Journal of Hyperbolic Differential Equations

Consider the focusing semilinear wave equation in ℝ³ with energy-critical nonlinearity

The optimal control problem for Burgers equation was first considered by Castro, Palacios and Zuazua. They proved the existence of a solution and proposed a numerical scheme to capture an optimal solution via the method of "alternate decent direction". In this paper, we introduce a new strategy for the optimal control problem for scalar conservation laws with convex flux. We propose a new cost function and by the Lax–Oleinik explicit formula for entropy solutions, the nonlinear problem is converted to a linear problem. Exploiting this property, we prove the existence of an optimal solution and, by a backward construction, we give an algorithm to capture an optimal solution.

We consider entropy solutions to the Cauchy problem for the isentropic compressible Euler equations in the spatially periodic case. In more than one space dimension, the methods developed by De Lellis–Székelyhidi enable us to show here failure of uniqueness on a finite time-interval for entropy solutions starting from any continuously differentiable initial density and suitably constructed bounded initial linear momenta.

We investigate the formation and propagation of singularities for the system of one-dimensional Chaplygin gas. Under suitable assumptions we construct a physically meaningful solution containing a new type of singularities called "delta-like" solution for this kind of quasilinear hyperbolic system with linearly degenerate characteristics. By a careful analysis, we study the behavior of the solution in a neighborhood of a blow-up point. The formation of this new kind of singularities is related to the envelop of different characteristic families, instead of characteristics of the same family in the standard situation. This shows that the blow-up phenomenon for systems with linearly degenerate characteristics is quite different from the problem of shock formation for the system with genuinely nonlinear characteristic fields. Different initial data can lead to different delta-like singularities: the delta-like singularity with point-shape and the delta-like singularity with line-shape.

This is the second (and last) part of a series in which we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in constant mean curvature and spatial harmonic (CMCSH) gauge, and we obtain a local well-posedness result in H^s with s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough Einstein metric g, we manage to implement the commuting vector field approach and prove a Strichartz estimate for the geometric wave equation □_g ϕ = 0 in a direct manner. This direct treatment would not work without gaining sufficient regularity on the background geometry. In this paper, we analyze the geometry of null hypersurfaces in rough Einstein spacetimes in terms of H^s data. We provide an integral control on the spatial supremum of the connection coefficients , ζ, which is crucially tied to the Strichartz estimates established in the first part.

We study a simple kinetic model with a conservative boundary condition which resembles the Maxwell diffuse boundary condition for the Boltzmann equation. We use a streamlined approach to construct the global pointwise structure of the full boundary data from the imposed boundary condition. Our estimates are strong enough to conclude the bifurcation in terms of the coefficients of the Broadwell model and the speed of the physical boundary.

We consider a variational kinetic formulation for weak, entropy solutions of scalar conservation laws due to Brenier. The solutions in this formulation are represented by a kinetic density function Y that solves a differential inclusion ∂_tY ∈ -A(Y) = -∂_vf ⋅ ∇_xY -∂ I_K(Y), where I_K is the indicator function of a closed, convex cone K. Under a certain "non-degeneracy" condition we determine a maximal monotone extension of A and use it to prove the existence of strong and weak solutions of the differential inclusion for a general, possibly degenerate, flux ∂_vf(v). Furthermore, we discuss several properties of strong solutions.

Non-uniform grids and mesh adaptation have become an important part of numerical approximations of differential equations over the past decades. It has been experimentally noted that mesh adaptation leads not only to locally improved solution but also to numerical stability of the underlying method. In this paper we consider nonlinear conservation laws and provide a method to perform the analysis of the moving mesh adaptation method, including both the mesh reconstruction and evolution of the solution. We moreover employ this method to extract sufficient conditions — on the adaptation of the mesh — that stabilize a numerical scheme in the sense of the entropy dissipation.