By Slavenas P.
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Pisa 13 (1959), 116–162. ; Mathematical Principles of Classical Fluid Mechanics. Encyclopedia of Physics VIII, 125–263. Springer-Verlag, Berlin, 1959. ; General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev. 91 (1963), 99–164. , Sˇ system of Navier-Stokes equations. Proc. Steklov Inst. Math. 125 (1973), 186–199. Ê H. it Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 63, 33–42 c 2005 Birkh¨ auser Verlag Basel/Switzerland Radiative Heat Transfer in Silicon Puriﬁcation A.
A. 2. Suppose that 2
2, n≥3 n = 1, 2. 2n n−2 , (8) If u0 ∈ H, and satisfying Cp p−2 2 2p E(0) p−2 <1 (9) then the solution u(t) ∈ H for each t ∈ [0, T ). Proof. Since I(u0 ) > 0, then there exists (by continuity) Tm < T such that I(u(t)) ≥ 0, ∀t ∈ [0, Tm ] ; this gives p−2 2p E(t) = ∇u(t) 2 2 p−2 2p 1 + I(t) ≥ p ∇u(t) 2 2 . (10) E(0), ∀t ∈ [0, Tm ] . (11) So, ∇u(t) 2 2 2p p−2 ≤ 2p p−2 E(t) ≤ We then use (7)–(9) and (11) to obtain, for each t ∈ [0, Tm ], u(t) ≤ Cp p p ≤ C p ∇u(t) 2p p−2 E(0) p−2 2 p 2 p−2 2 = C p ∇u(t) ∇u(t) 2 2 ∇u(t) 2 2 (12) < ∇u(t) 22 .
R is the ﬁxed point l = φp (l) of the map φp ). This requires sharp estimates at each stage of the proof. We succeed in proving these estimates and the desired result. 7). 3. 1) 30 H. Beir˜ ao da Veiga where [f ]−1 denotes the norm in H −1 . 3) it follows u(t) 2 L∞ (0,T ;L2 ) ≤ c u(0) 2 + ν0 u c ν0 + f 2 L2 (0,T ;H 1 ) 2 L2(0,T ;H0−1 ) + ν1 u p Lp (0,T ;W 1,p ) . 6) by ∂∂ ut and integrate in Ω. 3) dx . 4) (u · ∇) u ≤ D u 2p , 4n provided that p ≥ n+2 . 2. 2) shows that D u 4− p p 4− p p 2 + ν1 Du p p Du d dt p p 2 5 referred in Theorems ≤ .
A Possible Way to Discuss the Fundamental Principles of Relativity by Slavenas P.