## Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain.(English)Zbl 1248.76011

The author considers incompressible Euler equations in a bounded domain $$\Omega\subset\mathbb{R}^2$$, which may be multiply connected. He studies the regularity of the flow map. Precisely speaking, he assumes that the initial value of the solution belongs to $$L^2 (\Omega )$$, it has a bounded vorticity, and it is tangent to the boundary $$\partial\Omega$$. He proves that if $$\partial\Omega$$ is smooth, then the flow map is smooth in time. Furthermore, if $$\partial\Omega$$ has Gevrey regularity of degree $$M\geq 1$$, then the flow map has Gevrey regularity of degree $$M+2$$. Similar results were proved also by J.Y. Chemin, P. Serfati, and P. Gamblin. The author also considers the case of initial value with Hölder regularity, in this case, a similar result is true, modifying the Gevrey degree slightly.
Some generalizations are also considered. First, the case is studied when the initial value has an unbounded vorticity. It is shown that in this case one can prove a simlar result, modifying the function space with respect to space variables. Secondly, the author studies the case when $$\partial\Omega$$ has only $$C^2$$ regularity. In this case, if the initial value has a constant vorticity outside of a compact subset of $$\Omega$$, then the flow map has Gevrey regularity of a certain degree.

### MSC:

 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76B47 Vortex flows for incompressible inviscid fluids 35Q31 Euler equations
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### References:

 [1] Ambrosio, L., Transport equation and Cauchy problem for BV vector fields, Invent. math., 158, 2, 227-260, (2004) · Zbl 1075.35087 [2] Bahouri, H.; Chemin, J.-Y., Équations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. ration. mech. anal., 127, 2, 159-181, (1994) · Zbl 0821.76012 [3] Bahouri, H.; Chemin, J.-Y.; Danchin, R., Fourier analysis and nonlinear partial differential equations, (2011), Springer · Zbl 1227.35004 [4] P. Bernard, Some remarks on the continuity equation, Séminaire EDP de lʼEcole Polytechnique, 2008-2009. [5] Burch, C., The dini condition and regularity of weak solutions of elliptic equations, J. differential equations, 30, 3, 308-323, (1978) · Zbl 0362.35021 [6] Chemin, J.-Y., Sur le mouvement des particules dʼun fluide parfait incompressible bidimensionnel, Invent. math., 103, 3, 599-629, (1991) · Zbl 0739.76010 [7] Chemin, J.-Y., Régularité de la trajectoire des particules dʼun fluide parfait incompressible remplissant lʼespace, J. math. pures appl. (9), 71, 5, 407-417, (1992) · Zbl 0833.35112 [8] Chemin, J.-Y., Fluides parfaits incompressibles, Astérisque, 230, (1995) · Zbl 0829.76003 [9] Danchin, R., Évolution temporelle dʼune poche de tourbillon singulière, Comm. partial differential equations, 22, 5-6, 685-721, (1997) · Zbl 0882.35093 [10] DiPerna, R.J.; Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. math., 98, 3, 511-547, (1989) · Zbl 0696.34049 [11] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, Classics math., (2001), Springer-Verlag · Zbl 1042.35002 [12] Kato, T., On classical solutions of the two-dimensional nonstationary Euler equation, Arch. ration. mech. anal., 25, 188-200, (1967) · Zbl 0166.45302 [13] Kelliher, J., On the flow map for 2D Euler equations with unbounded vorticity, Nonlinearity, 24, 2599-2637, (2011) · Zbl 1222.76016 [14] Koebe, P., Abhandlungen zur theorie der konformen abbildung, Math. Z., 2, 1-2, 198-236, (1918) · JFM 46.0546.01 [15] Lin, C.C., On the motion of vortices in two dimensions, Appl. math. ser., vol. 5, (1943), University of Toronto Studies · Zbl 0063.03561 [16] Lacave, C.; Miot, E., Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex, SIAM J. math. anal., 41, 3, 1138-1163, (2009) · Zbl 1189.35259 [17] Gamblin, P., Système dʼeuler incompressible et régularité microlocale analytique, Ann. inst. Fourier (Grenoble), 44, 5, 1449-1475, (1994) · Zbl 0820.35111 [18] O. Glass, F. Sueur, T. Takahashi, Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid, preprint 2010, Ann. Sci. Ecole Norm. Sup., in press. · Zbl 1311.35217 [19] Kato, T., On the smoothness of trajectories in incompressible perfect fluids, (), 109-130 · Zbl 0972.35102 [20] Majda, A.; Bertozzi, A., Vorticity and incompressible flow, Cambridge texts appl. math., vol. 27, (2002) · Zbl 0983.76001 [21] Marchioro, C.; Pulvirenti, M., Mathematical theory of incompressible nonviscous fluids, Appl. math. sci., vol. 96, (1994) · Zbl 0789.76002 [22] Schochet, S., The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation, Comm. partial differential equations, 20, 5-6, 1077-1104, (1995) · Zbl 0822.35111 [23] Serfati, P., Équation dʼeuler et holomorphies à faible régularité spatiale, C. R. acad. sci. Paris, 320, 2, 175-180, (1995) · Zbl 0834.34077 [24] Serfati, P., Solutions $$C^\infty$$ en temps, n-log Lipschitz bornées en espace et équation dʼeuler, C. R. acad. sci. Paris, 320, 5, 555-558, (1995) · Zbl 0835.76012 [25] Serfati, P., Structures holomorphes à faible régularité spatiale en mécanique des fluides, J. math. pures appl. (9), 74, 2, 95-104, (1995) · Zbl 0849.35111 [26] Turkington, B., On the evolution of a concentrated vortex in an ideal fluid, Arch. ration. mech. anal., 97, 1, 75-87, (1987) · Zbl 0623.76013 [27] Vishik, M., Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. sci. école norm. sup. (4), 32, 6, 769-812, (1999) · Zbl 0938.35128 [28] Wolibner, W., Un theorème sur lʼexistence du mouvement plan dʼun fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z., 37, 1, 698-726, (1933) · Zbl 0008.06901 [29] Yudovich, V.I., Non-stationary flows of an ideal incompressible fluid, Z̆. vyčisl. mat. i mat. fiz., 3, 1032-1066, (1963) [30] Yudovich, V.I., Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. res. lett., 2, 1, 27-38, (1995) · Zbl 0841.35092
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