In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in}\quad \Omega\subset \mathbb{R}\times \mathbb{R}^{n} . $$ 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 $SBV_{loc}(\Omega)$.

1 aBianchini, Stefano1 aDe Lellis, Camillo1 aRobyr, Roger uhttp://hdl.handle.net/1963/491100415nas a2200121 4500008004100000245007100041210006900112260001000181100002200191700002300213700002100236856003600257 2011 en d00aStructure of level sets and Sard-type properties of Lipschitz maps0 aStructure of level sets and Sardtype properties of Lipschitz map bSISSA1 aAlberti, Giovanni1 aBianchini, Stefano1 aCrippa, Gianluca uhttp://hdl.handle.net/1963/465700413nas a2200121 4500008004100000245007000041210006800111260001000179100002200189700002300211700002100234856003600255 2011 en d00aA uniqueness result for the continuity equation in two dimensions0 auniqueness result for the continuity equation in two dimensions bSISSA1 aAlberti, Giovanni1 aBianchini, Stefano1 aCrippa, Gianluca uhttp://hdl.handle.net/1963/466301221nas a2200109 4500008004300000245005800043210005800101520086900159100002301028700002401051856003601075 2010 en_Ud 00aEstimates on path functionals over Wasserstein Spaces0 aEstimates on path functionals over Wasserstein Spaces3 aIn this paper we consider the class a functionals (introduced in [Brancolini, Buttazzo, and Santambrogio, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415-434] $\\\\mathcal{G}_{r,p}$ defined on Lipschitz curves $\\\\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\\\\mu$, give conditions in order to assure the existence of a curve $\\\\gamma$ such that $\\\\gamma(0)=\\\\mu$, $\\\\gamma(1)=\\\\delta_{x_0}$, and $\\\\mathcal{G}_{r,p}(\\\\gamma)<+\\\\infty$. To this end, new estimates on $\\\\mathcal{G}_{r,p}(\\\\mu)$ are given, and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions.1 aBianchini, Stefano1 aBrancolini, Alessio uhttp://hdl.handle.net/1963/358300371nas a2200097 4500008004300000245008300043210006900126100002300195700001900218856003600237 2010 en_Ud 00aOn the Euler-Lagrange equation for a variational problem : the general case II0 aEulerLagrange equation for a variational problem the general cas1 aBianchini, Stefano1 aGloyer, Matteo uhttp://hdl.handle.net/1963/255100300nas a2200097 4500008004300000245004100043210003700084100002300121700002200144856003600166 2010 en_Ud 00aThe Monge problem in geodesic spaces0 aMonge problem in geodesic spaces1 aBianchini, Stefano1 aCavalletti, Fabio uhttp://hdl.handle.net/1963/387301159nas a2200121 4500008004300000245006200043210005800105260001300163520078100176100002300957700002100980856003601001 2010 en_Ud 00aOn optimality of c-cyclically monotone transference plans0 aoptimality of ccyclically monotone transference plans bElsevier3 aAbstract. This note deals with the equivalence between the optimality of a transport plan for the Monge-Kantorovich problem and the condition of c-cyclical monotonicity, as an outcome of the construction in [7]. We emphasize the measurability assumption on the hidden structure of linear preorder, applied also to extremality and uniqueness problems. Resume. Dans la presente note nous decrivons brievement la construction introduite dans [7] a propos de l\\\'equivalence entre l\\\'optimalite d\\\'un plan de transport pour le probleme de Monge-Kantorovich et la condition de monotonie c-cyclique ainsi que d\\\'autres sujets que cela nous amene a aborder. Nous souhaitons mettre en evidence l\\\'hypothese de mesurabilite sur la structure sous-jacente de pre-ordre lineaire.1 aBianchini, Stefano1 aCaravenna, Laura uhttp://hdl.handle.net/1963/402302193nas a2200109 4500008004300000245008700043210006900130520180600199100002302005700001902028856003602047 2009 en_Ud 00aThe boundary Riemann solver coming from the real vanishing viscosity approximation0 aboundary Riemann solver coming from the real vanishing viscosity3 aWe study the limit of the hyperbolic-parabolic approximation $$ \\\\begin{array}{lll} v_t + \\\\tilde{A} ( v, \\\\, \\\\varepsilon v_x ) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in R^N\\\\\\\\ \\\\tilde \\\\beta (v (t, \\\\, 0)) = \\\\bar g \\\\\\\\ v (0, \\\\, x) = \\\\bar v_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nThe function $\\\\tilde \\\\beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\\\\tilde \\\\beta$ is not invertible.\\nThe data $\\\\bar g$ and $\\\\bar v_0$ are constant. When $\\\\tilde B$ is invertible, the previous problem takes the simpler form $$ \\\\left\\\\{ \\\\begin{array}{lll} v_t + \\\\tilde{A} \\\\big( v, \\\\, \\\\varepsilon v_x \\\\big) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in \\\\mathbb{R}^N\\\\\\\\ v (t, \\\\, 0) \\\\equiv \\\\bar v_b \\\\\\\\ v (0, \\\\, x) \\\\equiv \\\\bar{v}_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nAgain, the data $\\\\bar v_b$ and $\\\\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, 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 $\\\\tilde A$ can be 0.