The boundary Riemann solver coming from the real vanishing viscosity approximation

We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems: v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x = \epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx} The conservative case is, in particular, included in the previous formulation. We suppose that the solutions $v^{\epsilon}$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$. Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added Section 3.1.2. Minor changes in other section

SBV regularity of Systems of Conservation Laws and Hamilton-Jacobi Equation

We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper

On a quadratic functional for scalar conservation laws

We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. The proof is based on the introduction of a quadratic functional (t), decreasing at every interaction, and such that its total variation in time is bounded. Differently from other interaction potentials present in the literature, the form of this functional is the natural extension of the original Glimm functional, and coincides with it in the genuinely nonlinear case

Quadratic interaction functional for systems of conservation laws: a case study

We prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws $$\left\{ \begin{array}{c} u_t + \tilde f(u,v)_x = 0, \\ v_t - v_x = 0. \end{array} \right.$$ This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme \cite{anc_mar_11_CMP}. Our aim is to extend the analysis, done for scalar conservation laws \cite{bia_mod_13}, in the presence of transversal interactions among wavefronts of different families. The proof is based on the introduction of a quadratic functional $\mathfrak Q(t)$, decreasing at every interaction, and such that its total variation in time is bounded. %cancellations it variation is controlled by the total variation growths at most of the total variation of the solution multiplied by the amount of cancellation. The study of this particular system is a key step in the proof of the quadratic interaction estimate for general systems: it requires a deep analysis of the wave structure of the solution $(u(t,x),v(t,x))$ and the reconstruction of the past history of each wavefront involved in an interaction

Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension

The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points \u398 and a countable family of Lipschitz curves T{script} such that outside T{script} 2a \u398 the solution is continuous, and for all points in T{script}{set minus}\u398 the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems

Quadratic interaction functional for general systems of conservation laws

For the Glimm scheme approximation u_\e to the solution of the system of conservation laws in one space dimension \begin{equation*} u_t + f(u)_x = 0, \qquad u(0,x) = u_0(x) \in \R^n, \end{equation*} with initial data $u_0$ with small total variation, we prove a quadratic (w.r.t. \TV(u_0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux $f$ are made (apart smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely we obtain the following results: \begin{itemize} \item a new analysis of the interaction estimates of simple waves; \item a Lagrangian representation of the derivative of the solution, i.e. a map $\mathtt x(t,w)$ which follows the trajectory of each wave $w$ from its creation to its cancellation; \item the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; \item a new functional $\mathfrak Q$ controlling the variation in speed of the waves w.r.t. time. \end{itemize} This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution $D_{tt} \mathtt x(t,w)$ is a measure with total mass \leq \const \TV(u_0)^2

Constraints on Cosmological Parameters from the 500 deg² SPTPOL Lensing Power Spectrum

We present cosmological constraints based on the cosmic microwave background (CMB) lensing potential power spectrum measurement from the recent 500 deg² SPTPOL survey, the most precise CMB lensing measurement from the ground to date. We fit a flat ΛCDM model to the reconstructed lensing power spectrum alone and in addition with other data sets: baryon acoustic oscillations (BAO), as well as primary CMB spectra from Planck and SPTPOL. The cosmological constraints based on SPTPOL and Planck lensing band powers are in good agreement when analyzed alone and in combination with Planck full-sky primary CMB data. With weak priors on the baryon density and other parameters, the SPTPOL CMB lensing data alone provide a 4% constraint on σ₈Ω^(0.25)_m = 0.593 ± 0.025. Jointly fitting with BAO data, we find σ₈ = 0.779±0.023, Ω_m = 0.368^(+0.032)_(−0.037), and H₀ = 72.0^(+2.1)_(−2.5)kms⁻¹ Mpc⁻¹, up to 2σ away from the central values preferred by Planck lensing + BAO. However, we recover good agreement between SPTPOL and Planck when restricting the analysis to similar scales. We also consider single-parameter extensions to the flat ΛCDM model. The SPTPOL lensing spectrum constrains the spatial curvature to be Ω_K = −0.0007±0.0025 and the sum of the neutrino masses to be ∑m_ν < 0.23 eV at 95% C.L. (with Planck primary CMB and BAO data), in good agreement with the Planck lensing results. With the differences in the signal-to-noise ratio of the lensing modes and the angular scales covered in the lensing spectra, this analysis represents an important independent check on the full-sky Planck lensing measurement

SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x)

In this paper we prove the special bounded variation regularity of the gradient of a viscosity solution of the Hamilton-Jacobi equation partial derivative(t)u + H(t, x, D(x)u) = 0 in Omega subset of [0, T] x R-n under the hypothesis of uniform convexity of the Hamiltonian H in the last variable. This result extends the result of Bianchini, De Lellis, and Robyr obtained for a Hamiltonian H = H(D(x)u) which depends only on the spatial gradient of the solution