• ➱◆■❊

♥ ✐t✐❛

| u(ξ)| 2 dξ 1 2 < ∞

❲❡ s❤❛•• ❝♦♥st❛♥t•② ❜❡ ✉s✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ s✐♠♣•✐✜❡❞ ♥♦t❛t✐♦♥s✿ L ∞ T ( Ḣs ) ❞❡❢ = L ∞ ([0, T ], Ḣs ) ❛♥❞ L 2 T ( Ḣs+1 ) ❞❡❢ = L 2 ([0, T ], Ḣs+1 ).
▲❡t ✉s ❞❡✜♥❡ t❤❡ r❡•❡✈❛♥t ❢✉♥❝t✐♦♥ s♣❛❝❡ ✇❡ s❤❛•• ❜❡ ✇♦r❦✐♥❣ ✇✐t❤ ✐♥ t❤❡ s❡q✉❡•✿

X s T ❞❡❢ = L ∞ T ( Ḣs ) ∩ L 2 T ( Ḣs+1 ), ❡q✉✐♣♣❡❞ ✇✐t❤ u 2 X s T ❞❡❢ = u 2 L ∞ T ( Ḣs ) + u 2 L 2 T ( Ḣs+1 ) .
❚❤❡♦r❡♠ ✶✳✷✳ ▲❡t u 0 ❜❡ ✐♥ Ḣs ✱ ✇✐t❤

ρ s (T * ) = sup{ ρ > 0 | u 0 Ḣs < ρ =⇒ T * (u 0 ) > T * }. ❚❤❛♥❦s t♦ t❤✐s ❞❡✜♥✐t✐♦♥✱ ✇❡ ❞❡✜♥❡ t❤❡ ♥♦t✐♦♥ ♦❢ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥ ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s②st❡♠✳ ❉❡✜♥✐t✐♦♥ ✶✳✸✳ ✭♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥✮ ❲❡ s❛② t❤❛t u = N S(u 0 ) ✐s ❛ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥ ✐❢ u 0 s❛t✐s✜❡s t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s✿ u 0 Ḣs = ρ s (T * ) ❛♥❞ T * (u 0 ) = T * . ❚❤❡r❡❢♦r❡✱ u = N S(u 0 ) ✐s ❛ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥ ✐❢ ❛♥❞ ♦♥•② ✐❢ A σs c ✐s r❡❛❝❤❡❞✿ T * (u 0 ) u 0 σs Ḣs = A σs c ✳ ◗✉❡st✐♦♥✿ ■❢ ρ s (T * ) ✐s ✜♥✐t❡✱ ❞♦ s♦♠❡ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥s ❡①✐st ❄ ✹ ❊❯•➱◆■❊ P❖❯▲❖◆ ❲❡ ✇✐•• ♣r♦✈❡ ❛ str♦♥❣❡r r❡s✉•t✿ t❤❡ s❡t ♦❢ ✐♥✐t✐❛• ❞❛t❛ ❣❡♥❡r❛t✐♥❣ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥s✱ ❞❡♥♦t❡❞ ❜② M s (T * )✱ ✐s ♥♦t ♦♥•② ❛ ♥♦♥❡♠♣t② s✉❜s❡t ♦❢ Ḣs ✭✇❤✐❝❤✱ ✐♥ ♣❛rt✐❝✉•❛r✱ ❣✐✈❡s t❤❡ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥✮ ❜✉t ❛•s♦ ❝♦♠♣❛❝t ✐♥ ❛ s❡♥s❡ ✇❤✐❝❤ ✐s ❣✐✈❡♥ ✐♥ ❚❤❡♦r❡♠ ✶✳✸✳ ❲❡ ❞❡✜♥❡ t❤❡ s❡t M s (T * ) ❛s ❢♦••♦✇s M s (T * ) ❞❡❢ = u 0 ∈ Ḣs | T * (u 0 ) = T * ❛♥❞ u 0 Ḣs = ρ s (T * ) . ❚❤❡♦r❡♠ ✶✳✸✳ ❆ss✉♠✐♥❣ ❤②♣♦t❤❡s✐s H✳ ❋♦r ❛♥② ✜♥✐t❡ t✐♠❡ T * ✱ t❤❡ s❡t M s (T * ) ✐s ♥♦♥ ❡♠♣t② ❛♥❞ ❝♦♠♣❛❝t✱ ✉♣ t♦ tr❛♥s•❛t✐♦♥s✳ ❚❤✐s ♠❡❛♥s t❤❛t ❢♦r ❛♥② s❡q✉❡♥❝❡ (u 0,n ) n∈■◆ ♦❢ ♣♦✐♥ts ✐♥ t❤❡ s❡t M s (T * )✱ ❛ s❡q✉❡♥❝❡ (x n ) n∈■◆ ♦❢ ♣♦✐♥ts ♦❢ (■| 3 ) ■◆ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ V ✐♥ M s (T * ) ❡①✐st s✉❝❤ t❤❛t✱ ✉♣ t♦ ❛♥ ❡①tr❛❝t✐♦♥ lim n→+∞ ||u 0,n (• + x n ) -V || Ḣs = 0. ❚❤❡ s❡❝♦♥❞ r❡s✉•t ♦❢ t❤✐s ♣❛♣❡r st❛t❡s t❤❛t t❤❡ ❜•♦✇ ✉♣ r❛t❡ ♦❢ ❛ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥ ❝❛♥ ❜❡ ✉♥✐❢♦r♠❡•② ❝♦♥tr♦••❡❞ s✐♥❝❡ ✇❡ ❣❡t ❛ ♣r✐♦r✐ ❜♦✉♥❞ ♦❢ t❤❡s❡ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳✹✳ ✭❈♦♥tr♦• ♦❢ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥s✮ ❆ss✉♠✐♥❣ H✱ t❤❡r❡ ❡①✐sts ❛ ♥♦♥❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ F s : [0, A σs c [ → ■| + ✇✐t❤ lim r→A σs c F s (r) = +∞ s✉❝❤ t❤❛t ❢♦r ❛♥② ❞✐✈❡r❣❡♥❝❡ ❢r❡❡ ✈❡❝t♦r ✜❡•❞ u 0 ✐♥ Ḣs ✱ ❣❡♥❡r❛t✐♥❣ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥ ✭✐t ♠❡❛♥s T * (u 0 ) u 0 σs Ḣs = A σs c )✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥tr♦• ♦♥ t❤❡ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥ N S(u 0 ) ∀ T < T * (u 0 ), N S(u 0 ) X s T u 0 Ḣs F s (T 1 σs u 0 Ḣs ).
|❡♠❛r❦ ✶✳✷✳ ▲❡t ✉s ♣♦✐♥t ♦✉t t❤❛t t❤❡ q✉❛♥t✐t② T

