Global existence for a quasilinear wave equation outside of star-shaped domains

Hart F. Smith
2001 Journées Équations aux dérivées partielles  
in memory of tom wolff We prove global existence of small-amplitude solutions of quasilinear Dirichletwave equations outside of star-shaped obstacles in (3+1)-dimensions. We use a variation of the conformal method of Christodoulou. Since the image of the spacetime obstacle is not static in the Einstein diamond, our results do not follows directly from local existence theory as did Christodoulou's for the nonobstacle case. Instead, we develop weighted estimates that are adapted to the geometry.
more » ... d to the geometry. Using them and the energy-integral method we obtain solutions in the Einstein diamond minus the dime-dependent obstacle, which pull back to solutions in Minkowski space minus and obstacle. © 2002 Elsevier Science (USA) smooth, compact star-shaped obstacles K ... R 3 . Precisely, we shall consider smooth quasilinear systems of the form ( 1.1) which satisfy the so-called null condition [15] . The global existence for such equations in the absence of obstacles was established by Christodoulou [2] and Klainerman [11] using different techniques. We begin by describing our assumptions in more detail. We let a denote an N-tuple of functions, u=(u 1 , u 2 , ..., u N ). We assume that K is smooth and strictly star-shaped with respect to the origin. By this, we understand that in polar coordinates x=rw, (r, w) ¥ [0, .) × S 2 , we can write where f is a smooth positive function on S 2 . Thus, By quasilinearity, we mean that F(u, du, d 2 u) is linear in the second derivatives of u. We shall also assume that the highest order nonlinear terms are diagonal, by which we mean that, if we denote " 0 =" t , then A key assumption is that the nonlinear terms satisfy the null condition. Recall that even in the obstacle-free case there can be blowup in finite time for arbitrarily small data if this condition is not satisfied (see John [9]). The first part of the null condition is that the nonlinear terms are free of linear terms, Additionally, we assume that the quadratic terms do not depend on u, which means that we can write where Q is a quadratic form and where the remainder term R vanishes to third order at (u, du, d 2 u)=0; that is, R(p, q, r)=O((p 2 +q 2 ) r)+O((|p|+|q|) 3 ). (1.6) 156 KEEL, SMITH, AND SOGGE D be small, which would be the analog of Christodoulou's assumption in [2] . Additionally, the result should hold with s=0. 158 KEEL, SMITH, AND SOGGE
doi:10.5802/jedp.596 fatcat:bllxkvciezgr3btft6ji5obxui