. Hence, T such that, for any (a, z) in

L. Ambrosio, Transport equation and Cauchy problem for BV vector elds, Invent. math, pp.227-260, 2004.
DOI : 10.1007/s00222-004-0367-2

L. Ambrosio and G. Crippa, Existence, uniqueness, stability and dierentiability properties of the ow associated to weakly dierentiable vector elds, Lect. Notes Unione Mat. Ital, vol.5, 2008.

H. Beirão and . Veiga, A new regularity class for the Navier-Stokes equations in R n , Chinese Annals Math, pp.407-412, 1995.

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coecients, Nonlinear Analysis, Theory, Methods and Applications, p.891933, 1998.
DOI : 10.1016/s0362-546x(97)00536-1
URL : http://www.labomath.univ-orleans.fr/~james/Postscripts/dualpap.ps

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness Comm, Partial Di. Eq, vol.24, pp.11-12, 1999.

F. Bouchut, F. James, and S. Mancini, Uniqueness and weak stability for multidimensional transport equations with one-sided Lipschitz coecients, Ann. Scuola Norm. Sup. Pisa Cl. Sci, issue.5, pp.1-25, 2005.

T. Buckmaster, C. De-lellis, L. Székelyhidi, and V. Vicol, Onsager's conjecture for admissible weak solutions

L. Caarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the navier-stokes equations, Communications on Pure and Applied Mathematics, vol.8, issue.6, pp.771-831, 1982.
DOI : 10.1090/trans2/065/03

J. Chemin and P. Zhang, On the critical one component regularity for the 3D Navier-Stokes equations, Ann. sci. de l'ENS, vol.49, issue.1, pp.131-167, 2016.

J. Chemin and P. Zhang, Remarks on the Global Solutions of 3-D Navier-Stokes System with One Slow Variable, Communications in Partial Differential Equations, vol.5, issue.5, 2015.
DOI : 10.1112/jlms/s2-35.2.303

G. Crippa and S. Spirito, Renormalized Solutions of the 2D Euler Equations, Communications in Mathematical Physics, vol.179, issue.3, p.339, 2015.
DOI : 10.1007/s00205-005-0390-5

C. De-lellis and L. Székelyhidi, On Admissibility Criteria for Weak Solutions of the Euler Equations, Archive for Rational Mechanics and Analysis, vol.136, issue.3, p.260, 2010.
DOI : 10.1007/978-3-642-56200-6_1

N. Depauw, Non unicit?? des solutions born??es pour un champ de vecteurs BV en dehors d'un hyperplan, Comptes Rendus Mathematique, vol.337, issue.4, p.252, 2003.
DOI : 10.1016/S1631-073X(03)00330-3

R. J. Diperna and P. Lions, Ordinary dierential equations, transport theory and Sobolev spaces, Invent. math, pp.511-547, 1989.

R. J. Diperna and J. Lions, On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability, The Annals of Mathematics, vol.130, issue.2, 1989.
DOI : 10.2307/1971423

E. Fabes, B. Jones, and N. M. Riviere, The initial value problem for the Navier-Stokes equations with data in L p , Archive Rational Mechanics Analysis 45, pp.222-248, 1972.

C. Fabre and G. Lebeau, R??GULARIT?? ET UNICIT?? POUR LE PROBL??ME DE STOKES, Communications in Partial Differential Equations, vol.29, issue.1
DOI : 10.1137/S0036144598334588

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, Journal of Differential Equations, vol.62, issue.2, pp.186-212, 1986.
DOI : 10.1016/0022-0396(86)90096-3

P. Isett, A Proof of Onsager's Conjecture, 2016.

L. Iskauriaza and G. A. , -solutions of the Navier-Stokes equations and backward uniqueness, Russian Mathematical Surveys, vol.58, issue.2, pp.211-250, 2003.
DOI : 10.1070/RM2003v058n02ABEH000609

T. Kato, Nonstationary ows of viscous and ideal uids in R 3, Journal of functional analysis, p.9, 1972.

C. , L. Bris, and P. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coecients, Comm. Part. Di. Eq, vol.33, issue.79, pp.1272-1317, 2008.

P. , L. Floch, and Z. Xin, Uniqueness via the adjoint problems for systems of conservation laws, Comm. on Pure and Applied Maths., XLVI, issue.11, pp.1499-1533, 1993.