1 σs u 0 Ḣs ✐s s❝❛•✐♥❣ ✐♥✈❛r✐❛♥t❀ ✇❤✐❝❤ ✐s ♦❜✈✐♦✉s•② ♥❡❝❡ss❛r②✳ ❚❤❡ t✇♦ ♣r❡✈✐♦✉s t❤❡♦r❡♠s ❛r❡ t❤❡ ❛♥❛•♦❣✉❡ ♦❢ r❡s✉•ts✱ ♣r♦✈❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❙♦❜♦•❡✈ s♣❛❝❡ Ḣ 1 2 ✳ ❲❡ s❤❛•• ♥♦t r❡❝❛•• ❛•• t❤❡ st❛t❡♠❡♥ts ❡①✐st✐♥❣ ✐♥ t❤❡ •✐t❡r❛t✉r❡ ❝♦♥❝❡r♥✐♥❣ t❤❡ r❡❣✉•❛r✐t② ♦❢ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥s ✐♥ ❝r✐t✐❝❛• s♣❛❝❡s✱ s✉❝❤ ❛s
♦♥❝❡♣t ♦❢ ♠✐♥✐♠❛• ❜•♦✇✲✉♣ ❢♦r ❞❛t❛ ✐♥ ❙♦❜♦•❡✈ s♣❛❝❡ Ḣ 1 2 ✳ ❋✐rst•②✱ t❤❡② ❞❡✜♥❡❞ ❛ ❝r✐t✐❝❛• r❛❞✐✉s ρ 1 2 ρ 1 2 = sup ρ > 0 ❀ u 0 Ḣ 1 2 < ρ =⇒ T * (u 0 ) = +∞ . ❚❤❡♥✱ t❤❡② ✐♥tr♦❞✉❝❡❞ ❛ s✉❜s❡t M ♦❢ Ḣ 1 2 ✱ ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡ s❡t ♦❢ ♠✐♥✐♠❛•✲♥♦r♠ s✐♥❣✉•❛r✐t✐❡s ✭✇❡ s♣❡❛❦ ❛❜♦✉t ♠✐♥✐♠❛• ♥♦r♠ ✐♥ t❤❡ s❡♥s❡ ♦❢ u 0 Ḣ 1 2 ✐s ❡q✉❛• t♦ t❤❡ ❝r✐t✐❝❛• r❛❞✐✉s ρ 1 2 ✮ M = u 0 ∈ Ḣ 1 2 ❀ T * (u 0 ) < +∞ ❛♥❞ u 0 Ḣ 1 2 = ρ 1 2 . ❚❤❛♥❦s t♦ t❤❡s❡ ❞❡✜♥✐t✐♦♥s✱ ❲✳ |✉s✐♥ ❛♥❞ ❱✳ S✈❡rá❦ ♣r♦✈❡❞ t❤❛t ✐❢ t❤❡r❡ ❡①✐st ❡•❡♠❡♥ts ✐♥ t❤❡ s♣❛❝❡ Ḣ 1 2 ✇❤✐❝❤ ❞❡✈❡•♦♣ s✐♥❣✉•❛r✐t✐❡s ✐♥ ✜♥✐t❡ t✐♠❡ ✭✇❡ ❛ss✉♠❡ t❤❛t ❜•♦✇✲✉♣ ♦❝❝✉rs✮✱ t❤❡♥ s♦♠❡ ♦❢ t❤❡s❡ ❡•❡♠❡♥ts ❛r❡ ♦❢ ♠✐♥✐♠❛• Ḣ 1 2 ✲♥♦r♠ ✭❛♥❞ t❤✉s✱ t❤❡ s❡t M ✐s ♥♦♥❡♠♣t②✮ ❛♥❞ ❝♦♠♣❛❝t ✉♣ t♦ tr❛♥s•❛t✐♦♥s ❛♥❞ ❞✐•❛t✐♦♥s✳ ■t ♠❡❛♥s t❤❛t ❢♦r ❛♥② s❡q✉❡♥❝❡ (u 0,n ) n∈■◆ ♦❢ ♣♦✐♥ts ✐♥ t❤❡ s❡t M✱ ❛ s❡q✉❡♥❝❡ (λ n , x n ) n∈■◆ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ϕ ✐♥ M ❡①✐st s✉❝❤ t❤❛t✱ ✉♣ t♦ ❛♥ ❡①tr❛❝t✐♦♥✱ ✇❡ ❤❛✈❡ lim n→+∞ ||u 0,n -Λ λn,xn ϕ|| Ḣ 1 2 = 0. ▲❡t ✉s ♣♦✐♥t ♦✉t t❤❛t ■✳ •❛••❛❣❤❡r✱ •✳ ❑♦❝❤ ❛♥❞ ❋✳ P•❛♥❝❤♦♥ ❣❡♥❡r❛•✐③❡ ✐♥ ❬✶✻( t❤❡ r❡s✉•t ♦❢ ❲✳ |✉s✐♥ ❛♥❞ ❱✳ S✈❡rá❦ t♦ ❝r✐t✐❝❛• ▲❡❜❡s❣✉❡ ❛♥❞ ❇❡s♦✈ s♣❛❝❡s✱ s✉❝❤ ❛s L 3 ✳ ❈♦♥❝❡r♥✐♥❣ ❚❤❡♦r❡♠ ✶✳✹✱ ♦✉r ♠❛✐♥ s♦✉r❝❡ ♦❢ ✐♥s♣✐r❛t✐♦♥ ✐s ❛ r❡s✉•t ❡st❛❜•✐s❤❡❞ ❜② ■✳ •❛••❛❣❤❡r ✐♥ ❬✶✹(✳ •✐✈❡♥ ❛♥ ✐♥✐t✐❛• ❞❛t❛ u 0 ✐♥ t❤❡ ♦♣❡♥ ❜❛•• B ρ 1 2 ✳ ❚❤❡♥✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ρ 1 2 ✱ N S(u 0 ) ✐s ❛ ❣•♦❜❛• s♦•✉t✐♦♥ ❛♥❞ t❤✉s ❜❡•♦♥❣s t♦ t❤❡ s♣❛❝❡ L 4 (■| + , Ḣ1 )✱ t❤❛♥❦s t♦ t❤❡ ✐♠♣♦rt❛♥t ♣❛♣❡r ❬✶✺( ♦❢ ■✳ •❛••❛❣❤❡r✱ ❉✳ ■❢t✐♠✐❡ ❛♥❞ ❋✳ P•❛♥❝❤♦♥✳ ■♥ t❤✐s ✇❛②✱ t❤❡ ❜•♦✇ ✉♣ ✐♥ t❤❡ E ■ | + = L ∞ (■| + , Ḣ 1 2 ) ∩ L 2 (■| + , Ḣ 3 2 )✲♥♦r♠ ❞♦❡s ♥♦t ♦❝❝✉r✳ ❊✈❡♥ ❜❡tt❡r✿ ■✳ •❛••❛❣❤❡r ♣r♦✈❡❞ ✐♥ ❬✶✹( t❤❡ ❛ ♣r✐♦r✐ ❝♦♥tr♦• ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ✇✐t❤ ❞❛t❛ ✐♥ t❤❡ ♦♣❡♥ ❜❛•• B ρ 1 2 ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡r❡ ❡①✐sts ❛ ♥♦♥❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ F ❞❡✜♥❡❞ ❢r♦♠ [0, ρ 1 2 [ t♦ ■| + s✉❝❤ t❤❛t ❢♦r ❛♥② ❞✐✈❡r❣❡♥❝❡ ❢r❡❡ ✈❡❝t♦r ✜❡•❞ u 0 ✐♥ t❤❡ ♦♣❡♥ ❜❛•• B ρ 1 2 ✱ ✇❡ ❤❛✈❡ N S(u 0 ) E ■| + F ( u 0 Ḣ 1 2 ). ◆♦t❛t✐♦♥✳ ❲❡ s❤❛•• ❞❡♥♦t❡ ❜② C ❛ ❝♦♥st❛♥t ✇❤✐❝❤ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛♣❡r✱ ❛♥❞ ✇❤✐❝❤ ♠❛② ❝❤❛♥❣❡ ❢r♦♠ •✐♥❡ t♦ •✐♥❡✳ ❲❡ s❤❛•• ❛•s♦ ❞❡♥♦t❡ s♦♠❡t✐♠❡s x y t♦ ♠❡❛♥ t❤❡r❡ ❡①✐sts ❛♥ ❛❜s♦•✉t❡ ❝♦♥st❛♥t C > 0 s✉❝❤ t❤❛t x C y✳ ❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❛②✿ ■♥ s❡❝t✐♦♥ 2✱ ✇❡ r❡❝❛•• t❤❡ ❢✉♥❞❛♠❡♥t❛• t♦♦• ♦❢ t❤✐s ♣❛♣❡r ✿ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐♥ Ḣs ✳ ❚❤❡♥✱ ✇❡ ❣✐✈❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ ♠✐♥✐♠❛• ❜•♦✇ ✉♣ s♦•✉t✐♦♥s s❡t ✭❚❤❡♦r❡♠ ✶✳✸✮ ❛♥❞ ❝♦♥tr♦• ♦❢ ♦❢ s✉❝❤ s♦•✉t✐♦♥s ✭❚❤❡♦r❡♠ ✶✳✹✮✳ ❚❤❡s❡ t✇♦ r❡s✉•ts ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❝r✉❝✐❛• ❚❤❡♦r❡♠ ✷✳✷ ❛❜♦✉t t❤❡ •✐❢❡s♣❛♥ ♦❢ ❛ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ Ḣs ✳ ❙❡❝t✐♦♥ 3 ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✷✱ t❤❛♥❦s t♦ ❛ r❡❣✉•❛r✐③❛t✐♦♥ ♣r♦❝❡ss✳ ❋✐rst•②✱ ✇❡ ✇✐•• s❡❡ t❤❛t ✐t ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▲❡♠♠❛ ✸✳✶✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ str✉❝t✉r❡ ♦❢ ❛ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ❞❛t❛ ✐♥ Ḣs ✳ ❙❡❝♦♥❞•②✱ ✇❡ ✇✐•• ♣r♦✈✐❞❡ s♦♠❡ ❤❡•♣❢✉• t♦♦•s ✐♥ ♦r❞❡r t♦ ♣r♦✈❡ ▲❡♠♠❛ ✸✳✶✳ ■♥ s❡❝t✐♦♥ 4✱ ✇❡ ♣r♦✈❡ ▲❡♠♠❛ ✸✳✶✱ t❤❡ r❡s✉•t ♦♥ ✇❤✐❝❤ ❛•• ♦t❤❡rs ❛r❡ ❜❛s❡❞ ♦♥✳ ❚❤✐s s❡❝t✐♦♥ ✐s t❤❡ ♠♦st t❡❝❤♥✐❝❛• ♣❛rt ♦❢ t❤❡ ♣❛♣❡r✳ ■t r❡•✐❡s ♦♥ ❝•❛ss✐❝❛• ♣r♦❞✉❝t ❛♥❞ ♣❛r❛♣r♦❞✉❝t ❡st✐♠❛t❡s✱ ✇❤✐❝❤ ❛r❡ ❝♦••❡❝t❡❞ ✐♥ ❆♣♣❡♥❞✐① ❆ ❛♥❞ ❇✳ ❆❝❦♥♦✇•❡❞❣❡♠❡♥ts✳ ■ ❛♠ ✈❡r② ❣r❛t❡❢✉• t♦ ■✳ •❛••❛❣❤❡r ❢♦r ❢r✉✐t❢✉• ❞✐s❝✉ss✐♦♥s ❛r♦✉♥❞ t❤❡ q✉❡st✐♦♥ ♦❢ ♥♦♥✲s❝❛•❡ ✐♥✈❛r✐❛♥t s♣❛❝❡s ❛♥❞ t♦ P✳ •ér❛r❞ ❢♦r ♠❛♥② ❤❡•♣❢✉• ❝♦♠♠❡♥ts✳ ✷✳ Pr♦❢✐•❡s t❤❡♦r②✱ ❝♦♠♣❛❝t♥❡ss r❡s✉•t ❛♥❞ ❛♣♣•✐❝❛t✐♦♥ ❚❤✐s s❡❝t✐♦♥ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠s ✶✳✸ ❛♥❞ ✶✳✹✳ ❋♦••♦✇✐♥❣ ■✳ •❛••❛❣❤❡r ❬✶✹(✱ ❲✳ |✉s✐♥ ❛♥❞ ❱✳ S✈❡rá❦ ❬✷✾(✱ ❈✳ ❑❡♥✐❣ ❛♥❞ •✳ ❑♦❝❤ ❬✶✾( ❛♥❞ ■✳ •❛••❛❣❤❡r✱ •✳ ❑♦❝❤✱ ❋✳ P•❛♥❝❤♦♥ ❬✶✻(✱ ✇❡ s❤❛•• ✉s❡ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡♦r②✳ ❚❤❡ ♦r✐❣✐♥❛• ♠♦t✐✈❛t✐♦♥ ♦❢ t❤✐s t❤❡♦r② ✇❛s t❤❡ ❞❡s❝✐♣t✐♦♥ ♦❢ t❤❡ ❞❡❢❛✉•t ♦❢ ❝♦♠♣❛❝t♥❡ss ✐♥ ❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣s ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ t❤❡ ♣✐♦♥♥❡❡r✐♥❣ ✇♦r❦s ♦❢ P✳✲▲✳ ▲✐♦♥s ✐♥ ❬✷✻(✱ ❬✷✼( ❛♥❞ ❍✳ ❇r❡③✐s✱ ❏✳✲▼✳ ❈♦r♦♥ ✐♥ ❬✼(✳ ❍❡r❡✱ ✇❡ ✇✐•• ✉s❡ t❤❡ t❤❡♦r❡♠ ♦❢ P✳ •ér❛r❞ ❬✶✼(✱ ✇❤✐❝❤ ❣✐✈❡s✱ ✉♣ t♦ ❡①tr❛❝t✐♦♥s✱ t❤❡ str✉❝t✉r❡ ♦❢ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ Ḣs ✱ ✇✐t❤ s ❜❡t✇❡❡♥ 0 ❛♥❞ 3 2 • ▼♦r❡ ♣r❡❝✐s❡•②✱ t❤❡ ❞❡❢❛✉•t ♦❢ ❝♦♠♣❛❝t♥❡ss ✐♥ t❤❡ ❝r✐t✐❝❛• ❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣ Ḣs ⊂ L p ✐s ❞❡s❝✐❜❡❞ ✐♥ t❡r♠s ♦❢ ❛ s✉♠ ♦❢ r❡s❝❛•❡❞ ❛♥❞ tr❛♥s•❛t❡❞ ♦rt❤♦❣♦♥❛• ♣r♦✜•❡s✱ ✉♣ t♦ ❛ s♠❛•• t❡r♠ ✐♥ L p ✳ ❚❤❛t ✇❛s ❣❡♥❡r❛•✐③❡❞ t♦ ♦t❤❡r ❙♦❜♦•❡✈ s♣❛❝❡s Ḣs,p (■| d ) ✇✐t❤ 0 < s < d p ❜② ❙✳ ❏❛✛❛r❞ ✐♥ ❬✶✽(✱ t♦ ❇❡s♦✈ s♣❛❝❡s ❜② •✳ ❑♦❝❤ ✐♥ ❬✷✸( ❛♥❞ t♦ ❣❡♥❡r❛• ❝r✐t✐❝❛• ❡♠❜❡❞❞✐♥❣s ❜② ❍✳ ❇❛❤♦✉r✐✱ ❆✳ ❈♦❤❡♥ ❛♥❞ •✳ ❑♦❝❤ ✐♥ ❬✷(✳ ▲❡t ✉s ♥♦t✐❝❡ t❤❡ r❡❝❡♥t ✇♦r❦ ❬✺( ♦❢ ❍✳ ❇❛❤♦✉r✐✱ ▼✳ ▼❛•❞♦✉❞ ❛♥❞ ◆✳ ▼❛s♠♦✉❞✐ ❝♦♥❝❡r♥✐♥❣ t❤❡ •❛❝❦ ♦❢ ❝♦♠♣❛❝t♥❡ss ♦❢ t❤❡ ❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣ ♦❢ H 1 (■| 2 ) ✐♥ t❤❡ ❝r✐t✐❝❛• ❖r•✐❝③ s♣❛❝❡ L(■| 2 )✳ ❚❤❡♥ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ t❡❝❤♥✐q✉❡s ❤❛✈❡ ❜❡❡♥ ❛♣♣•✐❡❞ ✐♥ ♠❛♥② ✇♦r❦s ♦❢ ❡✈♦•✉t✐♦♥ ♣r♦❜•❡♠s s✉❝❤ ❛s t❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝② st✉❞② ♦❢ ✜♥✐t❡ ❡♥❡r❣② s♦•✉t✐♦♥s t♦ q✉✐♥t✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥s ♦♥ ■| 3 ✱ ❜② ❍✳ ❇❛❤♦✉r✐ ❛♥❞ P✳ •ér❛r❞ ❬✹(✳ ❈✳ ❑❡♥✐❣ ❛♥❞ ❋✳ ▼❡r•❡ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✷✵( t❤❡ ❜•♦✇ ✉♣ ♣r♦♣❡rt② ❢♦r t❤❡ ❡♥❡r❣② ❝r✐t✐❝❛• ❢♦❝✉s✐♥❣ ♥♦♥ •✐♥❡❛r ✇❛✈❡ ❡q✉❛t✐♦♥✳ Pr♦✜•❡ t❡❝❤♥✐q✉❡s t✉r♥❡❞ ♦✉t t♦ ❜❡ ❛•s♦ ❛ r❡•❡✈❛♥t t♦♦• ✐♥ t❤❡ st✉❞② ♦❢ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥s✳ ◆♦t✐❝❡ t❤✐s ❦✐♥❞ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇❛s st❛t❡❞ ❛♥❞ ❞❡✈❡•♦♣♣❡❞✱ ✐♥❞❡♣❡♥❞❡♥t•② ❢r♦♠ ❬✶✼(✱ ❜② ❋✳ ▼❡r•❡ ❛♥❞ ▲✳ ❱❡❣❛ ❬✷✽( ❢♦r L 2 ✲s♦•✉t✐♦♥s ♦❢ t❤❡ ❝r✐t✐❝❛• ♥♦♥ •✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ✐♥ 2D✱ ✐♥ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤❡ ✇♦r❦ ♦❢ ❏✳ ❇♦✉r❣❛✐♥ ❬✻(✳ ❚❤❡♥✱ ❙✳ ❑❡r❛❛♥✐ r❡✈✐s✐t❡❞ ✐♥ ❬✷✷( t❤❡ ✇♦r❦ ♦❢ ❍✳ ❇❛❤♦✉r✐ ❛♥❞ P✳ •ér❛r❞ ❬✹( ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❡♥❡r❣② ❝r✐t✐❝❛• ♥♦♥ •✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥s✳ ❈✳ ❑❡♥✐❣ ❛♥❞ ❋✳ ▼❡r•❡ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✷✶( t❤❡ ❣•♦❜❛• ✇❡••✲♣♦s❡❞♥❡ss✱ s❝❛tt❡r✐♥❣ ❛♥❞ ❜•♦✇ ✉♣ ♠❛tt❡r ❢♦r s✉❝❤ s♦•✉t✐♦♥s ✐♥ t❤❡ ❢♦❝✉s✐♥❣ ❛♥❞ r❛❞✐❛• ❝❛s❡✳ |❡♠❛r❦ ✷✳✶✳ ❯s✐♥❣ ♥♦t❛t✐♦♥ ✭✺✮✱ ✇❡ ❝❛♥ ♣r♦✈❡ ❡❛s✐•② t❤❛t t❤❡ L p ✭❛s ✇❡•• ❛s Ḣs ✮✲♥♦r♠ ✐s ❝♦♥s❡r✈❡❞ ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ u → Λ 3 p λ,x 0 u✳ ■t ♠❡❛♥s Λ 3 p λ,x 0 u = u ✳ ❚❤❡♦r❡♠ ✷✳✶✳ ▲❡t (u 0,n ) n∈■◆ ❜❡ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐♥ Ḣs ✳ ❚❤❡♥✱ ✉♣ t♦ ❛♥ ❡①tr❛❝t✐♦♥✿ ✲ ❚❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ✈❡❝t♦rs ✜❡•❞s✱ ❝❛••❡❞ ♣r♦✜•❡s (V j ) j∈■◆ ✐♥ Ḣs ✳ ✲ ❚❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ s❝❛•❡s ❛♥❞ ❝♦r❡s (λ n,j , x n,j ) n,j∈■◆ ✱ s✉❝❤ t❤❛t✱ ✉♣ t♦ ❛♥ ❡①tr❛❝t✐♦♥ ∀J 0, u 0,n (x) = J j=0 Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) ✇✐t❤ lim J→+∞ lim sup n→+∞ ψ J n L p = 0, ❛♥❞ p = 6 3 -2s • ❲❤❡r❡ (λ n,j , x n,j ) n∈■◆,j∈■◆ * ❛r❡ s❡q✉❡♥❝❡s ♦❢ (■| * + × ■| 3 ) ■◆ ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ ♦rt❤♦❣♦♥❛•✐t② ♣r♦♣❡rt②✿ ❢♦r ❡✈❡r② ✐♥t❡❣❡rs (j, k) s✉❝❤ t❤❛t j = k✱ ✇❡ ❤❛✈❡ ❡✐t❤❡r lim n→+∞ λ n,j λ n,k + λ n,k λ n,j = +∞ ♦r λ n,j = λ n,k ❛♥❞ lim n→+∞ |x n,j -x n,k | λ n,j = +∞. ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② J ✐♥ ■◆✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ♦rt❤♦❣♦♥❛•✐t② ♣r♦♣❡rt② ✭✾✮ u 0,n 2 Ḣs = J j=0 V j 2 Ḣs + ψ J n 2 Ḣs + •(1), ✇❤❡♥ n → +∞. ❆ ✜rst ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤✐s✱ ✐s ❚❤❡♦r❡♠ ✷✳✷ ❛❜♦✉t t❤❡ •✐❢❡s♣❛♥ ♦❢ ❛ ◆❙✲s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❜♦✉♥❞❡❞ ❞❛t❛ ✐♥ Ḣs ✳ ❚❤❡ ♣r♦♦❢ ♦❢ ✐t ✇✐•• ❜❡ ❣✐✈❡♥ ✐♥ s❡❝t✐♦♥ 3✳ ❚❤❡♦r❡♠ ✷✳✷✳ ▲❡t (u 0,n ) ❜❡ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ✐♥✐t✐❛• ❞❛t❛ ✐♥ Ḣs s✉❝❤ t❤❛t ✐ts ♣r♦✜•❡s ❞❡❝♦♠♣♦✲ s✐t✐♦♥ ✐s ❣✐✈❡♥ ❜② u 0,n (x) = J j=0 Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) ✇✐t❤ lim J→+∞ lim sup n→+∞ ψ J n L p = 0. ▲❡t ✉s ❞❡✜♥❡ J 1 ❛s t❤❡ s✉❜s❡t ♦❢ ✐♥❞✐❝❡s j ✐♥ ■◆✱ s✉❝❤ t❤❛t t❤❡ ♣r♦✜•❡ V j ✐s ♥♦♥✲③❡r♦ ❛♥❞ s✉❝❤ t❤❛t t❤❡ ❛ss♦❝✐❛t❡❞ s❝❛•❡ λ n,j ✐s ✐❞❡♥t✐❝❛••② ❡q✉❛• t♦ 1✳ ■❢ J 1 = ∅✱ t❤❡♥ lim inf n→+∞ T * (u 0,n ) = +∞✳ ■❢ J 1 = ∅✱ t❤❡♥ lim inf n→+∞ T * (u 0,n ) inf j∈J 1 T * (V j )✳ |❡♠❛r❦ ✷✳✷✳ ▲❡t ✉s ♣♦✐♥t ♦✉t s♦♠❡ ❢❛❝ts✳ ❋✐rst•②✱ ✐❢ T * (V j ) = +∞ ❢♦r ❛♥② j✱ t❤❡♥ lim inf n→+∞ T * (u 0,n ) = +∞✳ ❙❡❝♦♥❞•②✱ ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ J 1 ✐s ♥♦♥ ❡♠♣t②✱ t❤❡ q✉❛♥t✐t② inf j∈J 1 T * (V j ) ❡①✐sts ❛♥❞ ♦❜✈✐♦✉s•②✱ ✐❢ |J 1 | ✐s ✜♥✐t❡✱ ✇❡ ❣❡t ✐♠♠❡❞✐❛t❡•② t❤❛t inf j∈J 1 T * (V j ) = min j∈J 1 T * (V j )✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ |J 1 | ✐s ✐♥✜♥✐t❡✱ ✇❡ ❣❡t t❤❡ s❛♠❡ ❝♦♥❝•✉s✐♦♥✳ ■♥❞❡❡❞✱ ❜② ✈❡rt✉❡ ♦❢ ✭✾✮✱ t❤❡ s❡r✐❡ j 0 V j 2 Ḣs ✐s s✉♠♠❛❜•❡ ✭❛ ❢♦rt✐♦r✐ ✐❢ ✇❡ ❝♦♥s✐❞❡r ✐♥ t❤❡ s✉♠♠❛t✐♦♥ ✐♥t❡❣❡rs ❜❡•♦♥❣✐♥❣ t♦ J 1 ✮✱ ❛♥❞ t❤✉s lim j→+∞ V j Ḣs = 0✳ ❚❤❛♥❦s t♦ ■♥❡q✉❛•✐t② ✭✻✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t lim j→+∞ T * (V j ) = +∞ ❛♥❞ t❤✉s inf j∈J 1 T * (V j ) > 0 ❛♥❞ inf j∈J 1 T * (V j ) = min j∈J 1 T * (V j ). ❚❤✐s r❡s✉•t ❣✐✈❡s ✉s ❛♥ ✐♠♣♦rt❛♥t ✐♥❢♦r♠❛t✐♦♥✿ ✇❤❡♥❡✈❡r ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥✐t✐❛• ❞❛t❛ ✇❤✐❝❤ s❛t✐s✜❡s ♣r♦✜•❡s ❤②♣♦t❤❡s✐s ✭✐t ♠❡❛♥s ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐♥ Ḣs ✮✱ ✇❡ ❣❡t ❛♥ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ •✐❢❡s♣❛♥ ♦❢ t❤❡ ◆❙✲s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ s✉❝❤ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥✐t✐❛• ❞❛t❛✿ ✐t ♠❛✐♥•② ❞❡♣❡♥❞s ♦♥ t❤❡ •✐❢❡s♣❛♥ ♦❢ ♣r♦✜•❡s ✇✐t❤ ❛ ❝♦♥st❛♥t s❝❛•❡✳ ◆♦t❡ t❤❛t t❤❡ ♦rt❤♦❣♦♥❛•✐t② ♣r♦♣❡rt② ♦♥ s❝❛•❡s ❛♥❞ ❝♦r❡s ✐♥ ❚❤❡♦r❡♠ ✷✳✶ ✐♠♣•✐❡s ❡✐t❤❡r t❤❡ s❝❛•❡s ❛r❡ ❞✐✛❡r❡♥t ✭✐♥ t❤❡ s❡♥s❡ t❤❛t lim n→+∞ λ n,j λ n,k + λ n,k λ n,j = +∞✮ ♦r t❤❡ s❝❛•❡s ❛r❡ t❤❡ s❛♠❡ ✭λ n,j = λ n,k ✮✱ ❡q✉❛• t♦ ❛ ❝♦♥st❛♥t✱ ❛♥❞ t❤❡ ❝♦r❡s ❣♦ ❛✇❛② ❢r♦♠ ♦♥❡ ❛♥♦t❤❡r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t lim n→+∞ |x n,j -x n,k | λ n,j = +∞✳ ■♥ t❤❡ •❛st ❝❛s❡ ✇❤❡r❡ s❝❛•❡s ❛r❡ ❡q✉❛• t♦ ❛ ❝♦♥st❛♥t✱ ✇❡ s❤❛•• ❛ss✉♠❡ t❤❛t ✐t ✐s ♦♥❡✱ ✉♣ t♦ r❡s❝❛•✐♥❣ ♣r♦✜•❡s ❜② ❛ ✜①❡❞ ❝♦♥st❛♥t✳ ❚❤❡♦r❡♠ ✷✳✷ ❤❛s ❛ ❦❡② r♦•❡ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦♠♣❛❝t♥❡ss ❚❤❡♦r❡♠ ✶✳✸✿ t❤❡ s❡t M s (T * )✱ r❡❝❛••❡❞ ❜❡•♦✇✱ ✐s ♥♦♥ ❡♠♣t② ❛♥❞ ❝♦♠♣❛❝t✱ ✉♣ t♦ tr❛♥s•❛t✐♦♥s✳ M s (T * ) := u 0 ∈ Ḣs | T * (u 0 ) = T * ❛♥❞ u 0 Ḣs = ρ s (T * ) . ✷✳✶✳ Pr♦♦❢ ♦❢ t❤❡ ❝♦♠♣❛❝t♥❡ss ❚❤❡♦r❡♠ ✶✳✸✳ Pr♦♦❢✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ A σs c ✱ ✇❡ ❝♦♥s✐❞❡r ❛ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ (u 0,n ) n 0 s✉❝❤ t❤❛t lim n→+∞ T * (u 0,n ) u 0,n σs Ḣs = A σs c . ❯♣ t♦ ❛ r❡s❝❛•✐♥❣ ♣r♦❝❡ss✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ (u 0,n ) n 0 s❛t✐s✜❡s ✭✶✵✮ lim n→+∞ u 0,n Ḣs = ρ s (T * ) ❛♥❞ T * (u 0,n ) = T * . ■♥❞❡❡❞✱ ❝♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ (v 0,n ) n 0 ❞❡✜♥❡❞ ❛s v 0,n (x) ❞❡❢ = T * (u 0,n ) T * 1 2 u 0,n T * (u 0,n ) T * 1 2 x .
❚❤❡ r❡❛❞❡r ♥♦t✐❝❡s t❤❛t t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ s✉❝❤ ❛ s❡q✉❡♥❝❡

