Abstract
We study the boundary value problem for a linear first‐order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a unique L2‐solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss‐Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted conormal Sobolev spaces.
Highlights
Introduction and Main ResultsFor n ≥ 2, let Rn denote the n-dimensional positive half-spaceRn : x x1, x, x1 > 0, x : x2, . . . , xn ∈ Rn−1 .The boundary of Rn will be systematically identified with Rnx−1
In 2, the regularity of weak solutions to the characteristic BVP 1.2 - 1.3 was studied, under the assumption that the problem is strongly L2-well posed, namely, that a unique L2-solution exists for arbitrarily given L2-data and the solution obeys an a priori energy inequality without loss of regularity with respect to the data; this means that the L2-norms of the interior and boundary values of the solution are measured by the L2-norms of the corresponding data F, G
Agreeing with the notations set for the usual Sobolev spaces, for γ ≥ 1, Htman,γ Rn will denote the conormal space of order m equipped with the γ-depending norm u : 2
Summary
Rn : x x1, x , x1 > 0, x : x2, . . . , xn ∈ Rn−1. The boundary of Rn will be systematically identified with Rnx−1. In 2 , the regularity of weak solutions to the characteristic BVP 1.2 - 1.3 was studied, under the assumption that the problem is strongly L2-well posed, namely, that a unique L2-solution exists for arbitrarily given L2-data and the solution obeys an a priori energy inequality without loss of regularity with respect to the data; this means that the L2-norms of the interior and boundary values of the solution are measured by the L2-norms of the corresponding data F, G. Several problems, appearing in a variety of different physical contexts, such as fluid dynamics and magneto-hydrodynamics, exhibit a finite loss of derivatives with respect to the data, as considered by estimate 1.7 in the statement of Theorem 1.1 This is the case of some problems that do not satisfy the so-called uniform Kreiss-Lopatinskiı condition; see, for example, 3, 4.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
