arXiv subjects

Stefano Bianchini

Publications and source records attributed to Stefano Bianchini.

At least 37 records · Page 2Linked to original sources

arXiv2018

Characteristic boundary layers for mixed hyperbolic-parabolic systems in one space dimension, and applications to the Navier-Stokes and MHD equations

We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null conductivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to non-conservative systems.

math.AP↗

arXiv2017

Optimality of integrability estimates for advection-diffusion equations

We discuss $L^p$ integrability estimates for the solution $u$ of the advection-diffusion equation $\partial_t u + \mathrm{div} (bu) = Δu$, where the velocity field $b \in L^r_t L^q_x$. We first summarize some classical results proving such estimates for certain ranges of the exponents $r$ and $q$. Afterwards we prove the optimality of such ranges by means of new original examples.

math.AP↗

arXiv2016

On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension

We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. \\ In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f"=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

math.AP↗

arXiv2014

Quadratic interaction functional for general systems of conservation laws

For the Glimm scheme approximation $u_ε$ 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 \mathbb R^n, \end{equation*} with initial data $u_0$ with small total variation, we prove a quadratic (w.r.t. $\mathrm{TV}(u_0)$) 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: \newline - a new analysis of the interaction estimates of simple waves; \newline - 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; \newline - 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. \newline 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 $\leq \mathrm{const} \mathrm{TV}(u_0)^2$.

math.AP↗

arXiv2014

SBV regularity for Hamilton-Jacobi equations in $\mathbb R^n$

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in} Ω\subset \mathbb R\times \mathbb R^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(Ω)$.

math.AP↗

arXiv2014

The decomposition of optimal transportation problems with convex cost

Given a positive l.s.c. convex function $\mathtt c : \mathbb R^d \to \mathbb R^d$ and an optimal transference plane $\underlineπ$ for the transportation problem \begin{equation*} \int \mathtt c(x'-x) π(dxdx'), \end{equation*} we show how the results of \cite{biadan} on the existence of a \emph{Sudakov decomposition} for norm cost $\mathtt c= |\cdot|$ can be extended to this case. More precisely, we prove that there exists a partition of $\mathbb R^d$ into a family of disjoint sets $\{S^h_\mathfrak a\}_{h,\mathfrak a}$ together with the projection $\{O^h_\mathfrak a\}_{h,\mathfrak a}$ on $\mathbb R^d$ of proper extremal faces of $\mathrm{epi}\, \mathtt c$, $h = 0,\dots,d$ and $\mathfrak a \in \mathfrak A^h \subset \mathbb R^{d-h}$, such that - $S^h_\mathfrak a$ is relatively open in its affine span, and has affine dimension $h$; \item $O^h_\mathfrak a$ has affine dimension $h$ and is parallel to $S^h_\mathfrak a$; - $\mathcal L^d(\mathbb R^d \setminus \cup_{h,\mathfrak a} S^h_\mathfrak a) = 0$, and the disintegration of $\mathcal L^d$, $\mathcal L^d = \sum_h \int ξ^h_\mathfrak a η^h(d\mathfrak a)$, w.r.t. $S^h_\mathfrak a$ has conditional probabilities $ξ^h_\mathfrak a \ll \mathcal H^h \llcorner_{S^h_\mathfrak a}$; - the sets $S^h_\mathfrak a$ are essentially cyclically connected and cannot be further decomposed. \end{list} The last point is used to prove the existence of an optimal transport map. The main idea is to recast the problem in $(t,x) \in [0,\infty] \times \mathbb R^d$ with an $1$-homogeneous norm $\bar{\mathtt c}(t,x) := t \mathtt c(- \frac{x}{t})$ and to extend the regularity estimates of \cite{biadan} to this case.

math.CA↗

arXiv2014

Structure of entropy solutions to general scalar conservation laws in one space dimension

In this paper, we show that the entropy solution of a scalar conservation law is - continuous outside a $1$-rectifiable set $Ξ$, - up to a $\mathcal H^1$ negligible set, for each point $(\bar t,\bar x) \in Ξ$ there exists two regions where $u$ is left/right continuous in $(\bar t,\bar x)$. We provide examples showing that these estimates are nearly optimal. In order to achieve these regularity results, we extend the wave representation of the wavefront approximate solutions to entropy solution. This representation can the interpreted as some sort of Lagrangian representation of the solution to the nonlinear scalar PDE, and implies a fine structure on the level sets of the entropy solution.

math.AP↗

arXiv2014

Steady nearly incompressible vector fields in 2D: chain rule and renormalization

Given bounded vector field $b : \mathbb R^d \to \mathbb R^d$, scalar field $u : \mathbb R^d \to \mathbb R$ and a smooth function $β: \mathbb R \to \mathbb R$ we study the characterization of the distribution $\mathrm{div}(β(u)b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. In the case of $BV$ vector fields $b$ (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on so-called \emph{tangential set} of $b$. We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible $BV$ vector field $b$ and a bounded function $u$ for which this term is nonzero. For steady nearly incompressible vector fields $b$ (and under some further assumptions) in case when $d=2$ we provide complete characterization of $\mathrm{div}(β(u) b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. Our approach relies on the structure of level sets of Lipschitz functions on $\mathrm R^2$ obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution $u$ of $\partial_t u + b \cdot \nabla u=0$ is \emph{renormalized}, i.e. also solves $\partial_t β(u) + b \cdot \nabla β(u)=0$ for any smooth function $β: \mathbb R \to \mathbb R$. As a consequence we obtain new uniqueness result for this equation.