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, vol.63, issue.0, pp.193-248, 1934.
DOI : 10.1007/BF02547354
URL : http://doi.org/10.1007/bf02547354

N. Lerner, Transport equations with partially BV velocities, Ann. Sc. Norm. Super. Pisa Cl
URL : https://hal.archives-ouvertes.fr/hal-00021015

G. Lévy, On uniqueness for a rough transport???diffusion equation, Comptes Rendus Mathematique, vol.354, issue.8, pp.804-807, 2016.
DOI : 10.1016/j.crma.2016.05.003

G. Lévy, A uniqueness lemma with applications to regularization and uid mechanics, accepté à Comm, Cont. Math, 2016.

G. Lévy, On an anisotropic Serrin criterion for weak solutions of the Navier-Stokes equations, soumis arXiv :1702, p.2814, 2017.

W. F. Osgood, Beweis der Existenz einer Lösung der Dierentialgleichung dy dx = f (x, y) ohne

V. Scheer, An inviscid ow with compact support in space-time, J. Goem. Anal, vol.3, issue.4, pp.343-401, 1993.

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, vol.9, issue.1, pp.187-195, 1962.
DOI : 10.1007/BF00253344

A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Communications on Pure and Applied Mathematics, vol.50, issue.12, pp.1261-1286, 1997.
DOI : 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6

A. Shnirelman, Weak Solutions with Decreasing Energy??of Incompressible Euler Equations, Communications in Mathematical Physics, vol.210, issue.3, pp.541-603, 2000.
DOI : 10.1007/s002200050791

M. Struwe, On partial regularity results for the navier-stokes equations, Communications on Pure and Applied Mathematics, vol.62, issue.4, pp.437-458, 1988.
DOI : 10.1007/978-3-663-13911-9

H. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes ow to ideal ow in R 3, Trans. Amer. Math. Soc, vol.157, 1971.

E. Tadmor, On a new scale of regularity spaces with applications to Euler ?s equations, Nonlinearity, vol.14, 2001.

W. Wahl, Regularity of weak solutions of the Navier-Stokes equations, Proc. Symp. Pure Math, pp.497-503, 1986.

R. Band, G. Berkolaiko, and T. Weyand, Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs, Journal of Mathematical Physics, vol.25, issue.4, p.56, 2015.
DOI : 10.1090/conm/173/01831

R. Band and G. Lévy, Quantum graphs which optimize their spectral gap, soumis aux Annales de l'Institut Henri Poincaré, 2016.
DOI : 10.1007/s00023-017-0601-2

G. Berkolaiko, A Lower Bound for Nodal Count on Discrete and Metric Graphs, Communications in Mathematical Physics, vol.25, issue.100, p.278, 2008.
DOI : 10.1007/s00220-007-0391-3

G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical surveys and monographs, p.186, 2013.

G. Berkolaiko and W. Liu, Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph, Journal of Mathematical Analysis and Applications, vol.445, issue.1, 2016.
DOI : 10.1016/j.jmaa.2016.07.026

G. Buttazzo, B. Runi, and B. Velichkov, Shape optimization problems for metric graphs, ESAIM : Control, Optimisation and Calculus of Variations, p.20, 2014.

Y. Colin-de-verdière, Semi-Classical Measures on Quantum Graphs and the Gau?? Map of the Determinant Manifold, Annales Henri Poincar??, vol.55, issue.2, 2014.
DOI : 10.1215/S0012-7094-87-05546-3

R. Courant, Ein allgemeiner Satz zur Theorie der Eigenfuktionen selbstadjungierter Dierentialausdrücke, Nachr. Ges. Wiss. Göttingen Math Phys, pp.81-84, 1923.

L. M. Del-pezzo and J. D. Rossi, The rst eigenvalue of the p-laplacian on quantum graphs, Analysis and Mathematical Physics, pp.1-27, 2016.

E. Dinits, A. Karzanov, and M. Lomonosov, On the structure of a family of minimal weighted cuts in graphs, Studies in Discrete Math, pp.290-306, 1976.

P. Exner and M. Jex, On the ground state of quantum graphs with attractive ?-coupling, Phys. Letters A, p.276, 2012.

G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber, Bayer. Akad. Wiss. München , Math.-Phys. Kl, 1923.

