| Perturbation methods play an important part in the application of mathematical physics. Regular perturbation methods are invalid for some special physical problems.So kinds of singular perturbation technologies have to be used.In the study of irregular physical phe-nomena,singular perturbation theory is useful.However,as the scientists have verified,the pa-rameters of the solutions obtained by the irregular perturbation theory present no uniformly convergence phenomenon.The center problem of studying irregular perturbation theory is how to get the uniformly valid asymptotic expansions of the exact solutions.In the near 100 years,in order to solve the problem,mathematicians,engineers and physi-cists have developed many perturbation methods such as multiple scale method,telescopic 3d-coordinate method,matching gradual average method,WKB method, boundary-layer method, expansion method,centre manifold method and so on.But every method has some limitations so that wide applications of them have been affected.In 1950s,when large momentum be-havior in the photon spread of electrodynamics was processed,renormalization group theory was proposed. Since then,renormalization group method has been called the valid tool in the study of Kinetic equations.Renormalization group method is better than other perturbation methods.For renormalization group method can be directly expanded into perturbation and needn't have some additional knowledge such as the assumptions of incremental matching and perturbation sequence.Besides,renormalization group method,which is better than tradi-tional methods,can automatically produce asymptotic sequences in order to have some exact global solutions.Therefore,renormalization group method is more effective and easier to use than other methods.In recent years,the study of renormalization group method has got some development.â…¢inois they proposed to use the RG equation to get an asymptotic behavior of solutions of differ-ential equations including ones of singular and reductive perturbation problems in a unified way. Mathematically, the RG equdtion is used to improve the global behavior of the local solutions which were obtained in the perturbation theory. This fact suggests that the RG method may be formulated in a purely mathematical way without recourse to the notion of the RG. In the paper [10] showed that the RG method can be formulated in the classical theory of envelopes, and gave a proof why the RG equation can give a globally improved solution to ordinary differential equations. renormalization group method is more effective than other asymptotic perturbation methods,and it has been used in some partial differential equations,which have physical background,such as in the fields of constructing the approx-imate solutions of partial differential equations,proving convergence,giving the estimates of convergence and so on.Many physical variables in natural can be described as nonlinear and hyperbolic partial differential equations.But it is difficult to have the closed form solutions of almost equa-tions.Therefore,we have to determine some characteristics of the solutions and simplify the original equations.This paper can be divided into three parts, The purpose is to show that the renormal-ization group method for solving ordinary differential equations can be naturally extended to a class of partial differential equations. In the first part,we introduce some background and development of the study of some sigular perturbation methods. In the second part,the application of renormalization group method in some common partial differential equations is introduced.Considering:Dissipative nonlinear hyperbolic equationOne-dimensional damped Kuramoto-Shivashinsky equationTwo-dimensional Swift-Hohenberg equation Baltzamm equationIn the third part considering the model of slightly compressible fluidsWe introduce an asymptotic solution,which exists for an interval of time independent of the perturbation parameter and give the comprehensive introduction.
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