(v 0,n ) ❤❛s ❛ •✐❢❡s♣❛♥ ❡q✉❛• t♦ T * ✳ ❆s v 0,n σs Ḣs = T * (u 0,n ) T * u 0,n σs Ḣs ✱ ✐t s❡❡♠s ❝•❡❛r ♥♦✇ ✇❡ ❝❛♥ ❛ss✉♠❡ ✭✶✵✮✱ ❜② ✈❡rt✉❡ ♦❢ ❞❡✜♥✐t✐♦♥ ♦❢ ρ s (T * )✳ ❆s ❞❡✜♥❡❞✱ (u 0,n ) n 0 ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ♦❢ t❤❡ s❡t M s (T * )❀ ✐t ✐s ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐♥ Ḣs ❛♥❞ t❤✉s ✇❡ ❝❛♥ ❛♣♣•② ❚❤❡♦r❡♠ ✷✳✶✳ ❚❛❦✐♥❣ •✐♠✐t ✇❤❡♥ n → +∞ ✐♥ ✭✾✮✱ ✇❡ ❣❡t ∀J 0, ρ 2 s (T * ) J j=0 V j 2 Ḣs . ▲❡t ✉s ❛ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ t✇♦ ♥♦♥✲③❡r♦ ♣r♦✜•❡s ❛t •❡❛st✳ ❚❤❡♥ ✇❡ s❤♦✉•❞ ❤❛✈❡ ∀j ∈ {0, • • • , J}, V j 2 Ḣs < ρ 2 s (T * ). ❇② ❞❡✜♥✐t♦♥ ♦❢ ρ s (T * )✱ ✐t ♠❡❛♥s ❛•• ♣r♦✜•❡s V j ❣❡♥❡r❛t❡ s♦•✉t✐♦♥s ✇❤♦s❡ •✐❢❡s♣❛♥ s❛t✐s✜❡s ✭✶✶✮ T * (V j ) > T * , ∀j ∈ {0, • • • , J} ❆s T * (u 0,n ) = T * < ∞ ❢♦r ❛♥② n✱ ❚❤❡♦r❡♠ ✷✳✷ ✐♠♣•✐❡s t❤❛t J 1 = ∅✿ t❤❡r❡ ❡①✐sts ❛t •❡❛st ♦♥❡ ♣r♦✜•❡ ✇✐t❤ ❝♦♥st❛♥t s❝❛•❡✳ ▼♦r❡♦✈❡r✱ t❤❛♥❦s t♦ |❡♠❛r❦ ✷✳✷✱ ✇❡ ❤❛✈❡ inf j∈J 1 T * (V j ) = min j∈J 1 T * (V j )✳ ❈♦♠❜✐♥✐♥❣ t❤✐s ✇✐t❤ |❡•❛t✐♦♥ ✭✶✶✮ ✐♠♣•✐❡s t❤❛t T ❞❡❢ = inf j∈J 1 T * (V j ) > T * . ❇② ❤②♣♦t❤❡s✐s ♦♥ (u 0,n ) n∈■◆ ❛♥❞ t❤❛♥❦s t♦ ❚❤❡♦r❡♠ ✷✳✷✱ ✇❡ ❣❡t ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ s✐♥❝❡ ✇❡ ❤❛✈❡ lim inf n→+∞ T * (u 0,n ) = T * T > T * . ✽ ❊❯•➱◆■❊ P❖❯▲❖◆ ■t ♠❡❛♥s t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r j 0 s✉❝❤ t❤❛t t❤❡ ♣r♦✜•❡✱ V j 0 ❤❛s ❛ •✐❢❡s♣❛♥ ✇❤✐❝❤ s❛t✐s✜❡s T j 0 * T * ✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ρ s (T * )✱ ✐t ✐♠♣•✐❡s t❤❛t V j 0 2 Ḣs ρ 2 s (T * )✳ ❆♥❞✱ t❤❛♥❦s t♦ t❤❡ ♦rt❤♦❣♦♥❛• ♣r♦♣❡rt② ♦❢ t❤❡ Ḣs ✲♥♦r♠ ✭✾✮✱ ✇❡ ❞❡❞✉❝❡ t❤❡ ❡q✉❛•✐t② V j 0 2 Ḣs = ρ 2 s (T * ). ◆♦✇✱ ✇❡ ❤❛✈❡ •✉st t♦ ❝❤❡❝❦ t❤❛t T * = T j 0 * ✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② ♣r♦✈❡❞ ❛ ✜rst ✐♥❡q✉❛•✐t②✿ T j 0 * T * ✳ ❚❤❡ ♦t❤❡r ✇❛② ✐s ❣✐✈❡♥ ❜② ✭✽✮✿ ✇❡ ❤❛✈❡ ❛•✇❛②s t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥✿ T j 0 * V j 0 σs Ḣs A σs c ✳ ❚❤❛♥❦s t♦ t❤❡ r❡s✉•t V j 0 σs Ḣs = ρ σs s (T * ) = A σs c T * ✱ ✇❡ ❣❡t t❤❡ s❡❝♦♥❞ ✐♥❡q✉❛•✐t②✿ T j 0 * T * ✳ ❚❤✉s✱ t❤❡ s❡t M s (T * ) ✐s ♥♦♥ ❡♠♣t② ❛♥❞ t❤✉s✱ t❤❡r❡ ❡①✐sts s♦♠❡ ♠✐♥✐♠❛• ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥s✳ ❚❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ t❤❡ s❡t M s (T * ) ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❛❜♦✈❡ ✇♦r❦✳ ❚❤❛♥❦s t♦ ✭✾✮ ❛♥❞ V j 0 Ḣs = ρ s (T * )✱ ✇❡ ✐♥❢❡r t❤❛t ∀j = j 0 , V j = 0 ❛♥❞ lim n→+∞ ψ J n 2 Ḣs = 0. ❚❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥ ✐♠♣•✐❡s ✐♥ ♣❛rt✐❝✉•❛r t❤❛t j 0 ∈ J 1 ✳ ■♥❞❡❡❞✱ ✐❢ j 0 / ∈ J 1 ✱ t❤❡♥ J 1 = ∅ ❛♥❞ t❤✉s ✇❡ s❤♦✉•❞ ❤❛✈❡ T * = +∞✱ ✇❤✐❝❤ ✐s ❛❜s✉r❞✳ ❆s ❛ r❡s✉•t✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✐♥t❡❣❡r j 0 ∈ J 1 ✱ s✉❝❤ t❤❛t u 0,n (x) = V j 0 (x -x n,j 0 ) + ψ J n (x). ❚❤❡ ♣r♦♣❡rt② lim n→+∞ ψ l n 2 Ḣs = 0 ✐♠♣•✐❡s lim n→+∞ u 0,n (• +x j 0 ,n ) -V j 0 Ḣs = 0✳ ✷✳✷✳ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹✳ Pr♦♦❢✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❝r✐t✐❝❛• ❡•❡♠❡♥t u = N S(u 0 ) ✿ T 1 σs * (u 0 ) u 0 Ḣs = A c ✳ ❇② ✈❡rt✉❡ ♦❢ ❛ r❡s❝❛•✐♥❣✱ ✇❡ ❝❛♥ ❛s✉♠❡ t❤❛t u 0 Ḣs = 1 ❛♥❞ t❤✉s T 1 σs * (u 0 ) = A c ✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ s❡t N s T ❞❡❢ = N S(u 0 ) X T u 0 ✐♥ Ḣs s✉❝❤ t❤❛t u 0 Ḣs = 1 ❛♥❞ T < A σs c . ❚❤❡♦r❡♠ ✶✳✸ ❝•❛✐♠s t❤❛t t❤❡ s❡t N s T ✐s ♥♦♥❡♠♣t②✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t sup N s T ✐s ✜♥✐t❡ ❢♦r ❛♥② T ✳ ■❢ ♥♦t✱ ❛ s❡q✉❡♥❝❡ (u 0,n ) n 0 ✐♥ Ḣs ❡①✐sts✱ s✉❝❤ t❤❛t ❢♦r ❛♥② T < T * (u 0,n )✱ ✇❡ ❤❛✈❡ ✭✶✷✮ u 0,n Ḣs = 1, T * (u 0,n ) = A σs c ❛♥❞ lim n→+∞ N S(u 0,n ) X T = ∞. ❇② ❤②♣♦t❤❡s✐s✱ t❤❡ s❡q✉❡♥❝❡ (u 0,n ) n 0 ❜❡•♦♥❣s t♦ t❤❡ s❡t M s (T * )✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❡①✐st ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦r❡s (x n ) n∈■◆ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ V ✐♥ M s (T * ) s✉❝❤ t❤❛t✱ ✉♣ t♦ ❛♥ ❡①tr❛❝t✐♦♥✿ ✭✶✸✮ lim n→+∞ ||u 0,n (• + x n ) -V || Ḣs = 0. ❲❡ ❝❛♥ ♣r♦✈❡ ❡❛s✐•② t❤❛t✱ ❢♦r ❛♥② T < T * (V )✿ ✭✶✹✮ N S(u 0,n (• + x n )) = N S(V ) + R n ✇✐t❤ lim n→+∞ R n X T = 0 ■♥❞❡❡❞✱ ✇❡ ❞❡✜♥❡ R 0,n ❞❡❢ = u 0,n (• + x n ) -V. ❇❡❝❛✉s❡ ♦❢ ✭✶✸✮✱ t❤❡ s❡q✉❡♥❝❡ (R 0,n ) n 0 ❝♦♥✈❡r❣❡s t♦ 0 ✐♥ Ḣs ✲♥♦r♠✱ ❢♦r n •❛r❣❡ ❡♥♦✉❣❤✳ ▼♦r❡♦✈❡r✱ t❤❡ ❡rr♦r t❡r♠ R n s❛t✐✜❡s t❤❡ ❢♦••♦✇✐♥❣ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s s②st❡♠ ✭✶✺✮    ∂ t R n + R n • ∇R n -∆R n + R n • ∇N S(V ) + N S(V ) • ∇R n = -∇p div R n = 0 R n|t=0 = R 0,n . ❆♣♣•②✐♥❣ ❢♦rt❤❝♦♠✐♥❣ ❚❤❡♦r❡♠ ✻✳✶✱ ✇❡ ✐♥❢❡r t❤❛t✱ ❢♦r ❛♥② T < T * (V ) ❛♥❞ ❢♦r n •❛r❣❡ ❡♥♦✉❣❤ N S(u 0,n (• + x n )) X T N S(V ) X T + •(1). ❆s N S(u 0,n (• + x n )) X T = N S(u 0,n ) X T ✱ ✇❡ t❛❦❡ t❤❡ •✐♠✐t ✇❤❡♥ n → +∞ ✐♥ t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛•✐t② ❛♥❞ t❤✉s ✇❡ ❣❡t ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥✳ ✸✳ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✷ ❛♥❞ t♦♦• ❜♦① ❢♦r ▲❡♠♠❛ ✸✳✶ ❆•• t❤❡ ♣r❡✈✐♦✉s r❡s✉•ts ❛r❡ ❜❛s❡❞ ♦♥ ❚❤❡♦r❡♠ ✷✳✷✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r♦✈❡ t❤✐s t❤❡♦r❡♠✱ ✇❤✐❝❤ r❡•✐❡s ♦♥ ▲❡♠♠❛ ✸✳✶✳ ❚❤✐s •❛st ♦♥❡ ❣✐✈❡s t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥ ✐♥✐t✐❛• ❞❛t❛ ✇❤✐❝❤ ❤❛s ❛ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ■♥ ♦t❤❡rs ✇♦r❞s✱ ✇❡ ✇♦♥❞❡r ✐❢✱ ❣✐✈❡♥ t❤❡ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❞❛t❛✱ ✇❡ ❣❡t ❛ s✐♠✐•❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦♥ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ✐ts❡•❢✳ ▲❡♠♠❛ ✸✳✶ ❣✐✈❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✳ ▲❡t ✉s r❡❝❛•• t♦ t❤❡ r❡❛❞❡r t❤❛t t❤✐s q✉❡st✐♦♥ ❤❛s ❛•r❡❛❞② ❜❡❡♥ st✉❞✐❡❞ ❜② ■✳ •❛••❛❣❤❡r ✐♥ ❬✶✹( ✐♥ t❤❡ ❝❛s❡ ♦❢ ✐♥✐t✐❛• ❞❛t❛ ✐♥ t❤❡ ❙♦❜♦•❡✈ s♣❛❝❡ Ḣ 1 2 ❛♥❞ t❤❡ s❛♠❡ ❛✉t❤♦r ✇✐t❤ •✳ ❑♦❝❤✱ ❋✳ P•❛♥❝❤♦♥ ❬✶✻( ✐♥ ♦t❤❡rs ❝r✐t✐❝❛• s♣❛❝❡s ✭❡✳❣ s❝❛•❡❞ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s tr❛♥s❢♦r♠❛t✐♦♥✮✳ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❞✐✣❝✉•t② ✐s t❤❛t t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦•❡✈ s♣❛❝❡ Ḣs ✐s ♥♦t ❛ s❝❛•❡ ✐♥✈❛r✐❛♥t s♣❛❝❡ ✉♥❞❡r t❤❡ ♥❛t✉r❛• s❝❛•✐♥❣ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✱ t❤❡ ♠❡t❤♦❞ ❝♦♥s✐sts ✐♥ ❝✉tt✐♥❣ ♦✛ ❢r❡q✉❡♥❝✐❡s ♦❢ ♣r♦✜•❡s ❬✹( ✭s✉❝❤ ♣r♦✜•❡s ✇✐•• ❤❛✈❡ t❤❡ ✉s❡❢✉• ♣r♦♣❡rt② t♦ ❜❡•♦♥❣ t♦ ❛♥② Ḣs ✱ ❢♦r ❛♥② s✮✳ ■♥ ♣❛rt✐❝✉•❛r✱ ♣r♦✜•❡s s❝❛•❡❞ ❜② 0 ✭r❡s♣✳ ∞✮ ✇✐•• t❡♥❞ t♦ 0 ✐♥ s♦♠❡ ❙♦❜♦•❡✈ s♣❛❝❡s ✭♠♦r❡ ♣r❡❝✐s❡•② ✐♥ Ḣs 1 ✇✐t❤ s 1 < s✮✱ ✭r❡s♣✳ Ḣs 2 ✇✐t❤ s 2 > s✮ ❛♥❞ t❤❡r❡❢♦r❡✱ ✇✐•• ♥♦t ♣❡rt✉r❜ t❤❡ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ◆❙✲s♦•✉t✐♦♥✳ ✸✳✶✳ ❑❡② ▲❡♠♠❛ ❛♥❞ ❛♣♣•✐❝❛t✐♦♥✳ ▲❡t (u 0,n ) n 0 ❜❡ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ✐♥✐t✐❛• ❞❛t❛ ✐♥ Ḣs ✳ ❚❤❛♥❦s t♦ ❚❤❡♦r❡♠ ✷✳✶✱ (u 0,n ) n 0 ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s✱ ✉♣ t♦ ❛♥ ❡①tr❛❝t✐♦♥ u 0,n (x) = J j=0 Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x). ❇② ✈❡rt✉❡ ♦❢ ♦rt❤♦❣♦♥❛•✐t② ♦❢ s❝❛•❡s ❛♥❞ ❝♦r❡s ❣✐✈❡♥ ❜② ❚❤❡♦r❡♠ ✷✳✶✱ ✇❡ s♦rt ♣r♦✜•❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r s❝❛•❡s u 0,n (x) = j∈J 1 j J V j (x -x n,j ) + j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) ✭✶✻✮ ✇❤❡r❡ ❢♦r ❛♥② j ∈ J 1 ✱ ❢♦r ❛♥② n ∈ ■◆✱ λ n,j ≡ 1✳ ❲❡ ❝•❛✐♠ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ str✉❝t✉r❡ •❡♠♠❛ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥s✱ ✇❤✐❝❤ ♣r♦♦❢ ✇✐•• ❜❡ ♣r♦✈✐❞❡❞ ✐♥ s❡❝t✐♦♥ 4✳ ❚❤✐s •❡♠♠❛ ❤✐❣❤•✐❣❤ts t❤❡ s♣❡❝✐✜❝ r♦•❡ ♦❢ ♣r♦✜•❡s ✇✐t❤ ❝♦♥st❛♥t✲s❝❛•❡s✳ ▲❡♠♠❛ ✸✳✶✳ ✭Pr♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥✮ ▲❡t (u 0,n ) n 0 ❜❡ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ✐♥✐t✐❛• ❞❛t❛ ✐♥ Ḣs ✇❤✐❝❤ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s ❣✐✈❡♥ ❜② u 0,n (x) = J j=0 Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x). ❚❤❡♥✱ u ✐s ❛ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ u 0,n ✭❡✳❣ u = N S(u 0,n )✮ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡ ❡rr♦r t❡r♠ R J n ❞❡✜♥❡❞ ❜② R J n ❞❡❢ = N S(u 0,n ) -U app,J n ✐s ❛ s♦•✉t✐♦♥ ♦❢ t❤❡ ❜❡•♦✇ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥ ✭✶✼✮    ∂ t R J n + R J n .∇R J n -∆R J n + R J n • ∇U app,J n + U app,J n .∇R J n = -F J n -∇p J n div R J n = 0 R J n |t=0 = 0. ✇❤❡r❡ F J n ✐s ❛ ❢♦r❝✐♥❣ t❡r♠ ✇❤✐❝❤ ✇✐•• ❜❡ ❡①♣•✐❝✐t❡•② ❞❡t❛✐•❡❞ ✐♥ ✭✷✻✮ ❛♥❞ U app,J n (t, x) ❞❡❢ = j∈J 1 j J N S(V j )(t, x -x n,j ) + e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) . ▼♦r❡♦✈❡r✱ t❤❡ •✐❢❡s♣❛♥ τ J n ♦❢ t❤❡ ❡rr♦r t❡r♠ R J n s❛t✐s✐✜❡s ∀ε > 0, ∃J 0 ∃n J 0 ∀n n J , τ J n inf j∈J 1 T * (V j ) -ε. Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✷✳ ❈•❡❛r•②✱ ❚❤❡♦r❡♠ ✷✳✷ ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▲❡♠♠❛ ✸✳✶✳ ❆ss✉♠❡ ▲❡♠♠❛ ✸✳✶ ✐s ♣r♦✈❡❞✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ✐❢ t❤❡r❡ ✐s ♥♦ ♥♦♥ ③❡r♦ ♣r♦✜•❡ ✇✐t❤ ❝♦♥st❛♥t s❝❛•❡ ✭❡✳❣ J 1 = ∅✮✱ t❤❡ ✏♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥✑ ♦❢ t❤❡ s♦•✉t✐♦♥ ✐♥ ▲❡♠♠❛ ✸✳✶ ✐♠♣•✐❡s t❤❛t lim inf n→+∞ T * (u 0,n ) = +∞✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ J 1 = ∅✱ t❤❡ •✐❢❡s♣❛♥ ♦❢ s❡q✉❡♥❝❡ N S(u 0,n ) ✐s ❣✐✈❡♥ ❜② t❤❡ •✐❢❡s♣❛♥ ♦❢ ♣r♦✜•❡s✱ s❝❛•❡❞ ❜② t❤❡ ❝♦♥st❛♥t 1 ❛♥❞ T * (u 0,n ) inf j∈J 1 T * (V j )✳ ❚❤✐s ❡♥❞s ✉♣ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✷✳ ✸✳✷✳ ❚♦♦• ❜♦①✳ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ r❡❝❛•• s♦♠❡ ❜❛s✐❝ ❢❛❝ts ❛❜♦✉t ❤♦♠♦❣❡♥❡♦✉s ❇❡s♦✈ s♣❛❝❡s ❛♥❞ ✇❡ ♣r♦✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✇❡ ♥❡❡❞ t♦ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✶✳ ❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✶(✱ ❢r♦♠ ♣❛❣❡ 63✱ ❢♦r ❛ ❞❡t❛✐•❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t❤❡♦r② ❛♥❞ ❛♥❛•②s✐s ♦❢ ❤♦♠♦❣❡♥❡♦✉s ❇❡s♦✈ s♣❛❝❡s✳ ❉❡✜♥✐t✐♦♥ ✸✳✶✳ ▲❡t s ❜❡ ✐♥ ■|✱ (p, r) ✐♥ [1, +∞] 2 ❛♥❞ u ✐♥ S ′ ✳ ❆ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ u ✐s ❛♥ ❡•❡♠❡♥t ♦❢ t❤❡ ❇❡s♦✈ s♣❛❝❡ Ḃs p,r ✐❢ u s❛t✐✜❡s u Ḃs p,r ❞❡❢ = j∈Z 2 jrs || ∆j u|| r L p 1 r < ∞, ✇❤❡r❡ ∆j ✐s ❛ ❢r❡q✉❡♥❝✐❡s •♦❝❛•✐③❛t✐♦♥ ♦♣❡r❛t♦r ✭❝❛••❡❞ ▲✐tt•❡✇♦♦❞✲P❛•❡② ♦♣❡r❛t♦r✮✱ ❞❡✜♥❡❞ ❜② ∆j u(ξ) ❞❡❢ = F -1 ϕ(2 -j |ξ|) u(ξ) , ✇✐t❤ ϕ ∈ D([ 1 2 , 2])✱ s✉❝❤ t❤❛t j∈Z ϕ(2 -j t) = 1✱ ❢♦r ❛♥② t > 0✳ |❡♠❛r❦ ✸✳✶✳ ❲❡ ❤❛✈❡ t❤❡ ❡♠❜❡❞❞✐♥❣ Ḣs ⊂ Ḃs 2,2 ✳ ❚❤❡s❡ s♣❛❝❡s ❝♦✐♥❝✐❞❡ ✐❢ s < 3 2 ✳ ❚❤❡ ✜rst t❤✐♥❣ ✇❡ ❤❛✈❡ t♦ ♥♦t✐❝❡ ✐s t❤❡ ❢♦••♦✇✐♥❣✿ ❣✐✈❡♥ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ❞❛t❛ ✐♥ Ḣs ✭t❤✉s ✇❡ ❣❡t ❛ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤✐s s❡q✉❡♥❝❡✮✱ ❚❤❡♦r❡♠ ✷✳✶ ✐♠♣•✐❡s t❤❛t t❤❡ t❡r♠ ψ J n (x)✱ ✭✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ✐♥ Ḣs ✮✱ s❛t✐s✜❡s✿ lim J→+∞ lim sup n→+∞ ψ J n L p = 0. ■♥ ❢❛❝t✱ t❤❛♥❦s t♦ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ ❛r❣✉♠❡♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡ r❡♠❛✐♥✐♥❣ t❡r♠ ψ J n t❡♥❞s t♦ 0 ✐♥ ❝❡rt❛✐♥ ❇❡s♦✈ s♣❛❝❡s✳ ❚❤❛t ✐s t❤❡ ♣♦✐♥t ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳ ❋♦r ❛♥② 0 < θ < 1✱ •❡t p θ ❜❡ ❛ ♣♦s✐t✐✈❡ r❡❛• ♥✉♠❜❡r ❣✐✈❡♥ ❜② t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ r❡•❛t✐♦♥ 1 p θ = θ p + 1 -θ 2 • ❚❤❡♥✱ ✉♥❞❡r t❤❡ s❛♠❡ ❤②♣♦t❤❡s✐s ♦❢ ❚❤❡♦r❡♠ ✷✳✶✱ ✇❡ ❤❛✈❡✿ lim J→+∞ lim sup n→+∞ ψ J n Ḃs(1-θ) p θ ,p θ = 0• Pr♦♦❢✳ ■♥t❡r♣♦•❛t✐♦♥ ✐♥❡q✉❛•✐t② ✐♥ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ❛♥❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② t❤❡ ❢❛❝t♦r 2 js(1-θ) ❣✐✈❡ 2 js(1-θ) ∆j ψ J n L p θ ∆j ψ J n θ L p 2 js ∆j ψ J n L 2 1-θ . ❆♣♣•②✐♥❣ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ✐♥ t❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥✱ ✇❡ ❣❡t ψ J n Ḃs(1-θ) p θ ,p θ ψ J n θ Ḃ0 p,p ψ J n 1-θ Ḃs 2,2 . ❇❡❝❛✉s❡ p ✐s ❣r❡❛t❡r t❤❛♥ 2✱ L p ✐s ❝♦♥t✐♥✉♦✉s•② ✐♥❝•✉❞❡❞ ✐♥ Ḃ0 p,p ✳ |❡♠❛r❦ ✸✳✶ •❡❛❞s t♦ ✭✶✽✮ ψ J n Ḃs(1-θ) p θ ,p θ ψ J n θ L p ψ J n 1-θ Ḣs . ❇② ✈❡rt✉❡ ♦❢ ❚❤❡♦r❡♠ ✷✳✶✱ ✇❡ ❣❡t t❤❡ r❡s✉•t✳ ▲❡t ✉s ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡ (u 0,n ) n 0 ❛♥❞ ✐♥tr♦❞✉❝❡ s♦♠❡ ♥♦t❛t✐♦♥s✳ ▲❡t η > 0 ❜❡ t❤❡ ♣❛r❛♠❡t❡r ♦❢ r♦✉❣❤ ❝✉tt✐♥❣ ♦✛ ❢r❡q✉❡♥❝✐❡s✳ ❲❡ ❞❡✜♥❡ ❜② u η (x) ❛♥❞ uc η (x) t❤❡ ❡•❡♠❡♥ts ✇❤✐❝❤ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ❣✐✈❡♥ ❜② ✭✶✾✮ u η (ξ) = u(ξ)1 { 1 η |ξ| η} ❛♥❞ uc η (ξ) = u(ξ) 1 -1 { 1 η |ξ| η} . ❋r♦♠ t❤❡ ♣r♦✜•❡s ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✶✻✮✱ ✇❡ ✐♥❢❡r✱ t❤❛♥❦s t♦ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ♣r♦♣❡rt② ♦❢ s❝❛•❡s✱ t❤❛t ❛♠♦♥❣ ♣r♦✜•❡s V j s✉❝❤ t❤❛t j ❜❡•♦♥❣s t♦ c J 1 ✱ t❤❡r❡ ❛r❡ ♣r♦✜•❡s ✇✐t❤ s♠❛•• s❝❛•❡s ✭j ∈ J 0 ✮ ❛♥❞ •❛r❣❡ s❝❛•❡s ✭j ∈ J ∞ ✮✳ ❚❤❡s❡ ♣r♦✜•❡s ❛r❡ ❝✉t ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣❛r❛♠❡t❡r η✮✱ ✇✐t❤ r❡s♣❡❝t t♦ ♥♦t❛t✐♦♥s ✭✶✾✮ ❛♥❞ ✇❡ ❣❡t u 0,n (x) = j∈J 1 j J V j (x -x n,j ) + j∈J 0 j J Λ 3 p λ n,j ,x n,j V j η (x) + j∈J∞ j J Λ 3 p λ n,j ,x n,j V j η (x) + ψ J n,η (x) ✇❤❡r❡ ψ J n,η (x) ❞❡❢ = j∈J c 1 ≡J 0 ∪J∞ j J Λ 3 p λ n,j ,x n,j V j c η (x) + ψ J n (x), ✭✷✵✮ ✇✐t❤ ❢♦r ❛♥② j ∈ J 0 ✱ lim n→+∞ λ n,j = 0 ❛♥❞ ❢♦r ❛♥② j ∈ J ∞ ✱ lim n→+∞ λ n,j = +∞✳ ❋✐rst•②✱ ✇❡ ❝❤❡❝❦ t❤❡ r❡♠❛✐♥✐♥❣ t❡r♠ ψ J n,η ✐s st✐•• s♠❛•• ✐♥ Ḃs(1-θ) p θ ,p θ ✲♥♦r♠✱ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ s❡♥s❡✳ ❚❤❛t ✐s t❤❡ ♣♦✐♥t ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❜❡•♦✇✳ Pr♦♣♦s✐t✐♦♥ ✸✳✸✳ ▲❡t 0 < θ < 1✳ ❯♥❞❡r t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ r❡•❛t✐♦♥ 1 p θ = θ p + 1 -θ 2 ✱ ✇❡ ❤❛✈❡ lim J→+∞ lim η→+∞ lim sup n→+∞ ψ J n,η Ḃs(1-θ) p θ ,p θ = 0. Pr♦♦❢✳ ▲❡t 0 < θ < 1✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ ψ J n,η ❛♥❞ t❤❛♥❦s t♦ (a + b) 2 a 2 + b 2 ✱ ✇❡ ❤❛✈❡ ψ J n,η 2 Ḃs(1-θ) p θ ,p θ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j c η (x) 2 Ḃs(1-θ) p θ ,p θ + ψ J n 2 Ḃs(1-θ) p θ ,p θ . ✭✷✶✮ ❚❤❡ ❡♠❜❡❞❞✐♥❣ Ḣs ⊂ Ḃs(1-θ) p θ ,p θ ❛♥❞ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ♦❢ s❝❛•❡s ❛♥❞ ❝♦r❡s ✐♠♣•② ψ J n,η 2 Ḃs(1-θ) p θ ,p θ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j c η (x) 2 Ḣs + ψ J n 2 Ḃs(1-θ) p θ ,p θ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j c η (x) 2 Ḣs 2 Ḣs + •(1) + ψ J n 2 Ḃs(1-θ) p θ ,p θ . ✭✷✷✮ ❇② s❝❛•✐♥❣ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♥♦r♠ Ḣs ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ u → Λ 3 p λ n,j ,x n,j u✱ ✇❡ ❣❡t ψ J n,η 2 Ḃs(1-θ) p θ ,p θ ∞ j=0 V j c η (x) 2 Ḣs + •(1) + ψ J n 2 Ḃs(1-θ) p θ ,p θ , ✇❤❡♥ n → +∞. ❋♦r ❛♥② j 0✱ t❤❡ t❡r♠ V j c η (x) 2 Ḣs t❡♥❞s t♦ 0 ❢♦r η •❛r❣❡ ❡♥♦✉❣❤✱ ❜② ▲❡❜❡s❣✉❡ ❚❤❡♦r❡♠✳ ❚❤❡r❡❢♦r❡✱ ❛♣♣•②✐♥❣ ▲❡❜❡s❣✉❡ ❚❤❡♦r❡♠ ♦♥❝❡ ❛❣❛✐♥✱ ✇❡ ✐♥❢❡r t❤❛t lim η→+∞ ∞ j=0 V j c η (x) 2 Ḣs = 0✳ ❆s ❛ r❡s✉•t✱ ✇❡ t❛❦❡ ✶✷ ❊❯•➱◆■❊ P❖❯▲❖◆ ✐♥ ✜rst t❤❡ ✉♣♣❡r •✐♠✐t ♦❢ ψ J n,η 2 Ḃs(1-θ) p θ ,p θ ✱ ✇❤❡♥ n → +∞✳ ❚❤❡♥✱ ✇❡ t❛❦❡ t❤❡ •✐♠✐t ❢♦r η → +∞ ❛♥❞ ❛t t❤❡ •❛st✱ ❢♦r J → +∞✳ ❚❤❛♥❦s t♦ Pr♦♣♦s✐t✐♦♥ ✸✳✷✱ Pr♦♣♦s✐t✐♦♥ ✸✳✸ ✐s ♣r♦✈❡❞✳ ❆s ✐t ✇❛s ❛•r❡❛❞② ♠❡♥t✐♦♥♥❡❞ ♣r❡✈✐♦✉s•②✱ t❤❡ ♣♦✐♥t ♦❢ s✉❝❤ r♦✉❣❤ ❝✉tt✐♥❣ ♦✛ ✐♥ ❢r❡q✉❡♥❝✐❡s ✐s t❤❛t ♣r♦✜•❡s ✇❤✐❝❤ ❛r❡ s✉♣♣♦rt❡❞ ✐♥ t❤❡ ❛♥♥✉•✉s 1 { 1 η |ξ| η} ✱ ❜❡•♦♥❣ t♦ t❤❡ ❙♦❜♦•❡✈ s♣❛❝❡s Ḣs ✱ ❢♦r ❛♥② s > 0✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ❝❛♥ •♦♦❦ ❛t s✉❝❤ ♣r♦✜•❡s ✐♥ t❤❡ ❙♦❜♦•❡✈ s♣❛❝❡s s✉❝❤ ❛s Ḣs 1 ✇✐t❤ s 1 < s ❛♥❞ Ḣs 2 ✇✐t❤ s 2 > s✳ ❚❤❛t ✐s t❤❡ ♣♦✐♥t ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✿ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ s❝❛•❡ ✭❡✐t❤❡r s♠❛•• j ✐♥ J 0 ♦r •❛r❣❡ j ✐♥ J ∞ ✮✱ ♣r♦✜•❡s✱ tr❛♣♣❡❞ ✐♥ t❤❡ ❛♥♥✉•✉s✱ ❜❡❤❛✈❡ t❤❡✐rs❡•✈❡s ❛s ✏r❡♠❛✐♥✐♥❣ t❡r♠s✑✱ s❡❡♥ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ s♦•✈✐♥❣ ◆❛✈✐❡r✲❙t♦❦❡s✳ Pr♦♣♦s✐t✐♦♥ ✸✳✹✳ ❋♦r ❛♥② η > 0, s 1 < s, ❛♥❞ j ∈ J 0 , ❡✳❣ lim n→+∞ λ n,j = 0, t❤❡♥ lim n→+∞ Λ 3 p λ n,j ,x n,j V j η (x) Ḣs 1 = 0. ❋♦r ❛♥② η > 0, s 2 > s, ❛♥❞ j ∈ J ∞ , ❡✳❣ lim n→+∞ λ n,j = +∞, t❤❡♥ lim n→+∞ Λ 3 p λ n,j ,x n,j V j η (x) Ḣs 2 = 0. Pr♦♦❢✳ ▲❡t s 1 < s✳ ▲❡t j ∈ J 0 ❛♥❞ η > 0✳ ❉❡✜♥✐t✐♦♥ ♦❢ Ḣs 1 ✲♥♦r♠ ❛♥❞ ❛ ✈❛r✐❛❜•❡ ❝❤❛♥❣❡ ②✐❡•❞ Λ 3 p λ n,j ,x n,j V j η (x) 2 Ḣs 1 (■| 3 ) = ■| 3 |ξ| 2s 1 λ n,j 3(1-1 p ) V j η (λ n,j ξ) 2 dξ = λ 2(s-s 1 ) n,j ■| 3 |ξ| 2s 1 | V j η (ξ)| 2 dξ. ✭✷✸✮ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❢❛❝t♦r |ξ|✳ ❚❤❡ ❤②♣♦t❤❡s✐s ♦❢ t❤❡ r✐♥❣ ✐♠♣•✐❡s t❤❛t Λ 3 p λ n,j ,x n,j V j η (x) 2 Ḣs 1 = λ n,j 2(s-s 1 ) ■| 3 |ξ| 2s | V j η (ξ)| 2 1 |ξ| 2(s-s 1 ) dξ (η λ n,j ) 2(s-s 1 ) V j 2 Ḣs . ✭✷✹✮ ❆s λ n,j t❡♥❞s t♦ 0❀ t❤✐s ♣r♦✈❡s t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt r❡•✐❡s ♦♥ s✐♠✐•❛r ❛r❣✉♠❡♥ts ❛♥❞ t❤✉s t❤❡ ♣r♦♦❢ ✐s ♦♠✐tt❡❞✳ ✹✳ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✶ •✐✈❡♥ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ (u 0,n ) ✐♥ Ḣs ✇❤✐❝❤ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❚❤❡♦r❡♠ ✷✳✶✱ ✇❡ s❡❛r❝❤ s❡q✉❡♥❝❡s ❛ss♦❝✐❛t❡❞ s♦•✉t✐♦♥s N S(u 0,n )✱ ✉♥❞❡r t❤❡ ❢♦r♠ ♦❢ N S(u 0,n ) = U app,J n + R J n , ✇❤❡r❡ U app,J n ❞❡❢ = j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) , ✭✷✺✮ ◆♦t❡ t❤❛t ✐❢ J 1 = ∅✱ t❤❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❡r♠ U app,J n ✐s r❡❞✉❝❡❞ t♦ t❤❡ •✐♥❡❛r ♣❛rt U app,J n = e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) . P•✉❣❣✐♥❣ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥ •❡❛❞s t♦ t❤❡ ❢♦••♦✇✐♥❣ ♣❡rt✉r❜❡❞ ❡q✉❛t✐♦♥ ♦♥ t❤❡ ❡rr♦r t❡r♠ R J n ✭✷✻✮    ∂ t R J n + R J n .∇R J n -∆R J n + R J n • ∇U app,J n + U app,J n .∇R J n = -F J n -∇p J n div R J n = 0 R J n |t=0 = 0. ✇❤❡r❡ t❤❡ ❢♦r❝✐♥❣ t❡r♠ F J n ✐s ❣✐✈❡♥ ❜② F J n = 4 ℓ=1 F J,ℓ n ✱ ✇✐t❤ F J,1 n = 0 j,k J 1 ;j =k N S(V j )(t, • -x n,j )• ∇N S(V k )(t, • -x n,k ), F J,2 n = e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) • ∇ e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) , F J,3 n = e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) • ∇ j∈J 1 j J N S(V j )(t, • -x n,j ) , F J,4 n = j∈J 1 j J N S(V j )(t, • -x n,j ) • ∇ e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) . ✭✷✼✮ ▲❡t ✉s ❛❞♠✐t ❢♦r ❛ ✇❤✐•❡ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s✳ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❲✐t❤ ♥♦t❛t✐♦♥s ✭✸✼✮✱ t❤❡ s❡q✉❡♥❝❡ U app,J n ✐s ❜♦✉♥❞❡❞ ✐♥ t❤❡ s♣❛❝❡ X s T ✱ ✉♥✐❢♦r♠•② ✐♥ J ❛♥❞ η✱ U app,J n X s T < ∞, ∀ T < T ❞❡❢ = inf j∈J 1 T * (V j ). ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ✉s❡ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t inf j∈J 1 T * (V j ) = +∞ ✐❢ J 1 ✐s ❡♠♣t②✳ ▲❡t ✉s ❛❞♠✐t ❢♦r ❛ ✇❤✐•❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✹✳✷✳ lim J→+∞ lim sup n→+∞ F J n 2 L 2 T ( Ḣs-1 ) = 0. ❈♦♠♣•❡t✐♦♥ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✶✳ ▲❡t ε 0 > 0✳ ▲❡t T 0 ❜❡ t❤❡ t✐♠❡ ❞❡✜♥❡❞ ❜② T 0 ❞❡❢ = sup 0 < T < T | R J n (t) 2 L ∞ T ( Ḣs ) ε 0 . ❚❤❡r❡❢♦r❡✱ ❢♦r ❛♥② T < T 0 T ✱ ❚❤❡♦r❡♠ ✻✳✶ ✐♠♣•✐❡s R J n 2 X s T F J n 2 L 2 T ( Ḣs-1 ) exp ε 2 2s-1 0 T + T s-1 2 U app,J n 2 X s T + T U app,J n 4 2s-1 L ∞ T ( Ḣs ) . ✭✷✽✮ ❈♦♠❜✐♥✐♥❣ Pr♦♣♦s✐t✐♦♥s ✹✳✶ ❛♥❞ ✹✳✷✱ ▲❡♠♠❛ ✸✳✶ ✐s ♣r♦✈❡❞✳ ❚❤❡r❡❢♦r❡✱ t♦ ❝♦♠♣•❡t❡ t❤❡ ♣r♦♦❢✱ ✇❡ s❤❛•• ♣r♦✈❡ t❤❡ t✇♦ ❛❜♦✈❡ ♣r♦♣♦s✐t✐♦♥s✳ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ U app,J n ❛♥❞ ✈❡rt✉❡ ♦❢ (a + b) 2 2 a 2 + b 2 ✱ ✇❡ ❤❛✈❡ ✭✷✾✮ U app,J n 2 X s T 2 j∈J 1 j J N S(V j )(t, • -x n,j ) 2 X s T + e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) 2 X s T . ▲❡t ✉s ❢♦❝✉s ❢♦r ❛ ♠♦♠❡♥t ♦♥ t❤❡ ❤❡❛t t❡r♠ e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) ✳ ■t ✐s ✇❡••✲❦♥♦✇♥ t❤❛t ❛♥ Ḣs ✲❡♥❡r❣② ❡st✐♠❛t❡ ♦♥ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✐♠♣•✐❡s t❤❛t e t∆ u 2 X s T u 0 2 Ḣs ✱ ❢♦r ❛♥② u s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❞❛t❛ u 0 ✐♥ Ḣs ✳ ❆s ❛ r❡s✉•t✱ ✇❡ ❣❡t e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) 2 X s T j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) 2 Ḣs . ✶✹ ❊❯•➱◆■❊ P❖❯▲❖◆ ❚❤❡r❡❢♦r❡✱ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ②✐❡•❞s✱ ✉♣ t♦ tr✐❛♥❣✉•❛r ❛♥❞ (♦✉♥❣✬s ✐♥❡q✉❛•✐t❡s e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) 2 X s T u 0,n - j∈J 1 j J V j (• -x n,j ) 2 Ḣs 2 u 0,n 2 Ḣs + 2 j∈J 1 j J V j (• -x n,j ) 2 Ḣs . ▲❡t ✉s ❛❞♠✐t ❢♦r ❛ ✇❤✐•❡ t❤❡ ❢♦••♦✇✐♥❣ st❛t❡♠❡♥t ✭✸✵✮ ∀η > 0, j∈J 1 j J V j (• -x n,j ) 2 Ḣs = j∈J 1 j J V j 2 Ḣs + •(1), ✇❤❡♥ n → +∞. ❚❤❛♥❦s t♦ t❤❡ ♦rt❤♦❣♦♥❛•✐t② r❡•❛t✐♦♥ ✭✾✮✱ t❤❡ t❡r♠ j∈J 1 j J V j 2 Ḣs s❛t✐s✜❡s j∈J 1 j J V j 2 Ḣs u 0,n 2 Ḣs + •(1)✱ ❢♦r n •❛r❣❡ ❡♥♦✉❣❤✳ ❆s ❛ r❡s✉•t✱ ✭✸✶✮ ∀η > 0, e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (x) + ψ J n (x) 2 X s T u 0,n 2 Ḣs + •(1), ✇❤❡♥ n → +∞. ◆♦✇✱ •❡t ✉s ❝♦♠❡ ❜❛❝❦ t♦ ✭✷✾✮✳ ❚❤❛♥❦s t♦ t❤❡ ♣r❡✈✐♦✉s ❡st✐♠❛t❡ ✭✸✶✮✱ ✇❡ ✐♥❢❡r t❤❛t ∀η > 0, U app,J n 2 X s T j∈J 1 j J N S(V j )(t, • -x n,j ) 2 X s T + u 0,n 2 Ḣs + •(1), ✇❤❡♥ n → +∞. ❲❡ ❛❞♠✐t ❢♦r ❛ ✇❤✐•❡ t❤❡ ❢♦••♦✇✐♥❣ st❛t❡♠❡♥t✱ ❢♦r ❛♥② T < T ❞❡❢ = inf j∈J 1 T * (V j ) ❛♥❞ η > 0✳ ✭✸✷✮ j∈J 1 j J N S(V j )(t, • -x n,j ) 2 X s T j∈J 1 j J N S(V j )(t, • ) 2 X s T + •(1), ✇❤❡♥ n → +∞. ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❢♦r ❛♥② T < T ❞❡❢ = inf j∈J 1 T * (V j ) ∀η > 0, U app,J n 2 X s T C j∈J 1 j J N S(V j ) 2 X s T + u 0,n 2 Ḣs + •(1) , ✇❤❡♥ n → +∞. ❆s N S(V j η ) s♦•✈❡s ◆❙✲❡q✉❛t✐♦♥ ✇✐t❤ ✐♥✐t✐❛• ❞❛t❛ V j η ❜❡•♦♥❣✐♥❣ t♦ Ḣs ❛♥❞ s✐♥❝❡ t❤❡ t✐♠❡ T ✐s ❢❛r ❛✇❛② ❢r♦♠ t❤❡ ❜•♦✇ ✉♣ t✐♠❡✱ ✇❡ ✐♥❢❡r t❤❛t ❡❛❝❤ t❡r♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❜♦✉♥❞❡❞✱ ✉♥✐❢♦r♠•② ✐♥ J ❛♥❞ η✳ ◆♦✇ •❡t ✉s ♣r♦✈❡ ✭✸✵✮✳ ❈•❡❛r•② ✇❡ ❤❛✈❡✱ t❤❛♥❦s t♦ t❤❡ tr❛♥s•❛t✐♦♥s ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ Ḣs ✲♥♦r♠ j∈J 1 j J V j (• -x n,j ) 2 Ḣs = j∈J 1 j J V j (• -x n,j ) 2 Ḣs + 2 (j,k)∈J 1 ×J 1 j =k V j (• -x n,j ) V k (• -x n,k ) Ḣs = j∈J 1 j J V j 2 Ḣs + 2 (j,k)∈J 1 ×J 1 j =k |D| s V j (• -x n,j ) |D| s V k (• -x n,k ) L 2 , ✇❤❡r❡ |D| = √ -∆✳ ❚❤❡ ♦rt❤♦❣♦♥❛•✐t② ♦❢ ❝♦r❡s ✭❡✳❣✳ lim n→∞ |x n,j -x n,k | = +∞✮ ✐♠♣•✐❡s ✐♥ ♣❛rt✐❝✉•❛r t❤❛t t❤❡ t❡r♠ |D| s V k (x + (x n,j -x n,k )) ✇❡❛❦•② ❝♦♥✈❡r❣❡s t♦ 0 ✐♥ L 2 ❛♥❞ t❤✉s ✭♥♦t✐❝❡ t❤❛t |D| s V j η (x) ❜❡•♦♥❣s t♦ L 2 ✱ ❜② ❤②♣♦t❤❡s✐s✮ ∀η > 0, ∀(j, k) ∈ J 1 × J 1 , lim n→∞ ■ | 3 |D| s V j (x) |D| s V k (x + (x n,j -x n,k )) dx = 0, ✇❤✐❝❤ ❡♥❞s ✉♣ t❤❡ ♣r♦♦❢ ♦❢ st❛t❡♠❡♥t ✭✸✵✮✳ ❈♦♥❝❡r♥✐♥❣ st❛t❡♠❡♥t ✭✸✷✮✱ t❤❡ ♣r♦♦❢ ✐s s✐♠✐•❛r✳ ▲❡t ε > 0✳ ❆s ❢♦r ❛♥② T T -ε✱ N S(V j η ) ❜❡•♦♥❣s t♦ t❤❡ s♣❛❝❡ X s T ❞❡❢ = C T ( Ḣs ) ∩ L 2 T ( Ḣs+1 )✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ♠❛♣ t ∈ [0, T -ε] → N S(V j )(t, • ) ❜❡•♦♥❣s t♦ Ḣs ✳ Pr❡✈✐♦✉s ❝♦♠♣✉t❛t✐♦♥s ❤♦•❞ ❛♥❞✱ ❜② ✈❡rt✉❡ ♦❢ tr❛♥s•❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ Ḣs ✲♥♦r♠✱ ✇❡ ❣❡t ❢♦r ❛♥② t < T ❛♥❞ η > 0✱ j∈J 1 j J N S(V j )(t, • -x n,j ) 2 Ḣs = j∈J 1 j J N S(V j )(t, • ) 2 Ḣs + 2 (j,k)∈J 1 ×J 1 j =k |D| s N S(V j )(t, • -x n,j ) |D| s N S(V k )(t, • -x n,k ) L 2 . ✭✸✸✮ ❚❤❡♥✱ ❢♦r ❛♥② t ✐♥ [0, T -ε]✱ ✇❡ ❣❡t j∈J 1 j J N S(V j )(t, • -x n,j ) 2 L ∞ T ( Ḣs ) j∈J 1 j J N S(V j )(t, • ) 2 L ∞ T ( Ḣs ) + 2 (j,k)∈J 1 ×J 1 j =k Γ s,j,k ε,n , ✇❤❡r❡ Γ s,j,k ε,n ✐s ❞❡✜♥❡❞ ❜② Γ s,j,k ε,n ❞❡❢ = sup t∈[0, T -ε] |D| s N S(V j )(t, • -x n,j ) |D| s N S(V k )(t, • -x n,k ) L 2 = sup t∈[0, T -ε] ■ | 3 |D| s N S(V j )(t, • ) |D| s N S(V k )(t, • +(x n,j -x n,k )) dx. ✭✸✹✮ ❚❤❡ ♠❛♣ ψ : t ∈ [0, T -ε] → |D| s N S(V j )(t, • ) |D| s N S(V k )(t, • +(x n,j -x n,k )) ✐s ❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ ❝♦♠♣❛❝t [0, T -ε]✱ ✇✐t❤ ✈❛•✉❡ ✐♥ L 1 (■| 3 )✳ ❚❤✉s✱ ψ([0, T -ε]) ✐s ♣r❡❝♦♠♣❛❝t ✐♥ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ L 1 (■| 3 ) ❛♥❞ t❤✉s ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ❜② ❛ ✜♥✐t❡ ♦♣❡♥ ❜❛•• ✇✐t❤ ❛♥ ❛r❜✐tr❛r✐•② r❛❞✐✉s α > 0✳ ▲❡t α ❜❡ ❛ ♣♦s✐t✐✈❡ r❛❞✐✉s✳ ❚❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r N ✱ s✉❝❤ t❤❛t ❢♦r ❛♥② t ∈ [0, T -ε]✱ ψ(t) ❜❡•♦♥❣s t♦ N l=1 B ψ(t l ), α ✳ ❚❤✉s✱ ❢♦r ❛♥② t ❜❡•♦♥❣✐♥❣ t♦ t❤❡ ❝♦♠♣❛❝t [0, T -ε]✱ t❤❡r❡ ❡①✐sts ❛ t✐♠❡ t l s✉❝❤ t❤❛t ✭✸✺✮ ψ(t) L 1 (■| 3 ) α + ψ(t l ) L 1 (■| 3 ) . ❇② ✈❡rt✉❡ ♦❢ t❤❡ s✐♠♣•❡ ❢❛❝t f |f |✱ ✇❡ ✐♥❢❡r t❤❛t Γ s,j,k ε,n α + ψ(t l ) L 1 (■| 3 ) = α + ■ | 3 |D| s N S(V j )(t l , • ) |D| s N S(V k )(t l , • +(x n,j -x n,k )) dx. ◆♦✇✱ ✐♥ ♦r❞❡r t♦ ❝♦♥❝•✉❞❡✱ ✇❡ ♥♦t✐❝❡ t❤❛t ▲❡❜❡s❣✉❡ t❤❡♦r❡♠ ❝♦♠❜✐♥✐♥❣ ✇✐t❤ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ♣r♦♣❡rt② ♦❢ ❝♦r❡s ✐♠♣•② t❤❛t t❤❡ r✐❣❤t✲❤❛♥❞✲s✐❞❡ t❡♥❞s t♦ 0✱ ✇❤❡♥ n t❡♥❞s t♦ +∞ ✭s✐♥❝❡ ✇❡ ❝❛♥ ❝❤♦♦s❡ α ❛r❜✐tr❛r✐•② s♠❛••✮ ❛♥❞ t❤✉s✱ ✇❡ ❣❡t ∀η > 0, ∀(j, k) ∈ J 1 × J 1 , lim n→∞ sup t∈[0, T -ε] |D| s N S(V j )(t, • -x n,j ) | |D| s N S(V k )(t, • -x n,k ) L 2 = 0. ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ♣r♦✈❡❞ ❢♦r ❛♥② T < T ❛♥❞ η > 0✱ j∈J 1 j J N S(V j )(t, • -x n,j ) 2 L ∞ T ( Ḣs ) j∈J 1 j J N S(V j )(t, • ) 2 L ∞ T ( Ḣs ) + •(1), ✇❤❡♥ n → +∞. ❈♦♥❝❡r♥✐♥❣ t❤❡ L 2 T ( Ḣs+1 )✲♥♦r♠✱ ✇❡ ✇r✐t❡ ❡st✐♠❛t❡ ✭✸✸✮ ✐♥ Ḣs+1 ✲♥♦r♠✳ ❚❤❡♥✱ t❤❡ L 2 T ( Ḣs+1 )✲♥♦r♠ ♦❢ ❝r♦ss❡❞ t❡r♠s t❡♥❞s t♦ 0✱ t❤❛♥❦s t♦ ▲❡❜❡s❣✉❡ t❤❡♦r❡♠ ❛♥❞ ♦rt❤♦❣♦♥❛•✐t② ♦❢ ❝♦r❡s✳ ❉❡t❛✐•s ❛r❡ •❡❢t t♦ t❤❡ r❡❛❞❡r✳ ❋✐♥❛••②✱ ✇❡ ❣❡t ✭✸✷✮ j∈J 1 j J N S(V j )(t, • -x n,j ) 2 X s T j∈J 1 j J N S(V j )(t, • ) 2 X s T + •(1), ✇❤❡♥ n → +∞. ■♥ ♦r❞❡r t♦ ❝♦♠♣•❡t❡ t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶✱ ✇❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t t❤❡ t❡r♠ j∈J 1 j J N S(V j )(t, • ) 2 X s T ✐s ❜♦✉♥❞❡❞✱ ✉♥✐❢♦r♠•② ✐♥ J 1 ✭❛♥❞ t❤✉s ✐♥ J✮ ❛♥❞ η✳ ❚❤✐s ✇✐•• r❡s✉•t ❢r♦♠ |❡♠❛r❦ ✶✳✶ ❛♥❞ t❤❡ ♦rt❤♦❣✲ ♦♥❛•✐t② ♦❢ Ḣs ✲♥♦r♠ ✭✾✮ ✐♥ ♣r♦✜•❡ t❤❡♦r❡♠✳ ■♥❞❡❡❞✱ ❜② ✈❡rt✉❡ ♦❢ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ (u 0,n ) n 0 ✐♥ t❤❡ ❙♦❜♦•❡✈ s♣❛❝❡ Ḣs ✱ ✇❡ ❦♥♦✇ t❤❛t j∈J 1 j J V j 2 Ḣs ✐s ❜♦✉♥❞❡❞✳ ■t ♠❡❛♥s t❤❛t ∀ε > 0, ∃J * 1 ⊂ J 1 , ✇✐t❤ |J * 1 | < ∞ ∀ j ∈ J 1 \ J * 1 , V j Ḣs ε. ❇② ✈❡rt✉❡ ♦❢ |❡♠❛r❦ ✶✳✶✱ ✇❡ ✐♥❢❡r t❤❛t ❢♦r ❛♥② j ❜❡•♦♥❣✐♥❣ t♦ J 1 \ J * 1 ✱ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉✲ t✐♦♥s N S(V j ) ❛ss♦❝✐❛t❡❞ ✇✐t❤ s✉❝❤ ♣r♦✜•❡s V j s❛t✐s❢② N S(V j )(t, • ) X s T 2 V j Ḣs ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ✐♥❢❡r t❤❛t j∈J 1 j J N S(V j )(t, • ) 2 X s T j∈J * 1 j J N S(V j )(t, • ) 2 X s T + 4 j∈J 1 \J * 1 V j 2 Ḣs j∈J * 1 j J N S(V j )(t, • ) 2 X s T + 4 j∈J 1 j J V j 2 Ḣs j∈J * 1 j J N S(V j )(t, • ) 2 X s T + 4 lim sup n→+∞ u 0,n 2 Ḣs . ✭✸✻✮ ❆s ✇❡ ❛r❡ ♥♦t s♦ ❝•♦s❡ t♦ t❤❡ ❜•♦✇ ✉♣ t✐♠❡ ✭s✐♥❝❡ T < inf j∈J 1 T * (V j )✮✱ t❤❡ t❡r♠ j∈J * 1 j J N S(V j )(t, • ) 2 X s T ✐s ❜♦✉♥❞❡❞✱ ✉♥✐❢♦r♠•② ✐♥ J 1 ✭s✐♥❝❡ J * 1 ✐s ✜♥✐t❡ ❛♥❞ ❞❡♣❡♥❞s ♦♥•② ♦♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣r♦✜•❡s V j ✮✳ ❚❤✉s✱ t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶ ✐s ❝♦♠♣•❡t❡✳ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✷✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ s♠❛••♥❡ss r❡s✉•t ♦♥ t❤❡ ❢♦r❝✐♥❣ t❡r♠✱ ✇❡ s❤❛•• ♥❡❡❞ t♦ ✉s❡ t❤❡ r❡❣✉•❛r✐③❛t✐♦♥ ♣r♦❝❡ss ♠❡♥t✐♦♥♥❡❞ ✐♥ t❤❡ t♦♦• ❜♦① ♦❢ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ▲❡t ✉s r❡❝❛•• t❤❛t ✇❡ ❣❡t ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ s✉❝❤ ❛ ❞❛t❛✱ ✉♥❞❡r t❤❡ ❢♦r♠ ♦❢ N S(u 0,n ) = U app,J n + R J n , ✇❤❡r❡ U app,J n (t, • ) ❞❡❢ = j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ j∈J c 1 j J Λ 3 p λ n,j ,x n,j V j (• ) + ψ J n (• ) , ✭✸✼✮ ❆s ❛•r❡❛❞② ♠❡♥t✐♦♥♥❡❞ ✐♥ ✭✷✵✮✱ ♣r♦✜•❡s ❛r❡ s♦rt❡❞ ✇✐t❤ r❡s♣❡❝t ♦❢ t❤❡ s✐③❡ ♦❢ s❝❛•❡s✳ ▼♦r❡♦✈❡r✱ ✇❡ ❝✉t ♦✛ ❢r❡q✉❡♥❝✐❡s ♦❢ ♣r♦✜•❡s ✇✐t❤ s♠❛•• ❛♥❞ ❜✐❣ s❝❛•❡s ❛♥❞ t❤❡r❡❢♦r❡✱ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✼✮ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦••♦✇s U app,J n (t, • ) ❞❡❢ = j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ U 0 n,η + U ∞ n,η + ψ J n,η (• ) , ✭✸✽✮ ✇✐t❤ U 0 n,η ❞❡❢ = j∈J 0 j J Λ 3 p λ n,j ,x n,j V j η ❀ U ∞ n,η ❞❡❢ = j∈J∞ j J Λ 3 p λ n,j ,x n,j V j η ❛♥❞ ψ J n,η ❞❡❢ = j∈J 0 j J Λ 3 p λ n,j ,x n,j V j c η + j∈J∞ j J Λ 3 p λ n,j ,x n,j V j c η + ψ J n . ✭✸✾✮ ▲❡t ✉s ♣♦✐♥t ♦✉t t❤❛t t❤❡ ♠❛✐♥ ♣♦✐♥t ✐s t❤❛t✱ ❜② ✈❡rt✉❡ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✹✱ t❤❡ t❡r♠s U 0 n,η ❛♥❞ U ∞ n,η ❛r❡ s♠❛•• ✐♥ t❤❡ s❡♥s❡ t❤❛t✱ ❢♦r ❛♥② δ > 0✱ ❢♦r ❛♥② η > 0✱ lim n→+∞ ||U 0 n,η || Ḣs-δ = 0 ❛♥❞ lim n→+∞ ||U ∞ n,η || Ḣs+δ = 0✳ ❲❡ r❡❝❛•• ❛ ❜❛s✐❝ ♣r♦♣❡rt② ❞✉❡ t♦ ❞✐✈❡r❣❡♥❝❡ ❢r❡❡ ❝♦♥❞✐t✐♦♥✿ ❢♦r ❛♥② ✈❡❝t♦r ✜❡•❞ ✉✱ s♠♦♦t❤ ❡♥♦✉❣❤ ❛♥❞ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡✱ ✭✹✵✮ u • ∇v = div(u ⊗ v)• ❚❤❡ ♣r♦♣❡rt② ✭✹✵✮ ♣r♦✈✐❞❡s ✉s ❛♥♦t❤❡r ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❡①t❡r✐♦r ❢♦r❝❡ t❡r♠ F J n F J n = I J,1 n,η + I J,2 n,η + I J,3 n .
✭✹✶✮ ✇❤❡r❡

