Analyticity up to the boundary of solutions of nonlinear parabolic equations

Abstract

IntroduclionThe present paper is concerned with the analyticity of the solution u(x, t) of the nonlinear second-order analytic parabolic equationin a bounded cylindrical domain Q = R x I in R: x R,. Here R is assumed to be a bounded domain with analytic boundary aR, and the parabolicity means that the derivatives of F with respect to V : u form a positive definite matrix.We shall prove the analyticity of u, local in t and global in x up to the boundary aR, under suitable boundary conditions, especially the Dirichlet and the Neumann conditions.The analyticity in x is not new. For the local interior problem in x and t, it has been known (see Friedman [l]) that u is analytic in x and belongs to the second Gevrey class in t. The same local regularity for u up to the boundary aR was proved recently by Kinderlehrer and Nirenberg [3] under the same boundary conditions as in the present paper. Note that the solution u of the local problem is not necessarily analytic in t, even in the linear parabolic case. Our theorem states that the global conditions in x imply the local analyticity in t. In [3], Kinderlehrer and Nirenberg proved also the analyticity of u, local in t and global in x, in the case that R is an annulus.The purpose of the present paper is to extend their result to the general bounded domain R with analytic boundary.Our theorem is stated in Section 1. As in [4], the analyticity of the solution u is proved by estimating inductively the L2 norm of successive derivatives of u. In doing so, we consider the linearized problems satisfied by the derivatives of u. Since the right-hand sides of these problems depend on u and its derivatives, the induction scheme is somehow complicated. In Section 4, we state this scheme which was used in [3].

The dynamics of rotating waves in scalar reaction diffusion equations

“…[29]. It is known under much weaker conditions that z(u(t, ■)) is a nonincreasing function of t. Our assumption that all coefficients a, b and c are real analytic allows us to characterize those moments in time when z(u(t, ■)) drops.…”

“…PROOF. The solutions u and v are analytic on (0,r] x S1 [29]. Now consider w(t, x) = u(t, x) -v(t, x).…”

“…Since our nonlinearity is real analytic, solutions of (1.1) are real analytic functions for t > 0 and x e S1 (see [29]). …”

Local analyticity radii of solutions to the 3D Navier–Stokes equations with locally analytic forcing

“…Classical analyticity results for solutions to 3D NSE can be found in [2,14,23,31]. A pioneering work in the area of estimating analyticity radii was carried out by Foias and Temam in [10] using Fourier techniques and Gevrey spaces in an L 2 setting (see also [8,33]).…”

Global analyticity up to the boundary of solutions of the navier‐stokes equation

“…Setting F(V,u, u, x, t ) = f -(u -grad) u, we admit a slightly more general nonlinearity corresponding to F. Then the problem is reduced to the case of zero Dirichlet data & = O , a fact which is proved in subsection 1.2. The basic idea of the proof is the same as in [4] and [5]. As in [9], the analyticity of the solution is obtained by estimating its successive derivatives in terms of the I . '…”