On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

Abstract

Abstract. The classic problem of regularity of boundary points for higherorder partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener's (J. Math. Phys. Mass. Inst. Tech. 3:127-146, 1924) and Petrovskii's (Math. Ann. 109:424-444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960-1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat'ev and Maz'ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:where R(t) > 0 is a smooth function on [−1, 0) and R(t) → 0 + as t → 0 − . The zero Dirichlet conditions on the lateral boundary of Q0 and bounded initial data are posed:The boundary point (0, 0) is then regular (in Wiener's sense) if u(0, 0 − ) = 0 for any data u0, and is irregular otherwise. The proposed asymptotic blow-up approach shows that:(i) for the backward fundamental parabolae with R(t) = l(−t) 1/4 , the regularity of its vertex (0, 0) depends on the constant l > 0: e.g., l = 4 is regular, while l = 5 is not; (ii) for R(t) = (−t) 1/4 ϕ(− ln(−t)) with ϕ(τ ) → +∞ as τ → +∞, regularity/irregularity of (0, 0) can be expressed in terms of an integral Petrovskii-like (Osgood-Dini) criterion. E.g., after a special "oscillatory cut-off" of the boundary, the functioñ , a = constant ∈ (0, 3·23 ), together with typical ideas of boundary layers and blow-up matching analysis. Extensions to 2mth-order poly-harmonic equations in R N and other PDEs are discussed, and a partial survey on regularity/irregularity issues is presented.

The p-Laplace Equation in Domains with Multiple Crack Section via Pencil Operators

“…It should be mentioned that the problem of regularity at boundary points has been studied before using the so-called Wiener's regularity test/criterion via concepts of potential-theoretic Bessel (Riesz) capacities to strongly elliptic equations. In that sense the problem is formulated in terms of a divergence series of capacities, measuring the thickness of the complement of the domain near the point of the boundary at which the regularity is analysed (in this case the origin 0); see [8] for en extensive history of this problem applied to elliptic and parabolic equations, and the Kozlov-Maz'ya-Rossmann's monographs [17,18] for just elliptic problems. Therefore, following those arguments the principal extension of Wiener's-like capacity regularity test to a nonlinear degenerate p-Laplacian operator, with p ∈ (1, N ] was due to Maz'ya in 1970 [21], who extended it later on to have a sufficient capacity regularity condition to be optimal for any p > 1.…”

Boundary characteristic point regularity for Navier–Stokes equations: Blow-up scaling and Petrovskii-type criterion (a formal approach)

“…Concerning other problems and techniques of modern regularity theory, we refer to monographs [27,36,37,51,54] and [35], [46]- [53] as an update guide to elliptic regularity theory including higher-order equations, as well as to references/results in [28,38,41,40,12,69] and [17,20] for linear and semilinear parabolic PDEs.…”

“…Thus, the classic problem of regularity (in Wiener's sense, see [47]) of a boundary characteristic point for the NSEs problem (1.1), (1.2) is under consideration. u t = ∆u in Q 0 , the regularity problem of the characteristic vertex (0, 0) was optimally solved for dimensions N = 1 and 2 by Ivan Georgievich Petrovskii in 1934-35 [60,61] 3 , who introduced his famous Petrovskii's regularity criterion; see [17] and [20] for a full history and further developments in general parabolic theory. This is the so-called "2 √ log log-criterion" (see (2.8) and (2.9) below), which we are going to achieve, at least formally, for the 3D NSEs.…”

Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝN+1

“…The crucial point in this study is to establish an energy estimate similar to that proved in Proposition 4.1. In order to avoid this di culty, which does not appear in the second-order case, one must work in weighted Sobolev spaces, see [6,17]. These questions will be developed in forthcoming works.…”