I J,1 n,η = div 2 j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ U 0 n,η + U ∞ n,η + ψ J n,η (x) ⊗ e t∆ U 0 n,η + U ∞ n,η . I J,2 n,η = div 2 j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ U 0 n,η + U ∞ n,η + ψ J n,η (x) ⊗ e t∆ ψ J n,η . I J,3 n = F J,1 n = 0 j,k J 1 ;j =k N S(V j (t, • -x n,j ))• ∇N S(V k )(t, • -x n,k ). ✭✹✷✮ ❈♦♥❝❡r♥✐♥❣ I J,1 n,η ✱ ✇❡ ❛♣♣•② ✭✺✵✮ ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✺✱ ❢♦r ❛♥② δ > 0✱ s✉❝❤ t❤❛t s + δ < 3 2 ✱ I J,1 n,η L 2 T ( Ḣs-1 ) C T 1 2 (s-1 2 ) T -δ 2 ||U 0 n,η || Ḣs-δ + T δ 2 ||U ∞ n,η || Ḣs+δ × 2 j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ U 0 n,η + U ∞ n,η + ψ J n,η X s T C T 1 2 (s-1 2 ) T -δ 2 ||U 0 n,η || Ḣs-δ + T δ 2 ||U ∞ n,η || Ḣs+δ × 2 U app,J n X s T + e t∆ U 0 n,η + U ∞ n,η + ψ J n,η X s T .
✭✹✸✮ ❋r♦♠ ✭✸✶✮✱ ✇❡ ✐♥❢❡r t❤❛t

