Browsing by Author "Bianchini, Stefano"
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Item Characteristic boundary layers for mixed hyperbolic systems in one space dimension and applications to the Navier-Stokes and MHD equations(SISSA, 2018-10-16) Bianchini, Stefano; Spinolo, Laura; MathematicsWe 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.Item The decomposition of optimal transportation problems with convex cost(SISSA, 2014) Bianchini, Stefano; Bardelloni, MauroItem Differentiability in measure of the flow associated to a nearly incompressible BV vector field(SISSA, 2020-10-28) Bianchini, Stefano; De Nitti, Nicola; mathematicsItem Dissipative solutions to Hamiltonian systems(2022-09-12) Bianchini, Stefano; Leccese, Giacomo Maria; mathematicsItem Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux II(2016-06) Alberti, Giovanni; Bianchini, Stefano; Caravenna, Laura; MathematicsItem EXACT INTEGRABILITY CONDITIONS FOR COTANGENT VECTOR FIELDS(2021-07-28) Bianchini, Stefano; mathematicsItem EXISTENCE AND BLOW-UP FOR NON-AUTONOMOUS SCALAR CONSERVATION LAWS WITH VISCOSITY(2023-11-23) Bianchini, Stefano; Leccese, Giacomo MariaWe consider a question posed in [1], namely the blow-up of the PDE ut + (b(t, x)u1+k)x = uxx when b is uniformly bounded, Lipschitz and k = 2. We give a complete answer to the behavior of solutions when b belongs to the Lorentz spaces b ∈ Lp,∞, p ∈ (2,∞], or bx ∈ Lp,∞, p ∈ (1,∞].Item Forward untangling and applications to the uniqueness problem for the continuity equation(SISSA, 2020-05) Bianchini, Stefano; Bonicatto, PaoloItem Glimm interaction functional for BGK schemes(2005) Bianchini, Stefano; Mathematics; Functional Analysis and ApplicationsItem Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems(2008-12-15T12:03:09Z) Bianchini, Stefano; Spinolo, Laura; Mathematics; Functional Analysis and ApplicationsItem A Lagrangian approach for scalar multi-d conservation laws(2017-08) Bianchini, Stefano; Bonicatto, Paolo; Marconi, Elio; MathematicsItem Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian(SISSA, 2020-12) Bianchini, Stefano; Prerona, Dutta; Khai, Nguyen, T.; mathematicsItem The Monge problem in geodesic spaces(2010-06-07T11:02:41Z) Bianchini, Stefano; Cavalletti, Fabio; Mathematics; Functional Analysis and ApplicationsItem On Sudakov's type decomposition of transference plans with norm costs(SISSA, 2013) Bianchini, Stefano; Daneri, Sara; MathematicsItem On the sticky particle solutions to the multi-dimensional pressureless Euler equations(SISSA, 2020-05) Bianchini, Stefano; Daneri, SaraIn this paper we consider the multi-dimensional pressureless Euler system and we tackle the problem of existence and uniqueness of sticky particle solutions for general measure-type initial data. Although explicit counterexamples to both existence and uniqueness are known since [5], the problem of whether one can still find sticky particle solutions for a large set of data and of how one can select them was up to our knowledge still completely open. In this paper we prove that for a comeager set of initial data in the weak topology the pressureless Euler system admits a unique sticky particle solution given by a free flow where trajectories are disjoint straight lines. Indeed, such an existence and uniqueness result holds for a broader class of solutions de-creasing their kinetic energy, which we call dissipative solutions, and which turns out to be the compact weak closure of the classical sticky particle solutions. Therefore any scheme for which the energy is l.s.c. and is dissipated will converge, for a comeager set of data, to our solution, i.e. the free flow.Item On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension(SISSA, 2016) Bianchini, Stefano; Marconi, Elio; MathematicsWe 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.Item Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem(SISSA, 2007) Bianchini, Stefano; MathematicsItem Quadratic interaction functional for general systems of conservation laws(SISSA, 2014) Bianchini, Stefano; Modena, StefanoFor the Glimm scheme approximation u" to the solution of the system of conservation laws in one space dimension ut + f(u)x = 0; u(0; x) = u0(x) 2 Rn; with initial data u0 with small total variation, we prove a quadratic (w.r.t. Tot.Var.(u0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the ux 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: a new analysis of the interaction estimates of simple waves; a Lagrangian representation of the derivative of the solution, i.e. a map x(t;w) which follows the trajectory of each wave w from its creation to its cancellation; the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; a new functional Q controlling the variation in speed of the waves w.r.t. time. This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution Dttx(t;w) is a measure with total mass O(1)Tot.Var.(u0)2.Item Renormalization for autonomous nearly incompressible BV vector fields in 2D(SISSA, 2014-12) Bianchini, Stefano; Bonicatto, Paolo; Gusev, N.A.Given a bounded autonomous vector field $b \colon \R^d \to \R^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [4].Item Steady nearly incompressible vector elds in 2D: chain rule and renormalization(SISSA, 2014-08) Bianchini, Stefano; Gusev, N.A.; Mathematics