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On the number of singular points of weak solutions to the Navier-Stokes equations. (English) Zbl 1030.35133

The weak solutions of the three-dimensional Navier-Stokes equations for incompressible viscous fluids are studied in the paper. The author considers the system of equations: \(\partial_t v+ \operatorname {div}(v\otimes v)-\Delta v=f-\omega p\), \(\operatorname {div}v=0\) in the space-time cylinder \(Q_T\equiv \Omega\times ]0,T[\) where: \(\Omega\subset \mathbb{R}^3\), \(T\) – a positive parameter, \(v\) – the velocity field, \(p\) – the pressure, \(f\) – a given external force. The author is interested in the differentiability of functions \(v\) and \(p\), when \(v\in L_\infty (0,T,L_2 (\Omega,\mathbb{R}^3))\cap L_2(0,T,W_2^1 (\Omega,\mathbb{R}^3))\), \(p\in L_{3/2}(Q_T)\), \(f\in L_2(Q_T,\mathbb{R}^3)\). A point of the space-time cylinder \(Q_T\) is regular if the velocity field \(v\) is Hölder continuous in some neighborhood of this point; it is singular if it is not regular. The aim of the paper is to estimate the number of points in the set \(\Sigma(t_0)\cap \omega\) for any open subset \(\omega\subseteq \Omega\) at any moment \(t_0\in ]0,T[\). \(\Sigma\) is the set of singular points of a suitable weak solution.
The main result of the paper is contained in a theorem asserting that there exist a positive number \(\varepsilon_0\), depending only on \(\gamma\), an arbitrary positive constant, with the property that for any open subset \(\omega\subseteq \Omega\) and for any \(t_0\in ]0,T[\) the \[ \operatorname {card} \{\Sigma(t_0)\cap \omega\}\leq \varepsilon_0(\gamma) \limsup_{\mathbb{R}\to O} (1/\mathbb{R}^2) \int_{t_t-\mathbb{R}^2}^{t_0} dt \int_\omega |v(x,t)|^3 dx, \] \(\mathbb{R}\) is the radius of the ball around the point \(x_0\) as center. For the proof of this theorem, the author must prove a main lemma, as an intermediate step of the paper.

MSC:

35Q30 Navier-Stokes equations
35A20 Analyticity in context of PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35D99 Generalized solutions to partial differential equations
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References:

[1] Caffarelli, Comm Pure Appl Math 35 pp 771– (1982) · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[2] The mathematical theory of viscous incomprehensible flow. Second English edition, revised and enlarged. Mathematics and Its Applications, 2. Gordon and Breach, New York-London-Paris 1969.
[3] Mathematical problems of the dynamics of viscous incompressible fluids. 2nd edition, Nauka, Moscow, 1970.
[4] Ladyzhenskaya, J Math Fluid Mech 1 pp 356– (1999) · Zbl 0954.35129 · doi:10.1007/s000210050015
[5] Lin, Comm Pure Appl Math 51 pp 241– (1998) · Zbl 0958.35102 · doi:10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A
[6] Neustupa, J Math Fluid Mech 1 pp 309– (1999) · Zbl 0949.35107 · doi:10.1007/s000210050013
[7] Scheffer, Pacific J Math 66 pp 535– (1976) · Zbl 0325.35064 · doi:10.2140/pjm.1976.66.535
[8] Scheffer, Comm Math Phys 55 pp 97– (1977) · Zbl 0357.35071 · doi:10.1007/BF01626512
[9] Scheffer, Comm Math Phys 73 pp 1– (1980) · Zbl 0451.35048 · doi:10.1007/BF01942692
[10] Seregin, J Math Fluid Mech 1 pp 235– (1999) · Zbl 0961.35106 · doi:10.1007/s000210050011
[11] Taniuchi, Manuscripta Math 94 pp 365– (1997) · Zbl 0896.35106 · doi:10.1007/BF02677860
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