I J,1 n,η L 2 T ( Ḣs-1 ) C T 1 2 (s-1 2 ) T -δ 2 ||U 0 n,η || Ḣs-δ + T δ 2 ||U ∞ n,η || Ḣs+δ × U app,J n X s T + u 0,n Ḣs + •(1) , ✇❤❡♥ n → +∞. Pr♦♣♦s✐t✐♦♥s ✸✳✹ ❛♥❞ ✹✳✶ ✐♠♣•✐❡s t❤❛t I J,1 n,η t❡♥❞s t♦ 0 ✇❤❡♥ n t❡♥❞s t♦ ✐♥✜♥✐t② ∀ε > 0, ∃ ñ1 (ε, J, η), ∀n ñ1 (ε, J, η), I J,1 n,η L 2 T ( Ḣs-1 ) ε. ✶✽ ❊❯•➱◆■❊ P❖❯▲❖◆ ❈♦♥❝❡r♥✐♥❣ I J,2 n,η ✱ ✇❡ ❛♣♣•② t❤❡ ❡st✐♠❛t❡ ✭✹✾✮ ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✺ I J,2 n,η L 2 T ( Ḣs-1 ) C T 1 2 (s-1 2 ) ||ψ J n,η || Ḃs(1-θ) p θ ,p θ × 2 j∈J 1 j J N S(V j )(t, • -x n,j ) + e t∆ U 0 n,η + U ∞ n,η + ψ J n,η X s T C T 1 2 (s-1 2 ) ||ψ J n,η || Ḃs(1-θ) p θ ,p θ U app,J n X s T + u 0,n Ḣs + •(1) , ✇❤❡♥ n → +∞. ✭✹✹✮ ❚❤❛♥❦s t♦ Pr♦♣♦s✐t✐♦♥ ✸✳✸✱ ✇❡ ✐♥❢❡r ∀ε > 0, ∃ J(ε), ∀J J(ε), ∃η(J), ∃ ñ2 (J), ∀η η(J), ∀n ñ2 (J), I J,2 n,η L 2 T ( Ḣs-1 ) ε. ❈♦♥❝❡r♥✐♥❣ I J,3 n,η ✱ t❤❡ ❛r❣✉♠❡♥t r❡•✐❡s ♦♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ▲❡♠♠❛ ✺✳✻ ❛♣♣•✐❡❞ ✇✐t❤ ✇✐t❤ σ = s 2 + 3 4 ✱ ✇❤✐❝❤ ♣r♦♦❢ ✐s ❣✐✈❡♥ ✐♥ ❆♣♣❡♥❞✐① ❆✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣•✐❝✐t②✱ ✇❡ ♥♦t❡✿ Φ j = N S(V j ) ❛♥❞ Φ k = N S(V k )✳ |❡♠❛r❦ ✹✳✶✳ ❆s Φ j ❛♥❞ Φ k ❜❡•♦♥❣ t♦ t❤❡ s♣❛❝❡ L ∞ T ( Ḣs )∩L 2 T ( Ḣs+1 )✱ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ ❛r❣✉♠❡♥t ✐♠♣•✐❡s t❤❡② ❜❡•♦♥❣ t♦ t❤❡ s♣❛❝❡ L 4 T ( Ḣ s 2 + 3 4 )✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ u Ḣ s 2 + 3 4 u 1 2 (s+ 1 2 ) Ḣs u 1 2 ( 3 2 -s) Ḣs+1 . ❚❤❡♥✱ ❜② ✐♥t❡❣r❛t✐♥❣ ✐♥ t✐♠❡✱ ✇❡ ❞❡❞✉❝❡ t❤❛t u 4 L 4 T ( Ḣ s 2 + 3 4 ) T s-1 2 u 2(s+ 1 2 ) L ∞ T ( Ḣs ) u 3-2s L 2 T ( Ḣs+1 ) . ❚❤❛♥❦s t♦ t❤❡ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ❝♦♥❞✐t✐♦♥✱ ✇❡ ❤❛✈❡ Φ j .∇Φ k 2 Ḣs-1 = Φ j ⊗ Φ k 2 Ḣs ❛♥❞ t❤✉s Φ j .∇Φ k 2 L 2 T ( Ḣs-1 ) = T 0 Φ j ⊗ Φ k 2 Ḣs T 0 (Φ j -Φ j ε ) ⊗ Φ k 2 Ḣs + T 0 Φ j ε ⊗ (Φ k -Φ k ε ) 2 Ḣs + T 0 Φ j ε ⊗ Φ k ε 2 Ḣs . ❆s s 2 + 3 4 < 3 2 ✱ ❛ ♣r♦❞✉❝t r✉•❡ ✐♥ ❙♦❜♦•❡✈ s♣❛❝❡s ✐♠♣•✐❡s ✭✹✺✮ u v Ḣs C(s) u Ḣ s 2 + 3 4 v Ḣ s 2 + 3 4 • ❚❤❡r❡❢♦r❡✱ ✇❡ ✐♥❢❡r t❤❛t ✿ Φ j .∇Φ k 2 L 2 T ( Ḣs-1 ) T 0 (Φ j -Φ j ε ) 2 Ḣ s 2 + 3 4 Φ k 2 Ḣ s 2 + 3 4 + T 0 Φ j ε 2 Ḣ s 2 + 3 4 Φ k -Φ k ε 2 Ḣ s 2 + 3 4 + T 0 Φ j ε ⊗ Φ k ε 2 Ḣs . ❋✐♥❛••②✱ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t② ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ▲❡♠♠❛ ✺✳✻ ②✐❡•❞ Φ j .∇Φ k 2 L 2 T ( Ḣs-1 ) ε 2 Φ k 2 L 4 T ( Ḣ s 2 + 3 4 ) + ε 2 Φ j 2 L 4 T ( Ḣ s 2 + 3 4 ) + Φ j ε ⊗ Φ k ε 2 L 2 T ( Ḣs ) . ❚♦ ❝♦♥❝•✉❞❡✱ ✇❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t Φ j ε ⊗ Φ k ε 2 L 2 T ( Ḣs ) t❡♥❞s t♦ 0✱ ❢♦r ε s♠❛•• ❡♥♦✉❣❤✳ ❚❤✐s ✇✐•• ❝♦♠❡ ❢r♦♠ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ♦❢ ❝♦r❡s✳ ❇② ❞❡✜♥✐t✐♦♥✱ Φ j ε ✭r❡s♣✳ Φ k ε ✮ ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ Φ j ✭r❡s♣✳ Φ k ✮✳ ❇❡❝❛✉s❡ ♦❢ tr❛♥s•❛t✐♦♥s ❜② ❝♦r❡s✱ ✇❡ ❞❡✜♥❡ Φ j,n ε (t, x-x n,j ) ✭r❡s♣✳ Φ k,n ε (t, x-x n,k )✮ ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ Φ j (t, x -x n,j ) ✭r❡s♣✳ Φ k (t, x -x n,k )✮✳ ❆s Φ j,n ε ❛♥❞ Φ k,n ε ❛r❡ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ❛♥❞ ❝♦♥❝❡♥tr❛t❡❞ ❛r♦✉♥❞ x n,j ❛♥❞ x n,k ✱ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ ❝♦r❡s ✭ lim n→+∞ |x n,j -x n,k | = +∞✮ ✐♠♣•✐❡s t❤❡② ❛r❡ s✉♣♣♦rt❡❞ ♦♥ ❞✐s•♦✐♥t❡❞ ❝♦♠♣❛❝ts✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t❡r♠ Φ j ε ⊗ Φ k ε 2 L 2 T ( Ḣs ) ❝♦♥✈❡r❣❡s t♦ 0✱ ❢♦r n •❛r❣❡ ❡♥♦✉❣❤✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ∀ε > 0, ∃ñ(ε), ∀n ñ(ε), N S(V j (t, • -x n,j ))• ∇N S(V k )(t, • -x n,k ) L 2 T ( Ḣs-1 ) ε |J 1 | • ❚❤❡r❡❢♦r❡✱ ✇❡ ✐♥❢❡r t❤❛t✱ ∀ε > 0, ∃ñ(ε), ∀n ñ(ε)✱ I J,3 n,η L 2 T ( Ḣs-1 ) = 0 j,k J ;j =k (j,k)∈J 2 1 N S(V j (t, • -x n,j ))• ∇N S(V k )(t, • -x n,k ) L 2 T ( Ḣs-1 ) ε. ✭✹✻✮ ❚❤✐s ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✷✳ ✺✳ ❆♣♣❡♥❞✐① ❆✳ Pr♦❞✉❝t ❛♥❞ ♣❛r❛♣r♦❞✉❝t ❡st✐♠❛t❡s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❣✐✈❡ s♦♠❡ t②♣✐❝❛• ♣r♦❞✉❝t ❡st✐♠❛t❡s✱ ✐♥ ✇❤✐❝❤ s♣•✐tt✐♥❣ ❢r❡q✉❡♥❝② ❛••♦✇s ❢♦r ❛ ♠✉❝❤ ✜♥❡r ❝♦♥tr♦• ♦❢ t❤❡ ♣r♦❞✉❝t✳ ❚❤❡ ♠❛✐♥ t♦♦• ✐s t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛r❛❞✐✛❡r❡♥t✐❛• ❝❛•❝✉•✉s✳ ❋♦r ❛ ❞❡✲ t❛✐•❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ✐t✱ ✇❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✶(✱ ♣❛❣❡ 85✳ ❲❡ r❡❝❛•• t✇♦ ❢✉♥❞❛♠❡♥t❛• st❛t❡♠❡♥ts ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❚❤❡♦r❡♠ 2.47 ❛♥❞ 2.52 ✐♥ ❬✶(✮ ❛❜♦✉t ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛r❛♣r♦❞✉❝t ♦♣✲ ❡r❛t♦r T ✱ ❛♥❞ t❤❡ r❡♠❛✐♥❞❡r ♦♣❡r❛t♦r R✳ ❲❡ s❤❛•• ❝♦♥st❛♥t•② ❜❡ ✉s✐♥❣ t❤❡s❡ t✇♦ t❤❡♦r❡♠s ✐♥ t❤❡ s❡q✉❡•✳ ❚❤❡♦r❡♠ ✺✳✶✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❢♦r ❛♥② r❡❛• ♥✉♠❜❡r s ❛♥❞ ❛♥② (p, r) ✐♥ [1, ∞] 2 ✱ ✇❡ ❤❛✈❡ ❢♦r ❛♥② (u, v) ✐♥ L ∞ × Ḃs p,r ✱ T u v Ḃs p,r C 1+|s| u L ∞ v Ḃs p,r ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② (s, t) ✐♥ ■|×] -∞, 0[✱ (p, r 1 , r 2 ) ✐♥ [1, ∞] 3 ✱ ❛♥❞ (u, v) ✐♥ Ḃt ∞,r 1 × Ḃs p,r 2 ✱ ✇❡ ❤❛✈❡ T u v Ḃs+t p,r C 1+|s+t| -t u Ḃt ∞,r 1 v Ḃs p,r 2 ✇✐t❤ 1 r ❞❡❢ = min 1, 1 r 1 + 1 r 2 • ❚❤❡♦r❡♠ ✺✳✷✳ ❆ ❝♦♥st❛♥t C ❡①✐sts ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s✳ ▲❡t (s 1 , s 2 ) ❜❡ ✐♥ ■| 2 ❛♥❞ (p 1 , p 2 , r 1 , r 2 ) ✐♥ [1, ∞] 4 ✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t 1 p ❞❡❢ = 1 p 1 + 1 p 2 1 ❛♥❞ 1 r ❞❡❢ = 1 r 1 + 1 r 2 1. ■❢ s 1 + s 2 ✐s ♣♦s✐t✐✈❡✱ t❤❡♥ ✇❡ ❤❛✈❡ ❢♦r ❛♥② (u, v) ✐♥ Ḃs 1 p 1 ,r 1 × Ḃs 2 p 2 ,r 2 ✱ R(u, v) Ḃs 1 +s 2 p,r C 1+|s 1 +s 2 | s 1 + s 2 u Ḃs 1 p 1 ,r 1 v Ḃs 2 p 2 ,r 2 . ❆ •♦t ♦❢ r❡s✉•ts ♦❢ ❝♦♥t✐♥✉✐t② ♠❛② ❜❡ ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ t✇♦ ❛❜♦✈❡ ❚❤❡♦r❡♠s✳ ❋♦r ✐♥st❛♥❝❡✱ ✇❡ ❝❛♥ st❛t❡ t❤❡ ▲❡♠♠❛ ❜❡•♦✇✳ ▲❡♠♠❛ ✺✳✸✳ ✭Pr♦❞✉❝t r✉•❡ ✐♥ Ḣs ✮ ▲❡t u ❛♥❞ v ❜❡ t✇♦ ❢✉♥❝t✐♦♥s ✐♥ Ḣs ✇✐t❤ - 3 2 < s < 3 2 ✱ t❤❡♥ uv Ḣs C(s) u Ḣs v Ḣ 3 2 + u Ḣ 3 2 v Ḣs ❛♥❞ uv Ḣs C(s) u L ∞ ∩ Ḣ 3 2 v Ḣs . Pr♦♦❢✳ ❲❡ ❤❛✈❡ t♦ ❡st✐♠❛t❡ ❛ ♣r♦❞✉❝t ✐♥ ❙♦❜♦•❡✈ s♣❛❝❡ t❤✉s✱ ✇❡ s❤❛•• ✉s❡ t❤❡ ♣❛r❛❞✐✛❡r❡♥t✐❛• ❝❛•❝✉•✉s✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❛♥❦s t♦ t❤❡ ❇♦♥②✬s ♣❛r❛♣r♦❞✉❝t ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✇❡ ❣❡t u v = T u v + R(u, v) + T v u. ❚❤❡ t❡r♠ R(v, u) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐♥ Ḣs ✲♥♦r♠ ❡❛s✐•②✱ t❤❛♥❦s t♦ ❚❤❡♦r❡♠ ✺✳✷✱ R(u, v) Ḃs+ 3 2 1,1 C u Ḣs v Ḣ 3 2 .
❚❤❡r❡❢♦r❡✱ t❤❛♥❦s t♦ ❡♠❜❡❞❞✐♥❣s

