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Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. (English) Zbl 1098.35038
The aim of the paper is to give estimates of conservation or loss of regularity for the initial data of transport-diffusion equations \(\partial_tf+v\nabla f -\nu\Delta f= g\). The regularity is described in terms of inhomogeneous Besov spaces. Roughly speaking if \(\nabla v\) belongs to \(L^1(0,T;L^\infty)\) then the regularity of initial data is preserved. If \(\nabla v\) is less regular then the regularity may coarsen with time.

MSC:
35B45 A priori estimates in context of PDEs
35Q35 PDEs in connection with fluid mechanics
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[1] Bahouri, H. and Chemin, J.-Y.: Équations de transport relatives ‘ a des champs de vecteurs non lipschitziens et mécanique des fluides. Arch. Rational Mech. Anal. 127 (1994), 159-181. · Zbl 0821.76012
[2] Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246. · Zbl 0495.35024
[3] Chemin, J.-Y.: Perfect incompressible fluids. Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford Uni- versity Press, New York, 1998.
[4] Chemin, J.-Y.: Théor‘emes d’unicité pour le syst‘eme de Navier-Stokes tridimensionnel. J. Anal. Math. 77 (1999), 27-50. · Zbl 0938.35125
[5] Chemin, J.-Y. and Lerner, N.: Flot de champs de vecteurs non lip- schitziens et équations de Navier-Stokes. J. Differential Equations 121 (1995), 314-328. · Zbl 0878.35089
[6] Chemin, J.-Y. and Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33 (2001), 84-112. · Zbl 1007.76003
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