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Two-dimensional Navier-Stokes flow with measures as initial vorticity. (English) Zbl 0666.76052
The main results of this work are some existence theorems for Navier- Stokes eqs. in velocity and pressure variables or vorticity variable when the initial vorticity is a finite measure in \(R^ 2.\)
The authors construct a global solution for these eqs. and prove regularity for \(t>0\) as well as some decay estimates as \(t\to \infty\). They consider only two-dimensional flow. Their main result may be understood as an example of existence of solutions for nonlinear parabolic equations with measures as initial data (solutions which may have infinite energy).
They compare their results with those of T. Kato [Proc. Symp. Pure Math. 45, Pt. 2, 1-7 (1986; Zbl 0598.35093)], G. Ponce [Commun. Partial Differ. Equations 11, 483-511 (1986; Zbl 0594.35077)] or R. J. DiPerna and A. J. Majda [ibid. 40, No.3, 301-345 (1987)]. Some of their results are extensions. However the theorem of existence for Euler eqs. is new for initial data of class \(L^ p(R^ 2)\).
Reviewer: C.I.Gheorghiu

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
35K55 Nonlinear parabolic equations
35K99 Parabolic equations and parabolic systems
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[1] Aronson, D. G., Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 890–896 (1968). · Zbl 0153.42002
[2] Aronson, D. G., & J. Serrin, Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25, 81–122 (1967). · Zbl 0154.12001
[3] Benfatto, G., Esposito, R., & M. Pulvirenti, Planar Navier-Stokes flow for singular initial data. Nonlinear Anal. 9, 533–545 (1985). · Zbl 0621.76027
[4] Bergh, J., & J. Löfström, Interpolation Spaces, An Introduction. Berlin Heidelberg New York: Springer-Verlag 1976.
[5] Brézis, H., & A. Friedman, Nonlinear parabolic equations involving measures as initial data. J. Math. Pures et appl. 62, 73–97 (1983). · Zbl 0527.35043
[6] Dobrushin, R. L., Prescribing a system of random variables by conditional distributions. Theory Prob. Appl. 15, 458–486 (1970). · Zbl 0264.60037
[7] Fabes, E. B., Jones, B. F., & N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in L p Arch. Rational Mech. Anal. 45, 222–240 (1972). · Zbl 0254.35097
[8] Friedman, A., Partial Differential Equations of Parabolic Type. New Jersey: Prentice-Hall 1964. · Zbl 0144.34903
[9] Friedman, A., Partial Differential Equations. New York: Holt, Rinehart & Winston 1969. · Zbl 0224.35002
[10] Fujita, H., & T. Kato, On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal. 16, 269–315 (1964). · Zbl 0126.42301
[11] Giga, Y., & T. Miyakawa, Solutions in L r of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267–281 (1985). · Zbl 0587.35078
[12] Giga, Y., Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations 62, 186–212 (1986). · Zbl 0577.35058
[13] Gilbarg, D., & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin Heidelberg New York: Springer-Verlag 1983. · Zbl 0562.35001
[14] Kato, T., Strong Lp-solutions of the Navier-Stokes equation in R m, with applications to weak solutions. Math. Z. 187, 471–480 (1984). · Zbl 0545.35073
[15] Kato, T., Remarks on the Euler and Navier-Stokes equations in R 2. Nonlinear Functional Analysis and its Applications, F. E. Browder ed., Proc. of Symposia in Pure Math. 45, part 2, 1–8. Providence, RI: Amer. Math. Soc. 1986. · Zbl 0598.35093
[16] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon & Breach 1969. · Zbl 0184.52603
[17] Leray, J., Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. pures et appl., Serie 9, 12, 1–82 (1933). · Zbl 0006.16702
[18] Liu, T.-P., & M. Pierre, Source-solutions and asymptotic behavior in conservation laws. J. Differential Equations 51, 419–441 (1984). · Zbl 0545.35057
[19] Marchioro, C., & M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory. Commun. Math. Phys. 84, 483–503 (1982). · Zbl 0527.76021
[20] Marchioro, C., & M. Pulvirenti, Euler evolution for singular initial data and vortex theory. Commun. Math. Phys. 91, 563–572 (1983). · Zbl 0529.76023
[21] McGrath, F. J., Nonstationary planar flow of viscous and ideal fluids. Arch. Rational Mech. Anal. 27, 329–348 (1968). · Zbl 0187.49508
[22] McKean, H. P., Jr., Propagation of chaos for a class of nonlinear parabolic equations. Lecture series in differential equations, Session 7: Catholic Univ. 1967.
[23] Niwa, Y., Semilinear heat equations with measures as initial data. Preprint.
[24] Osada, H., & S. Kotani, Propagation of chaos for the Burgers equation. J. Math. Soc. Japan 37, 275–294 (1985). · Zbl 0603.60072
[25] Osada, H., Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27, 597–619 (1987). · Zbl 0657.35073
[26] Osada, H., Propagation of chaos for the two dimensional Navier-Stokes equations. Probabilistic Methods in Math. Phys., K. Ito & N. Ikeda eds., 303–334, Tokyo: Kinokuniya 1987. · Zbl 0645.76040
[27] Ponce, G., On two dimensional incompressible fluids. Commun. Partial Differ. Equations 11, 483–511 (1986). · Zbl 0594.35077
[28] Reed, M., & B. Simon, Methods of Modern Mathematical Physics Vol. I, II; New York: Academic Press 1972, 1975. · Zbl 0242.46001
[29] Sznitman, A. S., Propagation of chaos result for the Burgers equation. Probab. Th. Rel. Fields 71, 581–613 (1986). · Zbl 0597.60055
[30] Temam, R., Navier-Stokes Equations. Amsterdam: North-Holland 1977. · Zbl 0383.35057
[31] Turkington, B., On the evolution of a concentrated vortex in an ideal fiuid. Arch. Rational Mech. Anal. 97, 75–87 (1987). · Zbl 0623.76013
[32] Wahl, W. von, The Equations of Navier-Stokes and Abstract Parabolic Equations. Braunschweig: Vieweg Verlag 1985. · Zbl 0575.35074
[33] Weissler, F. B., The Navier-Stokes initial value problem in L p. Arch. Rational Mech. Anal. 74, 219–230 (1980). · Zbl 0454.35072
[34] Kato, T., & G. Ponce, Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces L sp(R 2). Rev. Mat. Iberoamericana 2, 73–88 (1986). · Zbl 0615.35078
[35] Baras, P., & M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures. Applicable Analysis 18, 111–149 (1984). · Zbl 0582.35060
[36] DiPerna, R. J., & A. J. Majda, Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 16, 301–345 (1987). · Zbl 0850.76730
[37] Cottet, G.-H., Équations de Navier-Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sc. Ser. 1, 303, 105–108 (1986). · Zbl 0606.35065
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