Ḃs+ 3 2 1,1 ֒→ Ḃs 2,1 ֒→ Ḃs 2,2 ❛♥❞ |❡♠❛r❦ ✸✳✶✱ ✇❡ ✐♥❢❡r t❤❛t ✭✹✼✮ R(u, v) Ḣs C u Ḣs v Ḣ 3 2 . ❈♦♥❝❡r♥✐♥❣ T v u ❛♥❞ T u v✱ ✇❡ ✉s❡ ♦♥❝❡ ❛❣❛✐♥ ❡st✐♠❛t❡s ♦❢ ❚❤❡♦r❡♠ ✺✳✶✱ ✇❤✐❝❤ ❣✐✈❡s T u v Ḣs C(s) u Ḃs-3 2 ∞,∞ v Ḃ 3 2 2,2 . ❇❡❝❛✉s❡ s -3 2 ✐s ♥❡❣❛t✐✈❡✱ ❇❡r♥st❡✐♥✬s ✐♥❡q✉❛•✐t② ❛♥❞ t❤❡ ❝•❛ss✐❝❛• ❡♠❜❡❞❞✐♥❣ ℓ 2 (Z) ⊂ ℓ ∞ (Z) ❣✐✈❡ Ḃs 2,2 ֒→ Ḃs-3 2 ∞,2 ֒→ Ḃs-3 2 ∞,∞ .
❚❤❡r❡❢♦r❡✱ ✇❡ ❞❡❞✉❝❡ t❤❛t

