Projective properties of divergence-free symmetric tensors, and new dispersive estimates in gas dynamics. (English) Zbl 07462041

Summary: The class of Divergence-free symmetric tensors is ubiquitous in Continuum Mechanics. We show its invariance under projective transformations of the independent variables. This action, which preserves the positiveness, extends Sophus Lie’s group analysis of Newtonian dynamics. When applied to models of gas dynamics – such as Euler system or Boltzmann equation, – in combination with Compensated Integrability, this yields new dispersive estimates. The most accurate one is obtained for mono-atomic gases. Then the space-time integral of \(t\rho^{\frac{1}{d}} p\) is bounded in terms of the total mass and moment of inertia alone.


35Q31 Euler equations
35B06 Symmetries, invariants, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B45 A priori estimates in context of PDEs
35F35 Systems of linear first-order PDEs
35Q20 Boltzmann equations
Full Text: DOI arXiv


[1] Bobylev, AV; Ibragimov, NKh, Intenconnectivity of symmetry properties for equations of dynamics, kinetic theory of gases, and hydrodynamics (in Russian), Mat. Mod., 1, 100-109 (1989) · Zbl 0974.76580
[2] Bobylev, AV; Vilasi, G., Projective invariance for classical and quantum systems, J. Group Theory Phys., 3, 101-115 (1995)
[3] De Lellis, C.; Székelyhidi, L., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195, 225-260 (2010) · Zbl 1192.35138
[4] DiPerna, R.; Lions, P-L, On the Cauchy problem for the Boltzmann equation: global existence and weak stability results, Ann. Math., 130, 321-366 (1990) · Zbl 0698.45010
[5] Illner, R.: Stellar dynamics and plasma physics with corrected potentials: Vlasov, Manev, Boltzman, Smoluchowski. In: Fields Institute Communications: Hydrodynamic limits and Related topics. vol. 27, pp. 95-108. AMS publications (2000) · Zbl 1067.85001
[6] Lions, P.-L., Masmoudi, N.: From the Boltzmann equation to the equations of incompressible fluid mechanics. II. Arch. Rat. Mech. Anal. 158, 195-211 (2001) · Zbl 0987.76088
[7] Pogorelov, A.V.: The Minkowski multidimensional problem. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; Halsted Press (Wiley), New York, Toronto, London (1978) · Zbl 0387.53023
[8] Serre, D.: Solutions classiques globales des équations d’Euler pour un fluide parfait incompressible. Ann. l’Inst. Fourier 47, 139-153 (1997) · Zbl 0864.35069
[9] Serre, D.: Divergence-free positive symmetric tensors and fluid dynamics. Ann. l’Inst. Henri Poincaré (analyse non linéaire) 35, 1209-1234 (2018) · Zbl 1393.35181
[10] Serre, D., Compensated integrability. Applications to the Vlasov-Poisson equation and other models of mathematical physics, J. Math. Pures Appl., 127, 67-88 (2019) · Zbl 1417.37235
[11] Serre, D.: A priori estimates from first principles in gas dynamics. Waves in Flows (Prague 2018). In: Bodnár, T., Galdi, G.P., Nečasová, S. (eds.) Advances in Mathematical Fluid Mechanics, pp. 1-47. Birkhäuser (2021) · Zbl 1478.76059
[12] Serre, D., Hard spheres dynamics: weak vs strong collisions, Arch. Rat. Mech. Anal., 240, 243-264 (2021) · Zbl 1476.70060
[13] Serre, D.; Silvestre, L., Multi-dimensional Burgers equation with unbounded initial data: Well-posedness and dispersive estimates, Arch. Rat. Mech. Anal., 234, 1391-1411 (2019) · Zbl 07114396
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.