T. Fleiner and A. Frank, A quick proof for the cactus representation of mincuts, 2009.

L. Friedlander, Propri??t??s extr??males des valeurs propres d'un graphe m??trique, Annales de l???institut Fourier, vol.55, issue.1, 2005.
DOI : 10.5802/aif.2095

L. Friedlander, Genericity of simple eigenvalues for a metric graph, Israel Journal of Mathematics, vol.98, issue.1, 2005.
DOI : 10.1007/BF02773531

S. Gnutzmann and U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics, Advances in Physics, vol.19, issue.5-6, p.55, 2006.
DOI : 10.1007/BF01298245

S. Gnutzmann, U. Smilansky, and J. Weber, Nodal counting on quantum graphs, Waves in random media, p.14, 2004.
DOI : 10.1088/0959-7174/14/1/011
URL : http://arxiv.org/pdf/nlin/0305020

G. Karreskog and P. , Kurasov et I. Trygg Kupersmidt, Schrödinger operators on graphs : symmetrization and Eulerian cycles, Proc. Amer, p.144, 2016.

J. B. Kennedy, P. Kurasov, G. Malenová, and D. Mugnolo, On the Spectral Gap of a Quantum Graph, Annales Henri Poincar??, vol.421, issue.9, pp.1-35, 2016.
DOI : 10.1016/j.laa.2006.07.020

J. B. Kennedy and D. Mugnolo, The Cheeger constant of a quantum graph, PAMM, vol.16, issue.1, 2016.
DOI : 10.12693/APhysPolA.124.1060

T. Kottos and U. Smilansky, Periodic Orbit Theory and Spectral Statistics for Quantum Graphs, Annals of Physics, vol.274, issue.1, p.274, 1999.
DOI : 10.1006/aphy.1999.5904
URL : http://www.chaos.gwdg.de/~tsamp/papers/graph2.ps.gz

E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann, p.94, 1925.
DOI : 10.1007/bf01208645

E. Krahn, Über Minimaleigenschaft der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu, vol.9, 1926.

P. Kuchment, Graph models for waves in thin structures, Waves in random media, p.12, 2002.

P. Kuchment and H. Zeng, Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph, Journal of Mathematical Analysis and Applications, vol.258, issue.2, p.258, 2001.
DOI : 10.1006/jmaa.2000.7415

P. Kuchment and H. Zeng, Asymptotics of spectra of Neumann laplacians in thin domains, Advances in dierential equations and mathematical physics, 2003.

P. Kurasov, On the Spectral Gap for Laplacians on Metric Graphs, Acta Physica Polonica A, vol.124, issue.6, p.124, 2013.
DOI : 10.12693/APhysPolA.124.1060

P. Kurasov, G. Malenová, and S. Naboko, Spectral gap for quantum graphs and their edge connectivity, Journal of Physics A: Mathematical and Theoretical, vol.46, issue.27, p.46, 2013.
DOI : 10.1088/1751-8113/46/27/275309
URL : http://arxiv.org/pdf/1302.5022

P. Kurasov and S. Naboko, Rayleigh estimates for dierential operators on graphs, J. Spectr. Theory, p.4, 2014.
DOI : 10.4171/jst/67

H. Nagamochi and T. Ibaraki, Algorithmic Aspects of Graph Connectivity, 2008.
DOI : 10.1017/CBO9780511721649

S. Nicaise, Spectre des réseaux topologiques nis, Bull. Sci. Math, vol.111, 1987.

J. Pauling, The diamagnetic anisotropy of armoatic molecules, J. Chem. Physics, vol.4, 1936.

O. Post, Spectral analysis of metric graphs and related spaces, Limits of graphs in group theory and computer science, pp.109-140, 2009.

J. Rohleder, Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc arXiv :1602.03864v3 [math.SP], 2016.
DOI : 10.1090/proc/13403

J. Rubinstein and M. Schatzman, Spectral and variational problems on multiconnected strips, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.325, issue.4, p.325, 1997.
DOI : 10.1016/S0764-4442(97)85620-0

J. Rubinstein and M. Schatzman, Asymptotics for thin superconducting rings, Journal de Math??matiques Pures et Appliqu??es, vol.77, issue.8, p.77, 1998.
DOI : 10.1016/S0021-7824(98)80009-3
URL : https://doi.org/10.1016/s0021-7824(98)80009-3