T u v Ḣs C(s) u Ḣs v Ḣ 3 2 . P❡r♠✉t✐♥❣ t❤❡ r♦•❡s ♦❢ u ❛♥❞ v ❛♥❞ ✉s✐♥❣ ✭✹✼✮ ❣✐✈❡s t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ r❡s✉•t✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ ▲❡♠♠❛ ✺✳✸ ✐s ❡❛s②✳ ❇② ✈❡rt✉❡ ♦❢ ❚❤❡♦r❡♠ ✺✳✶✱ ✇❡ ❤❛✈❡ T u v Ḣs C(s) u L ∞ v Ḣs ❛♥❞ T v u Ḣs C(s) v Ḣs u Ḣ 3 2 . ▼♦r❡♦✈❡r✱ ✐t s❡❡♠s ❝•❡❛r t❤❛t✱ ❞✉❡ t♦ ❚❤❡♦r❡♠ ✺✳✷ R(u, v) Ḃs+ 3 2 1,1 C v Ḣs u Ḣ 3 2 . ❚❤✐s •❡❛❞s t♦ t❤❡ ♣r♦♦❢ ♦❢ uv Ḣs C(s) u L ∞ ∩ Ḣ 3 2 v Ḣs ✳ |❡♠❛r❦ ✺✳✶✳ ▲❡t ✉s ♣♦✐♥t ♦✉t ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ ✐♥❡q✉❛•✐t②✿ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ s ✇❡ ❤❛✈❡ ✭✹✽✮ u Ḣ 3 2 C u s-1 2 Ḣs u 3 2 -s Ḣs+1 . ❚❤❡r❡❢♦r❡✱ ❝♦♠❜✐♥✐♥❣ t❤✐s ✇✐t❤ ▲❡♠♠❛ ✺✳✸✱ ✇❡ ❣❡t t❤❡ r❡s✉•t ❢♦••♦✇✐♥❣ ✇❤✐❝❤ ✇✐•• ❜❡ ❛ ❢r❡q✉❡♥t ✉s❡ •❛t❡r ♦♥✳ ❈♦r♦••❛r② ✺✳✹✳ ▲❡t u ❛♥❞ v ❜❡ ✐♥ Ḣs ✇✐t❤ 1 2 < s < 3 2 ✱ t❤❡♥ uv Ḣs C(s) u Ḣs v s-1 2 Ḣs v 3 2 -s Ḣs+1 + u s-1 2 Ḣs u 3 2 -s Ḣs+1 v Ḣs . Pr♦♣♦s✐t✐♦♥ ✺✳✺✳ ▲❡t 0 < θ < 1✳ ❯♥❞❡r t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ r❡•❛t✐♦♥ 1 p θ = θ p + 1 -θ 2 ✱ ✭✹✾✮ u ⊗ e t∆ r 0 L 2 T ( Ḣs ) C T 1 2 (s-1 2 ) u X s T ||r 0 || Ḃs(1-θ) p θ ,p θ . ✭✺✵✮ ❋♦r ❛♥② 1 2 < α < 3 2 , u ⊗ e t∆ r 0 L 2 T ( Ḣs ) C T 1 2 (α-1 2 ) u X s T ||r 0 || Ḣα . Pr♦♦❢✳ ▲❡t ✉s st❛rt ❜② ♣r♦✈✐♥❣ t❤❡ ✜rst ✐♥❡q✉❛•✐t②✳ ❇♦♥②✬s ♣❛r❛♣r♦❞✉❝t ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♠♣•✐❡s
u ⊗ e t∆ r 0 = T e t∆ r 0 u + R(e t∆ r 0 , u) + T u (e t∆ r 0 ).

❚❤❡ ✜rst t✇♦ t❡r♠s ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐♥ Ḣs ✲♥♦r♠ ❡❛s✐•②✳ ❚❤❛♥❦s t♦ ❚❤❡♦r❡♠ ✺✳✶✱ ✇❡ ❤❛✈❡

T e t∆ r 0 (u) Ḣs = Ḃs 2,2 C e t∆ r 0 Ḃs-3 2 ∞,∞ u Ḃ 3 2 2,2 . ▲❡t ✉s r❡❝❛•• t❤❛t 1 p θ ✐s ❞❡✜♥❡❞ ❜② 1 p θ = θ p + 1 -θ 2 ✱ ❢♦r ❛♥② θ ✐♥ ]0, 1[✳ ❆ ❝•❛ss✐❝❛• r❡s✉•t ❞✉❡ t♦ ❇❡r♥st❡✐♥✬s ✐♥❡q✉❛•✐t② ❣✐✈❡s t❤❡ ❢♦••♦✇✐♥❣ ❡♠❜❡❞❞✐♥❣ Ḃs(1-θ) p θ ,∞ ֒→ Ḃs-3 2 ∞,∞ ✳ ❚❤❡r❡❢♦r❡ ✇❡ ✐♥❢❡r t❤❛t T e t∆ r 0 (u) L 2 T ( Ḣs ) e t∆ r 0 L ∞ T ( Ḃs(1-θ) p θ ,∞ ) u L 2 T ( Ḣ 3 2 ) . ❚❤❡ •❛st t❡r♠ T u (e t∆ r 0 ) ✐s ♠♦r❡ ❞❡•✐❝❛t❡✳ ◆♦t❡ t❤❛t✱ ❤❡r❡✱ ❛s ✇❡ ✇♦r❦ •♦❝❛••② ✐♥ t✐♠❡✱ •♦✇ ❢r❡q✉❡♥❝✐❡s ❞♦ ♥♦t ♣•❛② ❛ ♠❛•♦r r♦•❡✱ ✉♥•✐❦❡ ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s✳ ❆s ❛ r❡s✉•t✱ ✇❡ ❤❛✈❡ t♦ ❤❛♥❞•❡ •♦✇ ❛♥❞ ❤✐❣❤
T u (e t∆ r 0 ) L 2 T ( Ḣs ) = T u (e t∆ r 0 ) L 2 T ( Ḃs 2,2 ) = 2 js ∆j T u (e t∆ r 0 ) L 2 T (L 2 ) ℓ 2 (Z) . ❲❡ s♣•✐t✱ ❛❝❝♦r❞✐♥❣ t♦ •♦✇ ❛♥❞ ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s T u (e t∆ r 0 ) L 2 T ( Ḣs ) 2 js ∆j T u (e t∆ r 0 ) L 2 T (L 2 ) 1 { √ T 2 j C} ℓ 2 (Z) + 2 js ∆j T u (e t∆ r 0 ) L 2 T (L 2 ) 1 { √ T 2 j C} ℓ 2 (Z) . ✭✺✷✮ ❆ ❝•❛ss✐❝❛• r❡s✉•t ✐♥ ▲✐tt•❡✇♦♦❞ P❛•❡② t❤❡♦r② ❣✐✈❡s t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ∆j T u (e t∆ r 0 ) = |j-j ′ | 4 Ṡj ′ -1 u ∆j ′ (e t∆ r 0 ). ❚❤❡r❡❢♦r❡✱ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ②✐❡•❞s ∆j T u (e t∆ r 0 ) L 2 |j-j ′ | 4 Ṡj ′ -1 u L q θ ∆j ′ (e t∆ r 0 ) L p θ ✇✐t❤ 1 2 = 1 p θ + 1 q θ . ■♥ ♣❛rt✐❝✉•❛r✱ ❇❡r♥st❡✐♥✬s ✐♥❡q✉❛•✐t② ✐♠♣•✐❡s Ṡj ′ -1 u L q θ j ′ -2 j ′′ =-∞ ∆j ′′ u L q θ j ′ -2 j ′′ =-∞ 2 3j ′′ ( 1 2 -1 q θ ) ∆j ′′ u L 2 = j ′ -2 j ′′ =-∞ 2 j ′′ ( 3 p θ -s) 2 j ′′ s ∆j ′′ u L 2 . ❆♣♣•②✐♥❣ (♦✉♥❣✬s ✐♥❡q✉❛•✐t②✱ ✇❡ ✐♥❢❡r t❤❡r❡ ❡①✐sts ❢♦r ❛♥② t✱ ❛ s❡q✉❡♥❝❡ (c j (t)) j∈Z ❜❡•♦♥❣✐♥❣ t♦ t❤❡ s♣❤❡r❡ ♦❢ ℓ q θ (Z)✱ s✉❝❤ t❤❛t Ṡj ′ -1 u L q θ C c j ′ (t) 2 j ′ ( 3 p θ -s) u(t) Ḃs 2,q θ . ❆s q θ 2✱ ℓ 2 (Z) ✐s ✐♥❝•✉❞❡❞ ✐♥ ℓ q θ (Z)✱ ✇❤✐❝❤ ✐♠♣•✐❡s t❤❛t Ṡj ′ -1 u L q θ C c j ′ (t) 2 j ′ ( 3 p θ -s) u(t) Ḃs 2,2
. T ( Ḣs ) r 0 Ḃs(1-θ)

p θ ,p θ . ❚❤✐s ❡♥❞s ✉♣ t❤❡ ♣r♦♦❢ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ✜rst ✐♥❡q✉❛•✐t② ♦❢ t❤❡ ♣r♦♣♦✲ s✐t✐♦♥ ✐s ♣r♦✈❡❞✳ ◆♦✇✱ •❡t ✉s ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ✐♥❡q✉❛•✐t②✱ ✇❤✐❝❤ ♣r♦♦❢ ✐s ✈❡r② ❝•♦s❡ t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳ ❲❡ ❣✐✈❡s ♦♥•② ♦✉t•✐♥❡s✳ ❚❤❛♥❦s t♦ ❇♦♥②✬s ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✇❡ ❤❛✈❡
u ⊗ e t∆ r 0 = T e t∆ r 0 u + R(e t∆ r 0 , u) + T u (e t∆ r 0 ).

