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.     

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.     

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     

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.

     

Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws

Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate m|x|. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function.We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.     

Scalar conservation laws with stochastic forcing

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.     

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.

     

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.     

Large time behaviour of solutions of balance laws with periodic initial data

We consider the Cauchy problem for a large class of scalar conservation laws with source term and periodic initial data. We show that the solutions can either tend uniformly to infinity or stay bounded in an interval containing at most one zero of the source term. In the latter case, depending on the properties of the zero, the solution tends uniformly to it or approaches an oscillating profile.     

Decay estimation in nonlinear hyperbolic system of conservation laws

In this paper we study some decay estimates in nonlinear hyperbolic system of conservation laws. This research is not only interesting in itself but also crucial in studying the large time behavior problem. By introducing a proper Glimm functional, we obtain some useful decay estimates which are proved helpful in obtaining decay rates of the admissible solutions to nonlinear hyperbolic conservation laws as t→∞.     

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.     

Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted intoL _loc estimates, following theLip convergence theory developed by Tadmor et al. Comparisons between the local truncation error and theL _loc -error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].     

Regularity of solutions to scalar conservation laws with a force

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.     

Solution of convex conservation laws in a strip

In this paper we consider scalar convex conservation laws in one space variable in a stripD =(x, t): 0 ≤x ≤1,t > 0 and obtain an explicit formula for the solution of the mixed initial boundary value problem, the boundary data being prescribed in the sense of Bardos-Leroux and Nedelec. We also get an explicit formula for the solution of weighted Burgers equation in a strip.     

Quasi-potentials of the entropy functionals for scalar conservation laws

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.     

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.     

1-Soliton solution and conservation laws of the generalizedDullin-Gottwald-Holm equation

This paper obtains the 1-soliton solution of the generalized Dullin-Gottwald-Holm equation by the aid of solitary wave ansatz. Subsequently, the conserved quantities are obtained by utilising the interplay between the multipliers and underlying Lie point symmetry generators of the equation.     

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-K?hler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.     

Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

In [H. Brézis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73–97.] Brézis and Friedman prove that certain nonlinear parabolic equations, with the δ-measure as initial data, have no solution. However in [J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186–196.] Colombeau and Langlais prove that these equations have a unique solution even if the δ-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais’ result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from [J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371–399.].