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ERC Starting Grant

SYSTEMS OF HYPERBOLIC CONSERVATION LAWS:
SINGULAR LIMITS, PROPERTIES OF SOLUTIONS AND CONTROL PROBLEMS


Principal Investigator

Stefano Bianchini, SISSA, Trieste

Members

Giovanni Alberti, University of Pisa
Fabio Ancona, University of Padova
Gianluca Crippa, University of Basel
Camillo De Lellis, University of Zuerich
Andrea Marson, University of Padova
Corrado Mascia, University of Roma

Research Program

The research program concerns various theoretic aspects of hyperbolic conservation laws.
In first place we plan to study the existence and uniqueness of solutions to systems of equations of mathematical physics with physic viscosity. This is one of the main open problems within the theory of conservation laws in one space dimension, which cannot be tackled relying on the techniques developed in the case where the viscosity matrix is the identity. Furthermore, this represents a first step toward the analysis of more complex relaxation and kinetic models with a finite number of velocities as for Broadwell equation, or with a continuous distribution of velocities as for Boltzmann equation.
A second research topic concerns the study of conservation laws with large data. Even in this case the basic model is provided by fluidodynamic equations. We wish to extend the results of existence, uniqueness and continuous dependence of solutions to the case of large (in BV or in L^infty) data, at least for the simplest systems of mathematical physics such as the isentropic gas dynamics.
A third research topic that we wish to pursue concerns the analysis of fine properties of solutions to conservation laws. Many of such properties depend on the existence of one or more entropies of the system. In particular, we have in mind to study the regularity and the concentration of the dissipativity measure for an entropic solution of a system of conservation laws.
Finally, we wish to continue the study of hyperbolic equations from the control theory point of view along two directions:
(i) the analysis of controllability and asymptotic stabilizability properties;
(ii) the study of optimal control problems related to hyperbolic systems.

Achievements

Hyperbolic Systems

The study of Hyperbolic Conservation Laws is a quite old subject, starting with the study of gas dynamics due to Riemann. Nowadays a quite complete theory is available, but it applies to general hyperbolic systems with small initial data. In physics, it is interesting to consider special systems with large data: an important example is the systems of gas dynamics. A first achievement is a short proof of the existence of solutions with large data and its extension to more general 2x2 genuinely nonlinear systems. An important topic is also the regularity of the solutions: the regularity results obtained are roughly speaking a justification of the pictures of the wave structure of solutions on finds in textbooks. Similar techniques have been used to prove SBV regularity for Hamilton-Jacobi equations. A sharp quantitative estimate of the compactness of solutions to scalar convex conservation laws has been obtained, answering to a question raised by Lax. An important improvement is the extension of a functional with stronger decay properties for general hyperbolic systems. This functional will allow new sharp estimates. A first application is the proof of sharp convergence estimates for the Glimm Scheme, a well known scheme for constructing solutions.

Transport Equation

We prove the well posedness for the one-dimensional ordinary differential equation, when no a priori assumptions on the compressibility of the velocity field are made. This result answers to questions raised by Ambrosio and by Perthame. An important analysis concerns the uniqueness for the transport equation in two space dimensions. In the simplest form, our result gives a characterization (i.e., provides a necessary and sufficient condition) of bounded, divergence-free vector fields on the plane such that the Cauchy problem for the associated transport equation has a unique bounded weak solution. Finally, this result can be applied to vector fields whose curl is a locally bounded signed measure. This answers (in the autonomous case) to a question raised by Lions, in connection with the study of vortex sheets for the two-dimensional Euler equation.

Euler Equations

The incompressible Euler equations were derived more than 250 years ago by Euler to describe the motion of an inviscid incompressible fluid. It is known since the pioneering works of Scheffer and Shnirelman that there are nontrivial distributional solutions to these equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. A celebrated theorem by Nash and Kuiper shows the existence of C^1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the threedimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. We pointed out that these two counterintuitive facts share many similarities. Our results (together with L\'aszl\'o Sz\'ekelyhidi) might be regarded as a first step towards the proof of a conjecture of Lars Onsager.

