# Bases in Banach spaces by Singer I. By Singer I.

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28) Obviously, this is not a classical solution. The discontinuity at x = 0 is a manifestation of the fact that we neglected viscous effects. Indeed if we restore the viscous term, the discontinuity becomes a viscous shock inside which the solution changes rapidly but remains smooth. 1. MATCHED ASYMPTOTICS where δ is the width of the shock. 30) δ 2 δ To balance terms, we must have δ ∼ ε. We see that the width of the shock is much smaller √ than the width of the boundary layer in incompressible flows (which is O( ε)).

41] J. D. W. Jacobsen, “Softening of nanocrystalline metals at very small grains,” Nature, vol. 391, pp. 561–563, 1998.  Y. Sone, Molecular Gas Dynamics, Birkh¨auser, Boston, Basel, Berlin, 2007.  H. Spohn, Large Scale Dynamics of Interacting Particles, Springer-Verlag, 1991.  S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York, 2002.  S. H. Tsien, Physical Mechanics (in Chinese), Science Press, 1962. -Y. -Y. -G. Han, “Electronic structure of impurity (oxygen)stacking-fault complex in nickel”, Phys.

43). 2. 55) where κ is the mean curvature of Γt . ]. 1) with boundary condition uε (0) = 0 and uε (1) = 1. 2) where A and B are constants determined by the boundary conditions: A = e1/ε /(e2/ε −1) and B = −e1/ε /(e2/ε −1). One thing we learn from this explicit solution is that it contains factors with 1/ε in the exponent. Power series expansion in ε will not be able to capture these terms. ]). This equation can no longer be solved explicitly. 4) 40 Since CHAPTER 2. 5) we have (εS0′′ + ε2 S1′′ ) + (S0′ + εS1′ )2 − V (x) = O(ε2 ).