J. Rubinstein and M. Schatzman, Variational Problems??on Multiply Connected Thin Strips I:??Basic Estimates and Convergence??of the Laplacian Spectrum, Archive for Rational Mechanics and Analysis, vol.160, issue.4, p.160, 2001.
DOI : 10.1007/s002050100164

K. Ruedenberg and C. W. Scherr, Free???Electron Network Model for Conjugated Systems. I. Theory, The Journal of Chemical Physics, vol.21, issue.9, 1953.
DOI : 10.1002/hlca.19490320705

Y. Saito, The limiting equation of the Neumann laplacians on shrinking domains, Electronic J. Di. Eq, p.31, 2000.

Y. Saito, Convergence of the Neumann Laplacian on shrinking domains, Analysis, vol.21, issue.2, p.21, 2001.
DOI : 10.1524/anly.2001.21.2.171

M. Schatzman, On the Eigenvalues of the Laplace Operator on a Thin Set with Neumann Boundary Conditions, Applicable Analysis, vol.4, issue.3-4, p.61, 1996.
DOI : 10.1007/978-3-642-96379-7

G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal, p.3, 1954.

H. F. Weinberger, An Isoperimetric Inequality for the N-Dimensional Free Membrane Problem, Indiana University Mathematics Journal, vol.5, issue.4, 1956.
DOI : 10.1512/iumj.1956.5.55021

H. Bahouri and J. , Chemin et R. Danchin, A frequency space for the Heisenberg group, 2016.

H. Bahouri, C. Fermanian-kammerer, and I. Gallagher, Phase space analysis and pseudodierential operators on the Heisenberg group, Astérisque, p.340, 2012.

H. Bahouri, C. Fermanian-kammerer, and I. Gallagher, Dispersive estimates for the Schr??dinger operator on step-2 stratified Lie groups, Analysis & PDE, vol.32, issue.3, pp.545-574, 2016.
DOI : 10.1090/S0002-9947-00-02750-1

R. Beals and P. Greiner, Calculus on Heisenberg manifolds, Annals of mathematics studies, vol.119, 1988.
DOI : 10.1515/9781400882397

L. Corwin and F. Greenleaf, Representations of nilpotent Lie groups and their applications Part 1 : basic theory and examples, Cambridge studies in advanced mathematics, 1990.

T. Coulhon, L. Salo-coste, and T. N. Varopoulos, Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, vol.100, 1992.

J. Faraut and K. Harzallah, Deux cours d'analyse harmonique, École d'été d'analyse harmonique de Tunis, Birkhäuser, 1984.

V. Fischer and M. Ruzhansky, A pseudo-dierential calculus on graded nilpotent Lie groups, Fourier analysis, Trends Math, pp.107-132, 1984.
DOI : 10.1007/978-3-319-02550-6_6
URL : http://arxiv.org/pdf/1209.2621

G. B. Folland, Harmonic analysis in phase space, Annals of mathematics studies, vol.122, 1989.
DOI : 10.1515/9781400882427

D. Geller, Fourier analysis on the Heisenberg group. I. Schwartz space, Journal of Functional Analysis, vol.36, issue.2, pp.205-254, 1980.
DOI : 10.1016/0022-1236(80)90100-7

W. Hebisch, Introduction to analysis on Lie groups, lecture notes (lectures given by W. Hebisch ), Mathematical notebooks, 2008.

L. Lavanya and S. Thangavelu, Revisiting the Fourier transform on the Heisenberg group, Publicacions Matem??tiques, vol.58, pp.47-63, 2014.
DOI : 10.5565/PUBLMAT_58114_03

F. W. Olver, Asymptotics and special functions, 1974.

J. Riordan, Combinatorial identities, 1968.

W. Rudin, Fourier analysis on groups, Interscience tracts in pure and applied mathematics, 1962.

E. M. Stein, Harmonic analysis, 1993.

M. E. Taylor, Noncommutative harmonic analysis, Mathematical surveys and monographs, American mathematical society Providence RI, vol.22, 1986.

S. Thangavelu, Harmonic analysis on the Heisenberg group, Progress in mathematics, Birkhäuser, p.159, 1998.

E. E. Tyrtyshnikov, A brief introduction to numerical analysis, 1997.
DOI : 10.1007/978-0-8176-8136-4

V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Cambridge studies in advanced mathematics, p.16, 1989.