Activities

IPERMIB 2013 - 15th Italian Meeting on Hyperbolic Equations

Publications

Journal articles

  1. F. Ancona & A. Marson, A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems , Archive for Rational Mechanics and Analysis 196 (2010)
  2. F. Ancona & A. Marson, Sharp convergence rate of the Glimm scheme for general nonlinear hyperbolic systems, Communications in Mathematical Physics, Vol. 302 (2011)
  3. F. Ancona, O. Glass & K.T. Nguyen, Lower compactness estimates for scalar balance laws, Communications on Pure and Applied Mathematics (2011)
  4. Luigi Ambrosio, Gianluca Crippa, Alessio Figalli & Laura V. Spinolo, Some new well-posedness results for continuity and transport equations, and applications to the chromatography system, SIAM J. Math. Anal. 41 (2009)
  5. Gianluca Crippa, Lagrangian flows and the one-dimensional Peano phenomenon for ODEs, Journal of Differential Equations 250 (2011)
  6. Giovanni Alberti, Stefano Bianchini & Gianluca Crippa, A uniqueness result for the continuity equation in two dimensions, Journal of the European Mathematical Society, to appear
  7. Giovanni Alberti, Stefano Bianchini & Gianluca Crippa, Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, to appear
  8. Gianluca Crippa & Laura V. Spinolo, An overview on some results concerning the transport equation and its applications to conservation laws, Communications on Pure and Applied Analysis 9 (2010)
  9. Mahir Hadzic, Orthogonality conditions and stability in the Stefan problem with surface tension, ARMA, to appear
  10. S. Bianchini, C. De Lellis, R. Robyr, SBV regularity for Hamilton-Jacobi equations in R^n, Arch. Ration. Mech. Anal. 200 (2011)
  11. C. De Lellis, E. Spadaro, Q-valued functions revisited, Memoirs of the AMS 211 (2011)
  12. Y. Brenier, C. De Lellis, L. Szekelyhidi Jr., Weak-strong uniqueness for measure-valued Solutions, Comm. Math. Phys. 305 (2011)
  13. C. De Lellis, R. Robyr, Hamilton-Jacobi equations with obstacles, Arch. Ration. Mech. Anal. 200 (2011)
  14. S. Bianchini, R.M. Colombo, F. Monti, 2x2 Systems Of Conservation Laws With L^infty Data, JDE 249 (2010)
  15. S. Bianchini, L. V. Spinolo, Invariant Manifolds For A Singular Ordinary Differential Equation, J. Differential Equations 250 (2011)
  16. S. Bianchini, L. Caravenna, On The Extremality, Uniqueness And Optimality Of Transference Plans, Bullettin of the Institute of Mathematics, Academia Sinica Taipei (2009)
  17. S. Bianchini, L. Caravenna, On The Optimality Of Transference Plans, C. R. Acad. Sci. Paris, Ser. I 348 (2010)
  18. S. Bianchini, A. Brancolini, Estimates On Path Functionals Over Wasserstein Spaces, SIAM Math. Anal. 42 (2010)
  19. S. Bianchini, D. Tonon, A Decomposition Theorem For Bv Functions, CPAA 10 (2011)

Conference proceedings

  1. F. Ancona & A. Marson, On the Glimm functional for general hyperbolic systems, Proceedings of the “8th AIMS International Conference (Dresden, Germany, 2010)” (2010)
  2. Francois Bouchut & Gianluca Crippa, Equations de transport \`a coefficient dont le gradient est donn\'e par une int\'egrale singuli\`ere, \'Equations aux D\'eriv\'ees Partielles (2009)
  3. Giovanni Alberti, Stefano Bianchini & Gianluca Crippa, Two-Dimensional Transport Equation with Hamiltonian Vector Fields, Proceedings of the International Conference on Hyperbolic Problems ``HYP2008'' (2009)
  4. Luigi Ambrosio, Gianluca Crippa, Alessio Figalli & Laura V. Spinolo, Existence and uniqueness results for the continuity equation and applications to the chromatography system, Nonlinear Conservation Laws and Applications (2011)
  5. Giovanni Alberti, Stefano Bianchini & Gianluca Crippa, Divergence-free vector fields in R^2, Proceedings of the Fifth International Conference on Differential and Functional Differential Equations (2010)
  6. S. Bianchini, SBV Regularity For Scalar Conservation Laws, Proceedings of the 13th conference on Hyperbolic Problems: Theory, Numerics and Applications (2010)
  7. S. Bianchini, F. Cavalletti, The Monge Problem In Geodesic Spaces, Proceedings of the IMA Summer programm: nonlinear conservation laws and applications (2010)

Books

  1. E. Acerbi, C. Arezzo, G. Crippa, C. De Lellis, G. Mingione, Proceedings of the Intensive Research Month on Hyperbolic Conservation Laws and Fluid Dynamics, Rivista di Matematica della Universita' di Parma (2012)
  2. F. Ancona, S. Bianchini. R.M. Colombo, G. Crippa, A. Marson, Proceedings of the Seventh Meeting on Hyperbolic Conservation Laws and Fluid Dynamics: Recent Results and Research Perspectives, Rivista di Matematica della Universita' di Parma (2010)
  3. G. Alberti, F. Ancona, S. Bianchini, G. Crippa, C. DeLellis, A. Marson, C. Mascia, Lectures notes originating from the Intensive Trimester Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations: Analysis and Control, AIMS Book Series on Applied Mathematics, to appear

Chapters in books

  1. S. Bianchini, M. Gloyer, Transport Rays And Applications To Hamilton-Jacobi Equations, Lecture Notes in Mathematics 2028 (2011)