m, there exists a solution of (1.1.14) such that has a compact support. But there are other models of self couplings for which the ansatz (1.1.16) is no longer valid, for instance (1.1.15) and F() = iIc1 + byS2 12 1.2. Objectives with nonzero b and c, c2 > 0. Esteban and Séré in [28] studied this more general nonlinearity. They proved by a variational method, there exists an infinity of solutions under the assumption G’(x).xOG(x), 6>1, xEIR for 1 < al,a2 < None of the approaches mentioned above yield a curve of solutions: the continuity of with respect to w, even the uniqueness of was unknown. These issues are important to study the stability of the standing waves. To our knowledge, Ounaies [56] was the first one to consider the regularity of the stationary solutions. He related the solutions to (1.1.14) to those of nonlinear Schrödinger equations. The ground states of Schrödinger equations generate a branch of solutions with small parameter s = m — w for nonlinear Dirac equations. He claimed that for F(s) = sla, 1 < < 2, is continuous w.r.t. w when w e (m — eo,m) for some sç > 0. For a thorough review on the linear and nonlinear Dirac equation, we refer to the work by Esteban, Lewin and Séré [30]. A basic question about standing waves is their stability, which has been studied for nonlinear Klein-Gordon equations and nonlinear Schrödinger equations. Grillakis, Shatah and Strauss [35, 36] proved a general orbital stability and instability condition in a very general setting, which can be applied to traveling waves of nonlinear PDEs such as Klein-Gordon, and Schrödinger and wave equations. Their assumptions allow the second vari ation operator to have only one simple negative eigenvalue, a kernel of di mension one and the rest of the spectrum to be positive and bounded away from zero. But this method cannot be applied to the Dirac operator di rectly. Contrary to —, the Dirac operator Dm = —i)’° y8j +0m is not bounded from below. However there are some partial results about the application of this method to the Dirac equation. Bogolubsky in [6] requires the positivity of the second variation of the energy functional as a necessary condition for stability. Werle [69], Strauss and Vázquez in [62] claim that the solitary waves are unstable, if the energy functional does not have a local minimum at the solitary waves. 1.2 Objectives In the following I describe the results and the methods used in my thesis for the above equations. 13 1.2. Objectives 1.2.1 Global well posedness and blow up for Landau-Lifshitz flow For the Landau-Lifshitz flow equation (1.1.4) where u : 1R2 x ,. §2, the well posedness vs. blowup is studied in my thesis [39, 40]. A good starting point to analyze the flow equation is to assume some symmetry. For maps u with equivariant symmetry u = em8Rv(r), the energy E(u) has a minimal energy 4irImI, which is attainable by a two-parameter family of harmonic maps: H5’ = e(mOh(r/s), E IR, s> 0 / \ / 2 ‘ hi(r) \ I r+r h(r)=( 0 1=1 0 \ h3(r) J \ The harmonic maps are static solutions of the evolution equation. The natural question is to consider the stability of the harmonic maps under the Landau-Lifshitz flow. Our first result concerns m-equivariant maps with energy near the min imal energy 47r1m1, E(uo) = 47r1m1 + , 0 < öo << 1. We have shown that there is no finite time blowup for ml 4. Further more the solutions converge to a specific family of harmonic maps in the space-time norm sense. Hence we say that the harmonic maps are asymp totically stable under the Landau-Lifshitz flow. This result is a rigorous verification of [10] where the authors showed no singularity formation in finite time by formal asymptotic analysis. Our main ingredients involved in the proof are the usage of two different coordinate systems. In an ap propriate orthonormal frame on the tangent plane, the coordinates of the tangent vector field u,. — JuRu satisfy a nonlinear heat-Schrödinger type equation. It leaves us with an equation with small L2 initial data. This equation is also coupled to a 2-dimensional dynamical system describing the dynamics of the scaling parameter s(t) and rotation parametero(t) of a nearby harmonic map H(s(t),(t)). A careful choice of these parameters must be made at each time to allow estimates. The key to prove convergence of the solutions is the space-time estimates for the linear operator of the nonlinear heat-Schrödinger type equation. This can be done because of the energy inequality, the positivity of the energy and the assumption of radial functions. 14 1.2. Objectives For harmonic map heat flow, i.e. /3 = 0, in the subclass of m-equivariant maps (see (1.1.8)), we proved that for m = 1, finite time singularities do occur for some initial data close to the energy of harmonic maps. This result is an adaptation of the blow up result in [18] for a disk domain D2 in R2. 1.2.2 Well-posedness and scattering of a model equation for Schrödinger maps Recall the model equation (1.1.11) m2sin2I4 2I (0,r)=o(r) where q(x, t) 1l2 x R —> C is a radial scalar function and m> 0 is an integer. We are interested in finite energy solutions which leads to sin Io(x)I = 0 both at IxI = 0 and lxI = oo. Therefore this yields the following boundary conditions = {o: [0, oc) — C, E(0) 0, then g(a + ) — g(a) — (26 + 1)IaI29u (CiIa29’+ C2uI201)2 where C1,02 depends on 0 and C1 = 0 if 0<0 Proof. We may assume that a> 0 in our proof. It is trivial if a = 0. So we assume that a 0. If a < 21a1, then a + °i <31o1 and g(a + a) — g(a) — (26 + 1)a29a

0 we have Illi2(D)