In this paper, we consider the focusing and defocusing energy-subcritical, nonlinear wave equation in R 1+d with radial initial data for d = 4, 5. We prove that if a solution remains bounded in the critical space on its interval of existence, then the solution exists globally and scatters at ±∞. The proof follows the concentration compactness/rigidity method initiated by Kenig and Merle, and the main obstacle is to show the nonexistence of nonzero solutions with a certain compactness property. A main novelty of this work is the use of a simple virial argument to rule out the existence of nonzero solutions with this compactness property rather than channels of energy arguments that have been proven to be most useful in odd dimensions. 4

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In this paper we consider global and non-global radial solutions of the focusing energycritical wave equation on R×R N where N ≥ 5 is odd. We prove that if the solution remains bounded in the energy space as you approach the maximal forward time of existence, then along a sequence of times converging to the maximal forward time of existence, the solution decouples into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume a bound on the evolution that rules out formation of multiple solitons, then this decoupling holds for all times approaching the maximal forward time of existence.

In this manuscript, we study the theory of conformal relativistic viscous hydrodynamics introduced in [4], which provided a causal and stable first-order theory of relativistic fluids with viscosity. The local well-posedness of its equations of motion has been previously established in Gevrey spaces. Here, we improve this result by proving local well-posedness in Sobolev spaces.

Abstract. In this paper, we initiate the study of finite energy equivariant wave maps from the (1 +3)-dimensional spacetime R × (R × S 2 ) → S 3 where the metric on R × (R × S 2 ) is given byThe constant time slices are each given by the Riemannian manifold M := R × S 2 with metricThe Riemannian manifold M contains two asymptotically Euclidean ends at r → ±∞ that are connected by a spherical throat of area 4π 2 at r = 0. The spacetime R × M is a simple example of a wormhole geometry in general relativity. In this work we will consider 1-equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree n. For each n, there exists a unique energy minimizing corotational harmonic map Q n : M → S 3 of degree n. In this work, we show that modulo a free radiation term, every corotational wave map of degree n converges strongly to Q n . This resolves a conjecture made by Bizon and Kahl in [3] in the corotational case.