By Giovanni P. Galdi, John G. Heywood, Rolf Rannacher
The mathematical idea of the Navier-Stokes equations provides nonetheless primary open questions that characterize as many demanding situations for the mathematicians. This quantity collects a sequence of articles whose aim is to provide new contributions and ideas to those questions, with specific regard to turbulence modelling, regularity of options to the initial-value challenge, circulation in zone with an unbounded boundary and compressible flow.
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Additional resources for Contributions to Current Challenges in Mathematical Fluid Mechanics (Advances in Mathematical Fluid Mechanics)
Unt1 satisfies aL ,un+1 = -(un . v)un+l - A2°un+1 - vpn+1 Clearly we have (u" - V)u"+', A2au"+' E L1(0,oo; B? 2°). 1 Since pn+1 satisfies -divVpn+1 = div((un . -2, ) from the Calderon Zygmund estimate [181. we have Vp"+' E L' (0, oo; Thus we readily have that u"+' is continuous with values in B2 -2* . By a 3E argument, u is continuous with values in BZ, 2°-`7 By the interpolation, we have C([0, oo); BZ 1) with = r - 2a, if 2 < a < a for 61 > 0. We still have -2a-dl, if0a< l to prove that u also belongs to L°°(v,oo; B21)nL'(Q, oo; B.
Springer-Verlag, New York - Heidelberg, 1974. 181 Hartman Ph. Ordinary Differential Equations. , New York, 1964. 191 Hormander L. Lectures on nonlinear hyperbolic differential equations. SpringerVerlag, Berlin, 1997. [101 Kolmogorov A. N. On inequalities for supremums of successive derivatives of a function on an infinite interval. N. 1 " Moscow, Nauka 1985. Engl. translation: Kluwer, 1991. , Grujic Z. Space Analyticity for the Navier-Stokes and Related Equations with Initial Data in L°. Journal of Functional Analysis 152, no.
LIlog6un+l IIL2 V)un)IIL2IIAg6un+1 11L-- Dividing both sides of the above inequality by IIt'. 1 2,1
Contributions to Current Challenges in Mathematical Fluid Mechanics (Advances in Mathematical Fluid Mechanics) by Giovanni P. Galdi, John G. Heywood, Rolf Rannacher