Renormalization group and asymptotics of solutions of nonlinear parabolic equations

Bricmont, Jean; Kupiainen, Antti; Lin, Guojian

doi:10.1002/cpa.3160470606

Cited by 179 publications

(246 citation statements)

References 18 publications

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“…In the context of Hamiltonian systems subject to small-amplitude Hamiltonian perturbations, it has been shown [47,48] that the RG method yields results equivalent to those obtained from canonical Hamiltonian perturbation theory, up to and including O( 2 ). Finally, for completeness, we note that RG has also been applied to derive reduced or amplitude equations for certain nonlinear partial differential equations; see [4][5][6]12,13,10,24,28]. …”

Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).

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Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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“…Recent development of the theories of pattern formation with dissipative structures gives a good example how to reduce complicated ordinary and partial differential equations to simple equations with slow variables, such as Landau-Stuart equation, the time-dependent Ginzburg-Landau equation and so on. [18] Some years ago, it was shown by an Illinois group [21,22,23] and Bricmont and Kupiainen [24]that the RG equations can be used for a global and asymptotic analysis of ordinary and partial differential equations, hence giving a reduction of evolution equations of some types. A unique feature of the Illinois group's method is to start with the naive perturbative expansion and allow secular terms to appear; the secular terms correspond to the logarithmically divergent terms in QFT.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, equating τ with the time t appearing in the original perturbative solution, they obtained global solutions of differential equations. Bricmont and Kupiainen [24] applied a scaling transformation (block transformation) to obtain asymptotic behavior of nonlinear diffusion equations in a rigorous manner.…”

Section: Introductionmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

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The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t 0 = t is naturally understood where t 0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

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“…4) We note that the effect of relaxation of various equations has been studied extensively, we take a moment to recall some known results. The problems of existence and uniqueness of solution of equation (1.4) have been studied by Escudero in [10].…”

Section: Introductionmentioning

confidence: 99%

“…These similarity variables have been introduced before for proving the convergence to self-similar solutions in the case of the parabolic equation u τ = u ξξ − |u| p−1 u (see [3,4,[7][8][9] and [21]). These techniques work even for a large similar class of equations (for more detail, see for example ( [22,28] and [32])).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic profiles for a class of perturbed Burgers equations in one space dimension

Dkhil¹,

Hamza²,

Mannoubi³

2018

OpMath

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Renormalization group and asymptotics of solutions of nonlinear parabolic equations

Cited by 179 publications

References 18 publications

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Asymptotic profiles for a class of perturbed Burgers equations in one space dimension

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