❚❤❡ ✜rst t✇

♦ t❡r♠s ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐♥ Ḣs ✲♥♦r♠ ❡❛s✐•②✱ t❤❛♥❦s t♦ ♠❛♣♣✐♥❣ ♦❢ ♣❛r❛♣r♦❞✉❝t ✐♥ t❤❡ ❇❡s♦✈ s♣❛❝❡s ✭❝❢ ❚❤❡♦r❡♠ ✺✳✶✮ T e t∆ r 0 u Ḃs 2,2 C e t∆ r 0 Ḃα-3 2 ∞,∞ u Ḃs+ 3 2 -α 2,2 . ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ❇❡r♥st❡✐♥✬s ▲❡♠♠❛ ❛♥❞ ♦❜✈✐♦✉s ❡♠❜❡❞❞✐♥❣ ℓ 2 (Z) ⊂ ℓ ∞ (Z) ❡♥s✉r❡ t❤❛t Ḃα 2,2 ֒→ Ḃα-3 2 ∞,2 ֒→ Ḃα-3 2 ∞,∞ ❛♥❞ t❤✉s e t∆ r 0 Ḃα-3 2 ∞,∞ e t∆ r 0 Ḃα 2,2 . ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❛s s s + 3 2 -α s + 1✱ u ❜❡•♦♥❣s t♦ Ḃs+ 3 2 -α 2,2 ✳ ■♥t❡r♣♦•❛t✐♦♥ ❛r❣✉♠❡♥t ②✐❡•❞s u Ḃs+ 3 2 -α 2,2 C u Ḣs+ 3 2 -α u α-1 2 Ḣs u 3 2 -α Ḣs+1 . ❇② ✐♥t❡❣r❛t✐♦♥ ✐♥ t✐♠❡ ❛♥❞ t❤❛♥❦s t♦ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✱ ✇❡ ❤❛✈❡ u 2 L 2 T ( Ḣs+ 3 2 -α ) T 0 u(t, • ) 2α-1 Ḣs u(t, • ) 3-2α Ḣs+1 dt T α-1 2 u 2α-1 L ∞ T ( Ḣs ) u 3-2α L 2 T ( Ḣs+1 ) . ❋✐♥❛••②✱ ✇❡ ❣❡t u L 2 T ( Ḣs+ 3 2 -α ) T 1 2 (α-1 2 ) u X s T . ❚❤❡r❡❢♦r❡✱ ✇❡ ❞❡❞✉❝❡ ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ t❡r♠ T e t∆ r 0 (u) L 2 T ( Ḣs ) ❛♥❞ R(e t∆ r 0 , u) L 2 T ( Ḣs ) ✳ T e t∆ r 0 u L 2 T ( Ḣs ) T 1 2 (α-1 2 ) u X s T r 0 Ḣα . R(e t∆ r 0 , u) L 2 T ( Ḣs ) T 1 2 (α-1 2 ) u X s T r 0 Ḣα . 1 2 (α-1 2 ) u L ∞ T ( Ḣs ) r 0 Ḃα+s-3 2 3 s ,2 . ❈♦♠❜✐♥✐♥❣ ✭✺✽✮ ❛♥❞ ✭✺✾✮ ✇✐t❤ t❤❡ ❢❛❝t t❤❛t Ḃα 2,2 ✐s ❡♠❜❡❞❞❡❞ ✐♥ Ḃα+s-3 2 3 s ,2 ✱ ✇❡ ❣❡t ✜♥❛••② ✭✻✵✮ T u (e t∆ r 0 ) L 2 T ( Ḣs ) T 1 2 (α-1 2 ) u L ∞ T ( Ḣs ) r 0 Ḣα . ❚❤✐s ❝♦♠♣•❡t❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ s❡❝♦♥❞ ✐♥❡q✉❛•✐t② ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥✳ ◆♦✇✱ •❡t ✉s st❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ •❡♠♠❛✳ ▲❡♠♠❛ ✺✳✻✳ ▲❡t 0 < σ < 3 2 ❛♥❞ ε > 0✳ ▲❡t a ❜❡ ❛♥ ❡•❡♠❡♥t ♦❢ L 4 T ( Ḣσ )✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C > 0✱ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐•② ♦❢ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ❢✉♥❝t✐♦♥s✱ a ε ✱ ✇❤✐❝❤ s❛t✐s✜❡s ❢♦r ❛♥② ♣♦s✐t✐✈❡ T lim ε→0 a -a ε L 4 T ( Ḣσ ) = 0 ❛♥❞ ✭✻✶✮ a ε L 4 T ( Ḣσ ) C a L 4 T ( Ḣσ ) . ✭✻✷✮ ❆❇❖❯❚ ❚❍❊ ❇❊❍❆❱■❖❯| ❖❋ |❊•❯▲❆| ◆❆❱■❊|✲❙❚❖❑❊❙ ❙❖▲❯❚■❖◆❙ ◆❊❆| ❚❍❊ ❇▲❖❲ ❯P ✷✺ Pr♦♦❢✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢✉♥❝t✐♦♥ a ε ❞❡✜♥❡❞ ❜② a ε = χ(ε• )a, ✇❤❡r❡ χ ✐s t❤❡ ✉s✉❛• ❢♦♥❝t✐♦♥ ♦❢ D(■| 3 ) ✇✐t❤ ✈❛•✉❡ 1 ♥❡❛r 0✳ ▲❡t ✉s st❛rt ❜② ♣r♦✈✐♥❣ ✭✻✷✮✳ ❉✉❡ t♦ ♣r♦❞✉❝t r✉•❡ ✐♥ ❙♦❜♦•❡✈ s♣❛❝❡s r❡❝❛••❡❞ ✐♥ ▲❡♠♠❛ ✺✳✸✱ ✇❡ ❤❛✈❡ a ε Ḣσ C(σ) χ(ε• ) L ∞ ∩ Ḣ 3 2 a Ḣσ C(σ) χ L ∞ ∩ Ḣ 3 2 a Ḣσ C(σ) a Ḣσ ✭✻✸✮ ◆♦✇ ✇❡ ♣r♦✈❡ ✭✻✶✮✳ ■♥ ♦r❞❡r t♦ ❛♣♣•② ▲❡❜❡s❣✉❡ ❚❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C✱ s✉❝❤ t❤❛t ✭✻✹✮ lim ε→0 a ε -a Ḣσ = 0 ❛♥❞ a ε -a Ḣσ C. ▲❡t ✉s ♥♦t✐❝❡ t❤❛t a ε -a Ḣσ ✐s ❜♦✉♥❞❡❞✱ t❤❛♥❦s t♦ ✭✻✸✮✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ lim ε→0 a ε -a Ḣσ = 0✱ ♦♥❡ ✇❛② ✐s t♦ ❛♣♣r♦❛❝❤ t❤❡ ❢✉♥❝t✐♦♥ a ❜② ❛♥ tr✉♥❝❛t❡❞ ❡•❡♠❡♥t A η ✇❤✐❝❤ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ❞❡✜♥❡❞ ❜② A η (ξ) = a(ξ)1 {η |ξ| 1 η } ✳ ■♥ t❤✐s ✇❛②✱ ❜② ✈❡rt✉❡ ♦❢ ▲❡❜❡s❣✉❡ ❚❤❡♦r❡♠✱ ✐t s❡❡♠s ❝•❡❛r t❤❛t ✭✻✺✮ lim η→0 A η -a Ḣσ = 0. ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ a ε -a Ḣσ = (1 -χ(ε • )) a Ḣσ (1 -χ(ε • )) (a -A η ) Ḣσ + (1 -χ(ε • )) A η Ḣσ . ❇② ✈❡rt✉❡ ♦❢ ▲❡♠♠❛ ✺✳✸✱ ✇❡ ❤❛✈❡ a ε -a Ḣσ 1 -χ(ε • ) Ḣ 3 2 ∩L ∞ a -A η Ḣσ + (1 -χ(ε • )) A η Ḣσ 1 + χ Ḣ 3 2 ∩L ∞ a -A η Ḣσ + (1 -χ(ε • )) A η Ḣσ . ◆♦✇✱ ✇❡ ❤❛✈❡ •✉st t♦ ♣r♦✈❡ t❤❛t lim ε→0 (1 -χ(ε • )) A η Ḣσ = 0✳ ❚❤✐s ❝♦♠❡s ❢r♦♠ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ ❛r❣✉✲ ♠❡♥t✳ ❋♦r ❛♥② 0 < σ < 3 2 ❛♥❞ σ < s < 3 2 ✱ (1 -χ(ε • )) A η Ḣσ (1 -χ(ε • )) A η 1-σ s L 2 (1 -χ(ε • )) A η σ s Ḣs (1 -χ(ε • )) A η 1-σ s L 2 1 + χ Ḣ 3 2 ∩L ∞ A η σ s Ḣs . ❚♦ ❝♦♥❝•✉❞❡✱ ✇❡ ❤❛✈❡ •✉st t♦ ♥♦t✐❝❡ t❤❛t t❤❡ t❡r♠ (1 -χ(ε • )) A η 1-σ s L 2 t❡♥❞s t♦ 0 ❢♦r ε s♠❛•• ❡♥♦✉❣❤✱ ❜② ✈❡rt✉❡ ♦❢ ▲❡❜❡s❣✉❡ ❚❤❡♦r❡♠✳ ❚❤❡ ♦t❤❡r t❡r♠ ✐s ♦❜✈✐♦✉s•② ❜♦✉♥❞❡❞✱ s✐♥❝❡ A η ❜❡•♦♥❣s t♦ ❛♥② ❙♦❜♦•❡✈ s♣❛❝❡s✱ ❢♦r ❛♥② ε > 0✱ t❤❛♥❦s t♦ tr✉♥❝❛t✉r❡ ♣r♦❝❡ss✳ ✻✳ ❆♣♣❡♥❞✐① ❇✳ ■♥ t❤✐s ❛♣♣❡♥❞✐①✱ ✇❡ ♣r♦✈❡ ❛ ❣❡♥❡r❛• ❚❤❡♦r❡♠ ❛❜♦✉t ❛♥ ❡st✐♠❛t❡ ✐♥ t❤❡ X s T ✲s♣❛❝❡ ♦❢ ❛ s♦•✉t✐♦♥ ♦❢ ❛ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s s②st❡♠✳ ❚❤❡ ♠❡t❤♦❞ ✐s st❛♥❞❛r❞✿ t❤❡ ✜rst st❡♣ ❝♦♥s✐sts ✐♥ ❡st❛❜•✐s❤✐♥❣ ❛♥ Ḣs ✲ ❡♥❡r❣② ❡st✐♠❛t❡✳ ❚❤❡♥✱ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥s ♦♥ s❝❛•❛r✲♣r♦❞✉❝t t❡r♠s •❡❛❞ t♦ ❛♥ ✐♥❡q✉❛•✐t② ♦♥ ✇❤✐❝❤ ✇❡ ❝❛♥ ❛♣♣•② •r♦♥✇❛••✬s •❡♠♠❛✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ❛♣♣•② t❤✐s ❚❤❡♦r❡♠ t♦ ♣r♦✈❡ t❤❛t t❤❡ ♠❛♣ u 0 → T * (u 0 ) ✐s ❛ •♦✇❡r s❡♠✐✲❝♦♥t✐♥♦✉s ❢✉♥❝t✐♦♥ ♦♥ Ḣs ✳ ❚❤❡♦r❡♠ ✻✳✶✳ ▲❡t q ❜❡ ❛♥ ❡•❡♠❡♥t ❜❡•♦♥❣✐♥❣ t♦ t❤❡ s♣❛❝❡ X s T ✱ ❞❡✜♥❡❞ ❜② ❢♦r ❛♥② T < T (q) = T q 2 X s T ❞❡❢ = q 2 L ∞ T ( Ḣs ) + q 2 L 2 T ( Ḣs+1 ) . ✷✻ ❊❯•➱◆■❊ P❖❯▲❖◆ ▲❡t r ❜❡ ❛ s♦•✉t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s s②st❡♠    ∂ t r + r.∇r -∆r + r • ∇q + q.∇r = -f -∇p div r = 0 r |t=0 = r 0 . ▲❡t ε 0 > 0✳ ▲❡t T 0 ❜❡ t❤❡ t✐♠❡ ❞❡✜♥❡❞ ❜② T 0 ❞❡❢ = sup 0 < T < T (q) | r(t) 2 L ∞ T ( Ḣs ) ε 0 . ❚❤❡♥✱ ❢♦r ❛♥② t T 0 ✱ ✇❡ ❤❛✈❡ r 2 X s T r 0 2 Ḣs + f 2 L 2 T ( Ḣs-1 ) exp ε 2 2s-1 0 T + T s-1 2 q 2 X s T + T q 4 2s-1 L ∞ T ( Ḣs ) . Pr♦♦❢✳ ❆ Ḣs s❝❛•❛r✲♣r♦❞✉❝t✱ t✐♠❡ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ②✐❡•❞ r 2 Xs T ❞❡❢ = r 2 Ḣs + 2 t 0 r(t ′ ) 2 Ḣs+1 dt ′ r 0 2 Ḣs + 2 t 0 (r • ∇r) | r Ḣs dt ′ + 2 t 0 (q • ∇r) | r Ḣs dt ′ + 2 t 0 (r • ∇q) | r Ḣs dt ′ + 2 t 0 f | r Ḣs dt ′ . ✭✻✻✮ ❲❡ ❛ss❡ss ❡❛❝❤ t❡r♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡❀ t❤❡ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ❝♦♥❞✐t✐♦♥ ✐♠♣•✐❡s (r • ∇r) | r Ḣs r • ∇r Ḣs-1 r Ḣs+1 r ⊗ r Ḣs r Ḣs+1 • ❚❤❛♥❦s t♦ ❈♦r♦••❛r② ✺✳✹✱ ✇❡ ✐♥❢❡r t❤❛t (r • ∇r) | r Ḣs C(s) r s+ 1 2 Ḣs r 5 2 -s Ḣs+1 • ❚❤❡♥✱ ✐♥t❡❣r❛t✐♥❣ ✐♥ t✐♠❡ ❛♥❞ ❛♣♣•②✐♥❣ (♦✉♥❣✬s ✐♥❡q✉❛•✐t② ab a p p + b p ′ p ′ , ✇✐t❤ 1 p + 1 p ′ = 1 ②✐❡•❞ t 0 (r • ∇r) | r Ḣs dt ′ C(s) t 0 r s+ 1 2 Ḣs r 5 2 -s Ḣs+1 dt ′ C(s) t 0 r 2 2s+1 2s-1 Ḣs dt ′ + 1 12 t 0 r 2 Ḣs+1 dt ′ . ✭✻✼✮ ◆♦✇ ✇❡ ❤❛✈❡ t♦ ❡st✐♠❛t❡ t 0 (r • ∇q) | r Ḣs dt ′ ❛♥❞ t 0 (q • ∇r) | r Ḣs dt ′ ✳ ❆❝t✉❛••②✱ t❤❛♥❦s t♦ t❤❡ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ❝♦♥❞✐t✐♦♥✱ ✐t ✐s ❡①❛❝t•② t❤❡ s❛♠❡ ❡st✐♠❛t❡ ❛♥❞ ✇❡ ❣❡t (r • ∇q) | r Ḣs r • ∇q Ḣs-1 r Ḣs+1 r ⊗ q Ḣs r Ḣs+1 . ❖♥❝❡ ❛❣❛✐♥✱ ❈♦r♦••❛r② ✺✳✹ ❣✐✈❡s t 0 (r • ∇q) | r Ḣs dt ′ C(s) t 0 r Ḣs q s-1 2 Ḣs q 3 2 -s Ḣs+1 r Ḣs+1 dt ′ + C(s) t 0 q Ḣs r s-1 2 Ḣs r 5 2 -s Ḣs+1 dt ′ . (♦✉♥❣✬s ✐♥❡q✉❛•✐t② ✐♠♣•✐❡s t 0 (r • ∇q) | r Ḣs dt ′ C(s) t 0 r 2 Ḣs q 2s-1 Ḣs q 3-2s Ḣs+1 dt ′ + C(s) t 0 q 4 2s-1 Ḣs r 2 Ḣs dt ′ + 2 12 t 0 r 2 Ḣs+1 dt ′ . ✭✻✽✮ ❙❛♠❡ ❛r❣✉♠❡♥ts ❣✐✈❡ ❛♥ ❡st✐♠❛t❡ ♦❢ ❡①t❡r✐♦r ❢♦r❝❡ t❡r♠ t 0 f | r Ḣs dt ′ t 0 f Ḣs-1 r Ḣs+1 dt ′ C t 0 f 2 Ḣs-1 dt ′ + 1 12 t 0 r 2 Ḣs+1 dt ′ . ✭✻✾✮ ❈♦♠❜✐♥✐♥❣ ■♥❡q✉❛•✐t✐❡s ✭✻✻✮✱ ✭✻✼✮✱ ✭✻✽✮ ❛♥❞ ✭✻✾✮✱ ✇❡ ❣❡t r 2 Xs T r 0 2 Ḣs + C t 0 f 2 Ḣs-1 dt ′ + 2 t 0 6 12 r 2 Ḣs+1 dt ′ + C(s) t 0 r 2 Ḣs r 4 2s-1 Ḣs + q 2s-1 Ḣs q 3-2s Ḣs+1 + q 4 2s-1 Ḣs dt ′ . ✭✼✵✮ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ t✐♠❡ T 0 ❞❡✜♥❡❞ ❜② T 0 ❞❡❢ = sup 0 < T < T | r(t) 2 L ∞ T ( Ḣs ) ε 0 . ❚❤❡r❡❢♦r❡✱ ❢♦r ❛♥② t T 0 ✱ ✇❡ ❤❛✈❡ r 2 X s T ❞❡❢ = r 2 Ḣs + t 0 r(t ′ ) 2 Ḣs+1 dt ′ r 0 2 Ḣs + t 0 f 2 Ḣs-1 dt ′ + t 0 r 2 Ḣs ε 2 2s-1 0 + q 2s-1 Ḣs q 3-2s Ḣs+1 + q 4 2s-1 Ḣs . ❚❤❛♥❦s t♦ •r♦♥✇❛••✬s •❡♠♠❛✱ ✇❡ ✐♥❢❡r t❤❛t ❢♦r ❛♥② T < T 0 T r 2 X s T r 0 2 Ḣs + f 2 L 2 T ( Ḣs-1 ) exp ε 2 2s-1 0 T + T s-1 2 q 2 X s T + T q 4 2s-1 L ∞ T ( Ḣs ) .

  t❤❡r❡ ❡①✐sts ❛ t✐♠❡ T ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s♦•✉t✐♦♥ N S(u 0 ) s✉❝❤ t❤❛t N S(u 0 ) ❜❡•♦♥❣s t♦ L ∞ T ( Ḣs ) ∩ L 2 T ( Ḣs+1 )✳ ▼♦r❡♦✈❡r✱ •❡t T * (u 0 ) ❜❡ t❤❡ ♠❛①✐♠❛• t✐♠❡ ♦❢ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❛ s♦•✉t✐♦♥✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ✭✻✮ T * (u 0 ) ✶✳✶✳ ❆s ❛ ❜②✲♣r♦❞✉❝t ♦❢ t❤❡ ♣r♦♦❢ ♦❢ P✐❝❛r❞✬s ❚❤❡♦r❡♠✱ ✇❡ ❣❡t ❛❝t✉❛••② ❢♦r ❢r❡❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt②✿ ✐❢ t❤❡ ✐♥✐t✐❛• ❞❛t❛ ✐s s♠❛•• ❡♥♦✉❣❤ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t c 0 ✱ s✉❝❤ t❤❛t T u 0 σs Ḣs c 0 ✮✱ t❤❡♥ ❛ ✉♥✐q✉❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✐t ❡①✐sts ✭•♦❝❛••② ✐♥ t✐♠❡✱ ✉♥t✐• t❤❡ ❜•♦✇ ✉♣ t✐♠❡ ❣✐✈❡♥ ❜② t❤❡ r❡•❛t✐♦♥ ✭✻✮✮ ❛♥❞ s❛t✐s✜❡s t❤❡ ❢♦••♦✇✐♥❣ •✐♥❡❛r ❝♦♥tr♦• ✭✼✮ ∀ 0 T c 0 u 0 σs Ḣs ✱ N S(u 0 )(t, • ) X s T 2 u 0 Ḣs . ❋♦r♠✉•❛ ✭✻✮ ✐♥✈✐t❡s ✉s t♦ ❝♦♥s✐❞❡r t❤❡ •♦✇❡r ❜♦✉♥❞❛r②✱ ❞❡♥♦t❡❞ ❜② A σs c ✱ ♦❢ t❤❡ •✐❢❡s♣❛♥ ♦❢ s✉❝❤ ❛ s♦•✉t✐♦♥ A σs c ❞❡❢ = inf T * (u 0 ) u 0 σs Ḣs | u 0 ∈ Ḣs ; T * (u 0 ) < ∞ . ❖❜✈✐♦✉s•②✱ A σs c ❡①✐sts ❛♥❞ ✐s ❛ ♣♦s✐t✐✈❡ r❡❛• ♥✉♠❜❡r ❛♥❞ ✇❡ ❛•✇❛②s ❤❛✈❡ t❤❡ ❢♦r♠✉•❛ ✭✽✮ T * (u 0 ) u 0 σs Ḣs A σs c . ❚❤r♦✉❣❤♦✉t t❤✐s ♣❛♣❡r✱ ✇❡ ♠❛❞❡ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❜•♦✇ ✉♣✱ ✇❤✐❝❤ ✐s st✐•• ❛♥ ♦♣❡♥ ♣r♦❜•❡♠✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ❝•❛✐♠ t❤❡ ❢♦••♦✇✐♥❣ ❤②♣♦t❤❡s✐s✳ ❍②♣♦t❤❡s✐s H✿ ❢♦r ❛♥② 1 2 < s < 3 2 ✱ ❛ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ✈❡❝t♦r ✜❡•❞ u 0 ❡①✐sts ✐♥ Ḣs s✉❝❤ t❤❛t t❤❡ •✐❢❡s♣❛♥ T * (u 0 ) ✐s ✜♥✐t❡✳ ▲❡t B ρ ❜❡ t❤❡ ♦♣❡♥ ❜❛•• ✐♥ Ḣs ❞❡✜♥❡❞ ❜② B ρ = {u 0 ∈ Ḣs / u 0 Ḣs < ρ}✳ ▲❡t T * > 0 ❜❡ ❛ ♣♦s✐t✐✈❡ r❡❛• ♥✉♠❜❡r✳ ❲❡ ❞❡✜♥❡ ❛ ❝r✐t✐❝❛• r❛❞✐✉s ❜② t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✉•❛ρ s (T * ) ✐♥ t❤✐s ✇❛② ❛♥❞ t❤❛♥❦s t♦ ✭✽✮✱ ✇❡ ❣❡t ❛♥ ❛♥♦t❤❡r ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝r✐t✐❝❛• r❛❞✐✉s

  r❡❢❡r ❢♦r ✐♥st❛♥❝❡ t❤❡ r❡❛❞❡r t♦ ❬✶✸( ❛♥❞ t♦ t❤❡ ❛rt✐❝•❡ ♦❢ ❈✳ ❑❡♥✐❣ ❡t •✳ ❑♦❝❤ ❬✶✾(✱ ✇❤❡r❡ t❤❡ ❛✉t❤♦rs ♣r♦✈❡ t❤❛t ◆❙✲s♦•✉t✐♦♥s ✇❤✐❝❤ r❡♠❛✐♥ ❜♦✉♥❞❡❞ ✐♥ t❤❡ s♣❛❝❡ Ḣ 1 2 ❞♦ ♥♦t ❜❡❝♦♠❡ s✐♥❣✉•❛r ✐♥ ✜♥✐t❡ t✐♠❡✳ ❈♦♥❝❡r♥✐♥❣ ❚❤❡♦r❡♠ ✶✳✸✱ ✇❡ ✇❡r❡ •❛r❣❡•② ✐♥s♣✐r❡❞ ❜② t❤❡ ❛rt✐❝•❡ ♦❢ ❲✳ |✉s✐♥ ❛♥❞ ❱✳ S✈❡rá❦ ❬✷✾(✱ ✐♥ ✇❤✐❝❤ t❤❡ ❛✉t❤♦rs s❡t ✉♣ t❤❡ ❦❡② ❝

  ❚❤✐s ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ❚❤❡♦r❡♠ ✻✳✶✳ Pr♦♣♦s✐t✐♦♥ ❜❡•♦✇ ✐s ✇❡••✲❦♥♦✇♥ ❛♥❞ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❚❤❡♦r❡♠ ✻✳✶✳ ❲❡ ♣❡rt✉r❜ ❛ ❞❛t❛ ❜② ❛ s♠❛•• t❡r♠ ❛♥❞ ✇❡ ❛r❡ ✐♥t❡r❡st✐♥❣ ✐♥ t❤❡ ❝♦♥s❡q✉❡♥❝❡ ♦♥ t❤❡ •✐❢❡s♣❛♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ s✉❝❤ ❛ ♣❡rt✉r❜❡❞ ❞❛t❛✳ ❚❤❡ •✐❢❡s♣❛♥ ♦❢ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ ❝❛♥ ♥♦t ❞❡❝r❡❛s❡ t♦♦ ♠✉❝❤✱ ❝♦♠♣❛r❡❞ t♦ t❤❡ •✐❢❡s♣❛♥ ♦❢ t❤❡ ♥♦♥✲♣❡rt✉r❜❡❞ ♦♥❡✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✻✳✷✳ ❚❤❡ ♠❛♣ u 0 → T * (u 0 ) ✐s ❛ •♦✇❡r s❡♠✐✲❝♦♥t✐♥♦✉s ❢✉♥❝t✐♦♥ ♦♥ Ḣs ❡✳❣✳ ∀ε > 0, ∃α > 0, ∀v 0 ✐♥ Ḣs s✉❝❤ t❤❛t v 0 Ḣs < α, t❤❡♥ T * (u 0 + v 0 ) T * (u 0 )ε. ▼♦r❡♦✈❡r✱ ✭✉♥❞❡r ♥♦t❛t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✻✳✶✮✱ ❛ ❝♦♥st❛♥t C > 0 ❡①✐sts s✉❝❤ t❤❛t ❢♦r ❛♥② T T * (u 0 )ε N S(u 0 + v 0 ) -N S(u 0 ) ❢✳ ▲❡t u 0 ❛♥❞ v 0 t✇♦ ❡•❡♠❡♥ts ✐♥ Ḣs ✳ ❲❡ ♦♣❡r❛t❡ ❛ s♠❛•• ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛ u 0 ❜② v 0 ✭t❤❡ ❛✐♠ ✐s t♦ q✉❛♥t✐❢② t❤✐s s♠❛••♥❡ss ❝♦♥❞✐t✐♦♥✮ ❛♥❞ ✇❡ ✇❛♥t t♦ ♣r♦✈❡ t❤❛t t❤❡ •✐❢❡s♣❛♥ ♦❢ t❤❡ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s s♦•✉t✐♦♥ N S(u 0 + v 0 ) ❝❛♥ ♥♦t ❜❡ ♠✉❝❤ •❡ss t❤❛♥ t❤❡ •✐❢❡s♣❛♥ ♦❢ N S(u 0 )✳ ❚❤❡ ♣r♦❝❡ss ✐s st❛♥❞❛r❞✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛♥ ❡rr♦r t❡r♠ R ❞❡✜♥❡❞ ❜② R(t, x) = N S(u 0 + v 0 ) -N S(u 0 ). ✷✽ ❊❯•➱◆■❊ P❖❯▲❖◆ ❈•❛ss✐❝❛• ❝♦♠♣✉t❛t✐♦♥s ