Journal of Hyperbolic Differential Equations

We prove that if the initial data do not belong to a certain subset of C^k, then the solutions of scalar conservation laws are piecewise C^k smooth. In particular, our initial data allow centered compression waves, which was the case not covered by Dafermos (1974) and Schaeffer (1973). More precisely, we are concerned with the structure of the solutions in some neighborhood of the point at which only a C^k+1 shock is generated. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except for a certain subset of 𝒮(ℝ) of the first category. It should be pointed out that this subset is smaller than those used in previous works. We point out that Thom's theory of catastrophes, which plays a key role in Schaeffer (1973), cannot be used to analyze the larger class of initial data considered in this paper.

We study travelling wave solutions describing non-isothermal subsonic phase transitions in a van der Waals fluid. We apply the center manifold method and establish the existence of viscosity-capillarity profiles for small viscosity, large heat conduction, and large heat capacity.

We construct a first order, physical space, parametrix for solutions to covariant, tensorial, wave equations on a general Lorentzian manifold. The construction is entirely geometric; that is both the parametrix and the error terms generated by it have a purely geometric interpretation. In particular, when the background Lorentzian metric satisfies the Einstein vacuum equations, the error terms, generated at some point p of the space-time, depend, roughly, only on the flux of curvature passing through the boundary of the past causal domain of p. The virtues of our specific geometric construction becomes apparent in applications to realistic problems. Though our main application is to General Relativity, which we discuss in [14], another simpler application shown here is to give a gauge invariant proof of the classical regularity result of Eardley–Moncrief [4, 5] for the Yang–Mills equations in ℝ¹⁺³.

Global weak solutions of bounded variation to systems of balance laws with non-local source are constructed by the method of vanishing viscosity. Suitable dissipativeness assumptions are imposed on the source terms to assure convergence of the method. Under these hypotheses, the total variation remains uniformly bounded and integrable in time, the vanishing viscosity solutions are uniformly stable in L¹ with respect to the initial data and converge to equilibrium as t → ∞. The motivation to study these systems is the observation that conservations laws with fading memory can be written in such form under appropriate conditions on the flux.

We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently "large" initial conditions, by giving a result which states that, for smooth "small" initial data, the solution remains at least Lipschitz continuous.

We derive an a posteriori error bound for Glimm's approximate solutions to convex scalar conservation laws containing only shock waves. Using Liu's wave-tracing method, we show that the L¹ norm of the error is bounded by a sum of residuals containing independent contributions from each wave in the approximate solution. We introduce a framework, similar to the method of characteristics, for the analysis of the local errors generated by wave interactions. The analysis allows for explicit cancellation among the errors created by a single wave and for error propagation along discontinuities.

We investigate the Cauchy problem for homogeneous equations of order m in the (t,x)-plane, with coefficients depending only on x. Assuming that the characteristic roots satisfy the condition

Benney introduced a general strategy for deriving systems of nonlinear partial differential equations associated with long- and short-wave solutions. The semi-linear Benney system was studied recently by Tsutsumi and Hatano. Here, we tackle the nonlinear version of it and using compensated compactness techniques, we prove the global existence of weak solutions to the Cauchy problem, in the case that the equation for the amplitude of the long wave is a quasilinear conservation law with flux f(v) = av² - bv³ where a, b are constants with b > 0.