Narrow Results

The Riemann problem for a hyperbolic system of conservation laws belonging to Temple class is considered within frame of $\alpha$-solutions based on the concept of solutions defined in the setting of a distributional product. We obtain all the possible discontinuous solutions within a convenient space of distributions that contains Dirac delta measures and discontinuous functions. In addition, an interesting discontinuous solution is found for certain Riemann initial conditions, in which the Dirac delta measure is involved in both of the two state variables simultaneously. As an application, a detailed example for the mentioned system but involving a linear damping term is provided, in which the trajectories of discontinuous solutions including contact discontinuity, shock wave and delta shock wave change in exponential forms.

For nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) we introduce a geometric framework in which solutions are sought as (continuous) parametrized graphs(t,s) ↦ (X,U)(t,s) satisfying ∂_sX ≥ 0, rather than (discontinuous) functions (t,x) ↦ u(t,x). On one hand, we generalize an idea by Dal Maso, LeFloch, and Murat who used a family of traveling wave profiles to define non-conservative products, and we define the notion of graph solution subordinate to a family of Riemann graphs. The latter naturally encodes the graph of the solution to the Riemann problem, which should be determined from an augmented model taking into account small-scale physics and providing an internal structure to the shock waves.

In a second definition, we write an evolution equation on the graphs directly and we introduce the notion of graph solution subordinate to a diffusion matrix, which merges together the hyperbolic equations (in the "non-vertical" parts of the graphs) with the traveling wave equation of the augmented model (in the "vertical" parts).

We consider the Cauchy problem within the class of graph solutions. The graph solution to the Cauchy problem is constructed by completion of the discontinuities of the entropy solution. The uniqueness is established by applying a general uniqueness theorem due to Baiti, LeFloch, and Piccoli. The proposed geometric framework illustrates the importance of the uniform distance between graphs to deal with solutions of nonlinear hyperbolic problems.

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schrödinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.

Conservation laws of the form ∂_tu + ∂_xf(x;u) = 0 with space-discontinuous flux f(x;⋅) = f_l(⋅)1_x<0 + f_r(⋅)1_x>0 were deeply investigated in the past ten years, with a particular emphasis in the case where the fluxes are "bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of f_l,r. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers are then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications.

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumption and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals' equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic -elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The autors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.