math.AP↗

arXiv2014

On Sudakov's type decomposition of transference plans with norm costs

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost $|\cdot|_{D^*}$ \[ \min \bigg\{\int |\mathtt T(x) - x|_{D^*} dμ(x), \ \mathtt T : \mathbb R^d \to \mathbb R^d, \ ν= \mathtt T_\# μ\bigg\}, \] with $μ$, $ν$ probability measures in $\mathbb R^d$ and $μ$ absolutely continuous w.r.t. $\mathcal L^d$. The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in $Z_\mathfrak a\times\mathbb R^d$, where $\{Z_\mathfrak a\}_{\mathfrak a\in\mathfrak A} \subset \mathbb R^d$ are disjoint regions such that the construction of an optimal map $\mathtt T_\mathfrak a : Z_\mathfrak a \to \mathbb R^d$ is simpler than in the original problem, and then to obtain $\mathtt T$ by piecing together the maps $\mathtt T_\mathfrak a$. In this paper we show how the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set $Z_\mathfrak a$ and then in $\mathbb R^d$. The strategy is sufficiently powerful to be applied to other optimal transportation problems.

math.CA↗

arXiv2013

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 \[ \begin{cases} u_t + \tilde f(u,v)_x = 0, \cr v_t - v_x = 0. \end{cases} \] This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. Our aim is to extend the analysis, done for scalar conservation laws, 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.

math.AP↗

arXiv2013

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 $\mathfrak Q(t)$, 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.

math.AP↗

arXiv2012

Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension

The paper gives an accurate description of the qualitative structure of an admissible BV solution to a strictly hyperbolic, piecewise genuinely nonlinear system of conservation laws. We prove that there are a countable set $Θ$ which contains all interaction points and a family of countably many Lipschitz curves $\T$ such that outside $\T\cup Θ$ $u$ is continuous, and along the curves in $\T$, u has left and right limit except for points in $Θ$. This extends the corresponding structural result in \cite{BL,Liu1} for admissible solutions. The proof is based on approximate wave-front tracking solutions and a proper selection of discontinuity curves in the approximate solutions, which converge to curves covering the discontinuities in the exact solution $u$.

math.AP↗

arXiv2012

SBV-like regularity for general hyperbolic systems of conservation laws

We prove the SBV regularity of the characteristic speed of the scalar hyperbolic conservation law and SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux function for general systems of conservation laws. More precisely, for the equation u_t + f(u)_x = 0, \quad u : \R^+ \times \R \to Ω\subset \R^N, we only assume the flux $f$ is $C^2$ function in the scalar case (N=1) and Jacobian matrix $Df$ has distinct real eigenvalues in the system case $(N\geq 2)$. Using the modification of the main decay estimate in Lau and localization method applied in \cite{R}, we show that for the scalar equation $f'(u)$ belongs to SBV, and for system of conservation laws the scalar measure \[\big(D_u λ_i(u) \cdot r_i(u) \big) \big(l_i(u) \cdot u_x \big)] has no Cantor part, where $λ_i$, $r_i$, $l_i$ are the $i$-th eigenvalue, $i$-th right eigenvector and $i$-th left eigenvector of the matrix $Df$.

math.AP↗

arXiv2011

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $μ_{i,\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $μ_{i,\mathrm{jump}}$ is positive.

math.AP↗

arXiv2011

The Monge Problem for distance cost in geodesic spaces

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X,dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1- dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

math.PR↗

arXiv2009

A connection between viscous profiles and singular ODEs

We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form $$ dV / dt = F(V) / z (V) . $$ Here $V \in R^d$ and the function F takes values into $R^d$ and is smooth. The real valued function z is as well regular: the equation is singular in the sense that z (V) can attain the value 0.

math.AP↗

arXiv2009

Invariant manifolds for a singular ordinary differential equation

We study the singular ordinary differential equation $$ \frac{d U}{d t} = f (U) / z (U) + g (U), $$ where $U \in R^N$, the functions $f \in R^N $ and $g \in R^N $ are of class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular in the sense that $z (U)$ can attain the value 0. We focus on the solutions of the singular ODE that belong to a small neighborhood of a point $\bar U$ such that $f (\bar U) = g (\bar U) = \vec 0$, $z (\bar U) =0$. We investigate the existence of manifolds that are locally invariant for the singular ODE and that contain orbits with a suitable prescribed asymptotic behaviour. Under suitable hypotheses on the set $\{U: z (U) = 0 \}$, we extend to the case of the singular ODE the definitions of center manifold, center stable manifold and of uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic-parabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation.

math.CA↗

arXiv2008

Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems

We are concerned with viscous profiles (travelling waves and steady solutions) for mixed hyperbolic-parabolic systems in one space variable. For a class of systems including the compressible Navier Stokes equation, these profiles satisfy a singular ordinary differential equation in the form \label{e:ab} dU / dt = F(U)/ z (U) . Here U takes values in $R^d$ and $F: R^d \to R^d$ is a regular function. The real valued function $z (U)$ is as well regular, but the equation is singular because $z (U)$ can attain the value 0. We focus on a small enough neighbourhood of a point $\bar U$ satisfying $F(\bar U) = \vec 0$, $z (\bar U) =0$. From the point of view of the applications to the study of hyperbolic-parabolic systems this means restricting to systems with small total variation. We discuss how to extend the notions of center manifold and of uniformly stable manifold. Also, we give conditions ensuring that if $z (U) \neq 0$ at $t=0$ then $z (U) \neq 0$ at every $t$. We provide an example showing that if $z(U)$ becomes zero in finite time then in general the solution $U$ of equation \eqref{e:ab} is not continuously differentiable.

math.AP↗