Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory

Self-consistent theory of rupture by progressive diffuse damage

We analyze a self-consistent theory of crack growth controlled by a cumulative damage variable d(t) dependent on stress history, in the quasistatic regime where the sound wave velocity is taken as infinite. Depending upon the damage exponent m, which controls the rate of damage dd/dt~sigma(m) as a function of local stress sigma, we find two regimes. For 0<m<2, the model predicts a finite-time singularity. This retrieves previous results by Zobnin for m=1 and by Bradley and Wu for 0<m<2. To improve on this self-consistent theory which neglects the dependence of stress on damage, we apply the functional renormalization method of Yukalov and Gluzman and find that divergences are replaced by singularities with exponents in agreement with those found in acoustic emission experiments. For m>/=2, the rupture dynamics is not defined without the introduction of a regularizing scheme. We investigate three regularization schemes involving, respectively, a saturation of damage, a minimum distance of approach to the crack tip, and a fixed stress maximum. In the first and third schemes, the finite-time singularity is replaced by a crack dynamics defined for all times but which is controlled by either the existence of a microscopic scale at which the stress is regularized or by the maximum sustainable stress. In the second scheme, a finite-time singularity is again found. In the first two schemes within this regime m>/=2, the theory has no continuous limit.

Renormalization-group theoretical reduction

It has been known for some time that singular perturbation and reductive perturbation can be unified from the renormalization-group (RG) theoretical point of view. However, renormalization-group approaches to singularly perturbed problems require explicit perturbation results, so they could be complicated practically. The approach proposed in this paper has considerably eliminated the need for explicit perturbation results, making the RG approach simpler than many conventional singular perturbation approaches. Consequently, we may assert that reductive extraction of global features of the problem is the essence of singular perturbation methods.

Renormalization-group theoretical reduction of the Swift-Hohenberg model

The Swift-Hohenberg model of the cellular pattern formation is exploited with a proto renormalization-group (RG) scheme. The method dispenses with the explicit perturbation solutions which are required in the standard RG approach. The RG equations obtained are the well-known reductive perturbation results such as a rotationally covariant amplitude equation and the nonlinear phase equations.

Three-dimensional pattern formation, multiple homogeneous soft modes, and nonlinear dielectric electroconvection

Patterns forming spontaneously in extended, three-dimensional, dissipative systems are likely to excite several homogeneous soft modes ( approximately hydrodynamic modes) of the underlying physical system, much more than quasi-one- (1D) and two-dimensional (2D) patterns are. The reason is the lack of damping boundaries. This paper compares two analytic techniques to derive the pattern dynamics from hydrodynamics, which are usually equivalent but lead to different results when applied to multiple homogeneous soft modes. Dielectric electroconvection in nematic liquid crystals is introduced as a model for 3D pattern formation. The 3D pattern dynamics including soft modes are derived. For slabs of large but finite thickness the description is reduced further to a 2D one. It is argued that the range of validity of 2D descriptions is limited to a very small region above threshold. The transition from 2D to 3D pattern dynamics is discussed. Experimentally testable predictions for the stable range of ideal patterns and the electric Nusselt numbers are made. For most results analytic approximations in terms of material parameters are given.

Reductive use of renormalization group

It has been recognized that singular perturbation and reductive perturbation can be unified from the renormalization group (RG) theoretical point of view. However, the recognition has been only formal in the sense that it has not given us any new insight nor provided any new technical advantage over the usual RG approach. With our approach, the proto RG method proposed here, we can clearly show that system reduction is the key to singular perturbation methods. The approach also makes the calculation of singular perturbation results more transparent than the conventional RG approach. Consequently, for example, a consistent and easy RG derivation of the rotational covariant Newell-Whitehead-Segel equation is possible.

Renormalization group for renormalization-group equations toward the universality classification of infinite-order phase transitions

We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.

Cavitation induced by explosion in an ideal fluid model

We discuss the problem of an explosion in the cubic-quintic superfluid model, in relation to some experimental observations. We show numerically that an explosion in such a model might induce a cavitation bubble for large enough energy. This gives a consistent view for rebound bubbles in superfluid and we identify the loss of energy between the successive rebounds as radiated waves. We compute self-similar solution of the explosion for the early stage, when no bubbles have been nucleated. The solution also gives the wave number of the excitations emitted through the shock wave.