Journal of Hyperbolic Differential Equations

We look for the optimal range of Lebesque exponents for which inhomogeneous Strichartz estimates are valid. It is known that this range is larger than the one given by admissible exponents for homogeneous estimates. We prove inhomogeneous estimates in this larger range adopting the abstract setting and interpolation techniques already used by Keel and Tao for the endpoint case of the homogeneous estimates. Applications to Schrödinger equations are given, which extend previous work by Kato.

The mapping properties of the time evolution operator E(t) of nonlinear hyperbolic scalar conservation laws is investigated. It is shown that this operator is Lipschitz in the Hausdorff metric in one space dimension whenever the flux is convex and one of the initial conditions satisfies a one-sided Lipschitz condition. The Hausdorff distance between the graphs of the solutions measures the closeness in L^∞ in the regions where the solutions are smooth, as well as the closeness between the locations of shocks. A similar result on Hausdorff stability is proved with respect to a perturbation of the flux function. These results complement the well known L¹ contractivity of the solution operator. They are used in a subsequent paper to prove new smoothness results for solutions to such conservation laws. Negative results are proved in the case of non-convex and genuinely multidimensional fluxes.

The smoothness of the solutions of 1D scalar conservation laws is investigated and it is shown that if the initial value has smoothness of order α in L^q with α > 1 and q = 1/α, this smoothness is preserved at any time t > 0 for the graph of the solution viewed as a function in a suitably rotated coordinate system. The precise notion of smoothness is expressed in terms of a scale of Besov spaces which also characterizes the functions that are approximated at rate N^-α in the uniform norm by piecewise polynomials on N adaptive intervals. An important implication of this result is that a properly designed adaptive strategy should approximate the solution at the same rate N^-α in the Hausdorff distance between the graphs.

In this work we are concerned with a local existence of certain semi-linear wave equations for which the initial data has minimal regularity. Assuming the initial data are in H^1+∊ and H^∊ for any ∊ > 0, we prove a local result by using a fixed point argument, the main ingredient being an a priori estimate for the quadratic nonlinear term uDu. The technique applies to the Yang–Mills equations in the Lorentz gauge.

We consider the scattering problem for the nonlinear Klein–Gordon Equation with long range nonlinearity in one dimension. We prove that for all prescribed asymptotic solutions there is a solution of the equation with such behavior, for some choice of initial data. In the case the nonlinearity has the good sign (repulsive) the result hold for arbitrary size asymptotic data. The method of proof is based on reducing the long range phase effects to an ODE; this is done via an appropriate ansatz. We also find the complete asymptotic expansion of the solutions.

We consider non-decreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L¹-contraction property shown by Kružkov.

We consider coupled systems of a second order hyperbolic equation and a first order kinetic equation

We study the behavior of solutions of the Dirac–Maxwell system (DM) in the nonrelativistic limit c → ∞, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c → ∞, sometimes also called post-Newtonian, yields a Schrödinger–Poisson system, where the spin and magnetic field no longer appear.

We prove that DM is locally well-posed for H¹ data (for fixed c), and that as c → ∞ the existence time grows at least as fast as log(c), provided the data are uniformly bounded in H¹. Moreover, if the datum for the Dirac spinor converges in H¹, then the solution of DM converges, modulo a phase correction, in C([0,T];H¹) to a solution of a Schrödinger–Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit.

The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Klein–Gordon–Maxwell system.

In this paper we consider the thin film approximation of a 1D scalar conservation law with strictly convex flux. We prove that the sequence of approximate solutions converges to the unique Kružkov solution.

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non-characteristic problem in [19], will be the content of a subsequent paper.