\\n Second, as pointed out before we take into account the possibility that $\\\\tilde B$ 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.1 aBianchini, Stefano1 aSpinolo, Laura uhttp://hdl.handle.net/1963/183100337nas a2200097 4500008004300000245006000043210005800103100002300161700001900184856003600203 2009 en_Ud 00aA connection between viscous profiles and singular ODEs0 aconnection between viscous profiles and singular ODEs1 aBianchini, Stefano1 aSpinolo, Laura uhttp://hdl.handle.net/1963/255500624nas a2200109 4500008004300000245007200043210006400115520025500179100002300434700002100457856003600478 2009 en_Ud 00aOn the extremality, uniqueness and optimality of transference plans0 aextremality uniqueness and optimality of transference plans3 aWe consider the following standard problems appearing in optimal mass transportation theory: when a transference plan is extremal; when a transference plan is the unique transference plan concentrated on a set A,; when a transference plan is optimal.1 aBianchini, Stefano1 aCaravenna, Laura uhttp://hdl.handle.net/1963/369200336nas a2200097 4500008004300000245005900043210005500102100002300157700002200180856003600202 2009 en_Ud 00aThe Monge problem for distance cost in geodesic spaces0 aMonge problem for distance cost in geodesic spaces1 aBianchini, Stefano1 aCavalletti, Fabio uhttp://hdl.handle.net/1963/370300382nas a2200097 4500008004300000245009400043210006900137100002300206700001900229856003600248 2008 en_Ud 00aInvariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems0 aInvariant Manifolds for Viscous Profiles of a Class of Mixed Hyp1 aBianchini, Stefano1 aSpinolo, Laura uhttp://hdl.handle.net/1963/340000781nas a2200133 4500008004100000020002200041245006700063210006700130260001300197520035900210100002300569700001900592856003600611 2008 en d a978-3-642-21718-000aTransport Rays and Applications to Hamilton–Jacobi Equations0 aTransport Rays and Applications to Hamilton–Jacobi Equations bSpringer3 aThe aim of these notes is to introduce the readers to the use of the Disintegration Theorem for measures as an effective tool for reducing problems in transport equations to simpler ones. The basic idea is to partition Rd into one dimensional sets, on which the problem under consideration becomes one space dimensional (and thus much easier, hopefully).1 aBianchini, Stefano1 aGloyer, Matteo uhttp://hdl.handle.net/1963/546300645nas a2200121 4500008004300000245011200043210006900155520019600224100002300420700002200443700002200465856003600487 2007 en_Ud 00aAsymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy0 aAsymptotic behaviour of smooth solutions for partially dissipati3 aWe study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition.1 aBianchini, Stefano1 aHanouzet, Bernard1 aNatalini, Roberto uhttp://hdl.handle.net/1963/178000302nas a2200085 4500008004300000245006100043210005300104100002300157856003600180 2007 en_Ud 00aOn the Euler-Lagrange equation for a variational problem0 aEulerLagrange equation for a variational problem1 aBianchini, Stefano uhttp://hdl.handle.net/1963/179200408nas a2200097 4500008004100000245011900041210006900160260001000229100002300239856004800262 2007 en d00aPerturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem0 aPerturbation techniques applied to the real vanishing viscosity bSISSA1 aBianchini, Stefano uhttp://preprints.sissa.it/handle/1963/3531500319nas a2200085 4500008004300000245006900043210006200112100002300174856003600197 2006 en_Ud 00aOn Bressan\\\'s conjecture on mixing properties of vector fields0 aBressans conjecture on mixing properties of vector fields1 aBianchini, Stefano uhttp://hdl.handle.net/1963/180600286nas a2200085 4500008004300000245004900043210004900092100002300141856003600164 2006 en_Ud 00aGlimm interaction functional for BGK schemes0 aGlimm interaction functional for BGK schemes1 aBianchini, Stefano uhttp://hdl.handle.net/1963/177001264nas a2200121 4500008004300000245006600043210006600109260002600175520086100201100002301062700002101085856003601106 2005 en_Ud 00aVanishing viscosity solutions of nonlinear hyperbolic systems0 aVanishing viscosity solutions of nonlinear hyperbolic systems bAnnals of Mathematics3 aWe consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$.1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/307400845nas a2200109 4500008004100000245007300041210006900114260001800183520047500201100002300676856003600699 2003 en d00aA note on singular limits to hyperbolic systems of conservation laws0 anote on singular limits to hyperbolic systems of conservation la bSISSA Library3 aIn this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. \\nUnder the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation.1 aBianchini, Stefano uhttp://hdl.handle.net/1963/154200816nas a2200121 4500008004300000245005800043210005600101260004800157520040900205100002300614700002100637856003600658 2002 en_Ud 00aA center manifold technique for tracing viscous waves0 acenter manifold technique for tracing viscous waves bAmerican Institute of Mathematical Sciences3 aIn this paper we introduce a new technique for tracing viscous travelling profiles. To illustrate the method, we consider a special 2 x 2 hyperbolic system of conservation laws with viscosity, and show that any solution can be locally decomposed as the sum of 2 viscous travelling profiles. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data.1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/307500788nas a2200121 4500008004100000245008600041210006900127260001300196520037700209100002300586700002100609856003600630 2002 en d00aOn a Lyapunov functional relating shortening curves and viscous conservation laws0 aLyapunov functional relating shortening curves and viscous conse bElsevier3 aWe study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law.1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/133700355nas a2200109 4500008004100000245005500041210004800096260001800144100002300162700002400185856003600209 2002 en d00aOn the Stability of the Standard Riemann Semigroup0 aStability of the Standard Riemann Semigroup bSISSA Library1 aBianchini, Stefano1 aColombo, Rinaldo M. uhttp://hdl.handle.net/1963/152800359nas a2200109 4500008004300000245004000043210003800083260004800121100002300169700002100192856003600213 2001 en_Ud 00aA case study in vanishing viscosity0 acase study in vanishing viscosity bAmerican Institute of Mathematical Sciences1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/309100342nas a2200097 4500008004100000245006700041210006400108260001300172100002300185856003600208 2001 en d00aA Glimm type functional for a special Jin-Xin relaxation model0 aGlimm type functional for a special JinXin relaxation model bElsevier1 aBianchini, Stefano uhttp://hdl.handle.net/1963/135501077nas a2200109 4500008004100000245010100041210006900142260001800211520067900229100002300908856003600931 2001 en d00aStability of L-infinity solutions for hyperbolic systems with coinciding shocks and rarefactions0 aStability of Linfinity solutions for hyperbolic systems with coi bSISSA Library3 aWe consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\\\\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \\\\mathcal{S}_t u_0$, defined on initial data $u_0 \\\\in L^\\\\infty$. The semigroup $\\\\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\\\\text{loc}}$ topology. Moreover $\\\\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.1 aBianchini, Stefano uhttp://hdl.handle.net/1963/152300394nas a2200109 4500008004300000245005900043210005900102260004300161100002300204700002100227856003600248 2000 en_Ud 00aBV solutions for a class of viscous hyperbolic systems0 aBV solutions for a class of viscous hyperbolic systems bIndiana University Mathematics Journal1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/319400824nas a2200109 4500008004300000245008300043210006900126260002300195520043700218100002300655856003600678 2000 en_Ud 00aThe semigroup generated by a Temple class system with non-convex flux function0 asemigroup generated by a Temple class system with nonconvex flux bKhayyam Publishing3 aWe consider the Cauchy problem for a nonlinear n × n system of conservation laws of Temple class, i.e. with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.1 aBianchini, Stefano uhttp://hdl.handle.net/1963/322100416nas a2200097 4500008004100000245010100041210006900142260004800211100002300259856003600282 2000 en d00aOn the shift differentiability of the flow generated by a hyperbolic system of conservation laws0 ashift differentiability of the flow generated by a hyperbolic sy bAmerican Institute of Mathematical Sciences1 aBianchini, Stefano uhttp://hdl.handle.net/1963/127400360nas a2200097 4500008004300000245007800043210006900121260001300190100002300203856003600226 1999 en_Ud 00aExtremal faces of the range of a vector measure and a theorem of Lyapunov0 aExtremal faces of the range of a vector measure and a theorem of bElsevier1 aBianchini, Stefano uhttp://hdl.handle.net/1963/337000387nas a2200109 4500008004100000245006900041210006900110260001800179100002300197700002100220856003600241 1999 en d00aVanishing viscosity solutions of hyperbolic systems on manifolds0 aVanishing viscosity solutions of hyperbolic systems on manifolds bSISSA Library1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/123800352nas a2200109 4500008004300000245005500043210005100098260001300149100002300162700002100185856003600206 1999 en_Ud 00aThe vector measures whose range is strictly convex0 avector measures whose range is strictly convex bElsevier1 aBianchini, Stefano1 aMariconda, Carlo uhttp://hdl.handle.net/1963/3546