Journal of Hyperbolic Differential Equations

We compare inviscid stability conditions obtained by Lewicka for large-amplitude shock wave patterns with "slow eigenvalue", or low-frequency, stability conditions obtained by Lin and Schecter through a vanishing viscosity analysis of the Dafermos regularization. Under the structural condition that scattering coefficients for each component wave are positive, we show that BV and L¹ inviscid stability are equivalent to respective versions of low-frequency Dafermos-regularized stability. When scattering coefficients appear with different signs, the conditions are in general distinct. We give various examples demonstrating this phenomenon and indicating the subtle role of cancellation in linearized behavior in the presence of negative scattering coefficients.

We study the propagation of singularities for semi-linear wave equations □u = F(u, Du) satisfying the null condition, and improve an earlier result of ours. Microlocal Sobolev regularity in propagates along null bicharacteristics, when a suitable condition is imposed on s and r. We provide here optimal conditions on s and r, and improve existing results to include values up to n/2 < s ≦ r < 3s - n when the nonlinearity F(u, Du) satisfies the null condition.

We consider a semi-classical nonlinear Schrödinger equation. For initial data causing focusing at one point in the linear case, we study a nonlinearity which is super-critical in terms of asymptotic effects near the caustic. We prove the existence of infinitely many phase shifts appearing at the approach of the critical time. This phenomenon is suggested by a formal computation. The rigorous proof shows a quantitatively different asymptotic behavior. We explain these aspects, and discuss some problems left open.

We consider the cubic defocusing Schrödinger equation in two space dimensions and prove that if the initial data are in H^1/2, then there exists a global solution in time. The proof combines the argument from [5] with some new correlation estimates for the Schrödinger equation.

We establish global regularity for the logarithmically energy-supercritical wave equation □u = u⁵log(2 + u²) in three spatial dimensions for spherically symmetric initial data, by modifying an argument of Ginibre, Soffer and Velo for the energy-critical equation. This example demonstrates that critical regularity arguments can penetrate very slightly into the supercritical regime.

The existence and the properties of isolated solutions to the relativistic Vlasov–Maxwell system with initial data on the backward hyperboloid are investigated. Isolated solutions of Vlasov–Maxwell can be defined by the condition that the particle density is compactly supported on the initial hyperboloid and by imposing the absence of incoming radiation on the electromagnetic field. Various consequences of the mass-energy conservation laws are derived by assuming the existence of smooth isolated solutions which match the inital data. In particular, it is shown that the mass-energy of isolated solutions on the backward hyperboloids and on the surfaces of constant proper time are preserved and equal, while the mass-energy on the forward hyperboloids is non-increasing and uniformly bounded by the mass-energy on the initial hyperboloid. Moreover the global existence and uniqueness of classical solutions in the future of the initial surface is established for the one-dimensional version of the system.

We prove that the Cauchy problem for the Dirac–Klein–Gordon equations in two space dimensions is locally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor, and an associated range of spaces of positive index for the meson field. In particular, we can go below the charge norm, that is, the L² norm of the spinor. We hope that this can have implications for the global existence problem, since the charge is conserved. Our result relies on the null structure of the system, and bilinear space-time estimates for the homogeneous wave equation.

Chemotaxis models are typically able to develop blow-up in finite times. For some specific models, we obtain some estimates on the set of concentration of cells (defined roughly as the points where the density of cells is infinite with a non-vanishing mass). More precisely we consider models without diffusion for which the cells' velocity decreases if the concentration of the chemical attractant becomes too large. We are able to give a lower bound on the Hausdorff dimension of the concentration set, one in the "best" situation where the velocity exactly vanishes for too large concentrations of attractant. This in particular implies that the solution may not form any Dirac mass.

The initial boundary value problem for the Euler–Poisson equations of the two-dimensional compressible rotating plasma flow with large data in L^∞ is studied in the isothermal case. The shock capturing method is used to construct the approximate solution. The uniform estimate and the H^-1 estimate of the entropy dissipation measures are obtained, and the compensated compactness method is applied to show the convergence of the approximate solution. The nonlocal effect of the Poisson equation is analyzed. The limit of the approximate solution is a weak entropy solution. Therefore the global weak entropy solution in L^∞ to the Euler–Poisson equations for rotating plasma flow is constructed and the global existence is established.