Journal of Hyperbolic Differential Equations

We investigate the two-dimensional Schrödinger equation with repulsive inverse square potential, and we prove the following homogeneous endpoint Strichartz estimate:

We consider numerical schemes which combine non-uniform, adaptively redefined spatial meshes with entropy conservative schemes for the evolution step for shock computations. We observe that the resulting adaptive schemes yield approximations free of oscillations in contrast to known fully discrete entropy conservative schemes on uniform meshes. We conclude that entropy conservative schemes are transformed to entropy diminishing schemes when combined with the proposed geometrically driven mesh adaptivity.

We prove the existence of a large class of one-parameter families of cosmological solutions to the Einstein–Euler equations that have a Newtonian limit. This class includes solutions that represent a finite, but otherwise arbitrary, number of compact fluid bodies. These solutions provide exact cosmological models that admit Newtonian limits but, are not, either implicitly or explicitly, averaged.

We show global persistence of solutions with small data for the model equation □u = u⋅∇u + u³, on R^1+d, d ≥ 5, subject to the Coulomb gauge condition . In particular, this covers the important case of the Yang–Mills problem.

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

We consider the initial-boundary value problem for scalar conservation laws on the strip (0, ∞) × [0, 1] with strictly convex smooth flux of a superlinear growth. We show that an associated Hamilton–Jacobi equation with initial and (appropriately defined) boundary conditions has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. Each of these functions, in turn, can be expressed using Lax's formula. The traces of the gradients V_x satisfy generalized boundary conditions (as in LeFloch (1988)) in a pointwise manner when the initial and boundary data are continuous and in a weak sense when they are discontinuous. It is also shown that V_x is continuous almost everywhere, and a result concerning the traces of the sign of f′(V_x(t, ⋅)) is proven.

A semiglobal stability result for a class of purely radiative vacuum spacetimes arising from hyperboloidal data is given. This result generalizes semiglobal existence results for Minkowski-like spacetimes to the case where the reference solution contains gravitational radiation. The analysis makes use of the extended conformal field equations and a gauge based on conformal geodesics so that the location and structure of the conformal boundary of the perturbed solutions is known a priori.

Building on work by Lindblad, Coutand and Shkoller, and the present author, we prove that the equations of motion of an inviscid, incompressible, self-gravitating fluid with free boundary are well-posed.