We study Wave Maps from R2+1 to the hyperbolic plane H2 with smooth compactly supported initial data which are close to smooth spherically symmetric initial data with respect to some H1+μ, μ > 0. We show that such Wave Maps don’t develop singularities in finite time and stay close to the Wave Map extending the spherically symmetric data(whose existence is ensured by a theorem of Christodoulou-T...

The shadowing property is to find an exact solution to an iterated map that remains close to an approximate solution. In this article, using shadowing property, we show the stability of the following equation in normed group: 4n−2Cn/2−1rf ∑n j 1 xj/r n ∑ ik∈{0,1}, ∑n k 1 ik n/2 r2f ∑n i 1 −1 ik xi/r 4n·n−2Cn/2−1 ∑n i 1 f xi , where n ≥ 2, r ∈ R r2 / n and f is a mapping. And we prove that the e...

We consider the semilinear elliptic system { −∆u+m1(x)u = fu(x, u, v) x ∈ Ω, −∆v +m2(x)v = fv(x, u, v) x ∈ Ω, with the boundary conditions ∂u ∂n = λg(x, u) and ∂v ∂n = μh(x, v), where Ω ⊂ RN is a bounded smooth domain, λ, μ > 0 and the functions f , g, h, m1 and m2 satisfy some suitable conditions. Using the fibering map and by extracting the Palais-Smale sequences in the Nehari manifold, we pr...

For the Schrödinger flow from R2 × R+ to the 2-sphere S2, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blowup time (if any), and that they blow up if and only if the length scale of the neare...

We prove existence of an optimal transport map in the MongeKantorovich problem associated to a cost c(x, y) which is not finite everywhere, but coincides with |x− y|2 if the displacement y − x belongs to a given convex set C and it is +∞ otherwise. The result is proven for C satisfying some technical assumptions allowing any convex body in R2 and any convex polyhedron in R, d > 2. The tools are...

images from 6-pt IDEAL reconstruction, and the R2* map (c) from 6-pt T2*-IDEAL reconstruction. The echo spacing was 1.3ms (1.1π). Uniform water-fat decomposition was achieved. Figure 1 (left): IDEAL cost functions for two sets of echo times. The purpose of the IDEAL iteration is to search for the minimum (0Hz in this example) of the cost function curves. To avoid converging to local minima, the...