laws with functional solutions

A two-dimensional hyperbolic system of nonlinear conservation laws is considered for any piecewise constant initial data having two discontinuity rays with the origin as vertex. One kind of new waves, which is labeled the Dirac-contact wave, appears in the solution. The entropy conditions for the Dirac-contact waves are given. The solutions on the Dirac-contact waves can be viewed as the bounded linear functionals onC ₀ ^∞ (R ² ×R ₊). Supported by CNSF and a grant from Academia Sinica Author’s current address: CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France     

Stratified solutions for systems of conservation laws

We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.

     

Periodic solutions of conservation laws constructed through Glimm scheme

We present a periodic version of the Glimm scheme applicable to special classes of systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in , for Bakhvalov's class, and in , in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in with , for some constant which is for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, , are uniformly bounded in , for all 0$">, where is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the -systems with , , , which, for , model isentropic gas dynamics in Lagrangian coordinates.

     

Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations

We prove the uniqueness of Riemann solutions in the class of entropy solutions in for the system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global -stability of the Riemann solutions even in the class of entropy solutions in with arbitrarily large oscillation for the system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under perturbation of the Riemann initial data, as long as the corresponding solutions are in and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions , piecewise Lipschitz in , for any 0$">.

     

Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis

We consider hysteresis perturbations of a system of ODEs which possesses an asymptotically stable periodic solution z_∗. Where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity, it is shown that the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of z_∗. The main result is a theorem on the stability of this continuum.     

On decay of periodic entropy solutions to a scalar conservation law

We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector.     

Continuous dependence on the initial and flux functions for solutions of conservation laws

We prove the continuous dependence on the initial and flux functions for the entropy solutions to the Cauchy problem for conservation laws. Accordingly, we can show that the continuous dependence on the flux function for the entropy solutions depends only on the sup norm, not on the Lipschitz norm.     

Well-posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition

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.     

Symmetry reduction, exact solutions and conservation laws of the Sawada-Kotera-Kadomtsev-Petviashvili equation

In this paper, by applying a direct symmetry method, we obtain the symmetry reduction, group invariant solution and many new exact solutions of SK-KP equation, which include Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and so on. At last, we also give the conservation laws of SK-KP equation.