Journal of Hyperbolic Differential Equations

We prove new estimates in Besov spaces or for transport-diffusion equations. Comparing with previous works, there is no restriction on the index p (hence our estimates apply in Hölder spaces) and the transport vector-field need not be divergence-free. Besides, similar estimates can be proved for the non-stationary Stokes equations with convection. We expect our results to be useful for studying compressible or incompressible fluid mechanics with high Reynolds number.

This paper is concerned with the circumstances under which the dissipative character of a one-dimensional scalar conservation law may be described by a formalism strictly analogous to that arising naturally in the dynamics of nonlinearly elastic materials. It is shown that this occurs if and only if the entropy density, entropy flux pair associated with the conservation law takes a particular form. We compare the admissibility condition associated with this special entropy with other admissibility criteria such as those of Lax, Oleinik and regularization theory. Using the special entropy, we consider the Riemann problem for an example in which genuine nonlinearity fails and a kinetic relation is needed to determine a unique solution.

We show how to construct globally defined multipeakon solutions of the Camassa–Holm equation. The construction includes in particular the case with peakon-antipeakon collisions. The solutions are conservative in the sense that the associated energy is constant for almost all times. Furthermore, we construct a new set of ordinary differential equations that determines the multipeakons globally. The system remains globally well-defined.

For a fluid-radiation mixture, i.e. a relativistic gas consisting of material particles and photons, we derive a 14D causal evolution system of first-order symmetric hyperbolic form for a set of basic physical gas-state variables, which is defined relative to a network of observers with arbitrary 4-velocities. This system is obtained by taking moments of the relativistic Boltzmann equation (the so-called central moments) and by truncating and closing the resulting system of moment equations by means of an appropriate (5+9)-moment closure prescription. The material medium under consideration is assumed to be a nonbarotropic perfect gas described by a local Jüttner equilibrium distribution function. With regards to the photon gas, we employ the modified Grad-type approach, which begins by expanding the phase density (i.e. the number density of photons) about a local Planck equilibrium distribution and including the trace-free anisotropic pressure in the expansion. The main advantage of using the above (5+9)-moment closure prescription is that the matter fluid's peculiar velocity and the radiative heat flux are incorporated into the model in a non-perturbative manner, thereby allowing virtually arbitrarily large values for the individual components of these dynamical variables. Within the framework of a reduced 3 + 1 orthonormal frame formalism, we also explain how to choose the initially arbitrary 4-velocity vector field, as this is crucial for the determination of the full 38D causal matter-radiation-gravity system.

We are interested in properties of the multidimensional Euler equations for compressible fluids. Rarefaction waves are the unique solutions that may contain vacuum states in later time, in the context of one-dimensional Riemann problem, even when the Riemann initial data are away from the vacuum. For the multidimensional Euler equations describing isentropic or adiabatic fluids, we prove that plane rarefaction waves and vacuum states are stable within a large class of entropy solutions that may contain vacuum states. Rarefaction waves and vacuum states are also shown to be global attractors of entropy solutions in L^∞, provided initial data are L^∞ ∩ L¹ perturbations of Riemann initial data. Our analysis applies to entropy solutions with arbitrarily large oscillation, and no bounded variation regularity is required.

We study the relaxation of kinetic BGK models involving a high-field term within the transport operator. They lead us to multidimensional scalar conservation laws with a flux-function which is perturbed with respect to classical relaxation. The proof of the relaxation limit makes modified Maxwellians to appear. We consider "pseudo distribution functions" which can take negative values and we introduce appropriate admissible states. This is a first step towards adapting this analysis to quantum kinetic BGK models.

We are concerned with the initial value problem for a damped wave equation with a nonlinear convection term which is derived from a semilinear hyperbolic system with relaxation. We show the global existence and asymptotic decay of solutions in W^1,p (1 ≤ p ≤ ∞) under smallness condition on the initial data. Moreover, we show that the solution approaches in W^1,p (1 ≤ p ≤ ∞) the nonlinear diffusion wave expressed in terms of the self-similar solution of the Burgers equation as time tends to infinity. Our results are based on the detailed pointwise estimates for the fundamental solutions to the linearlized equation.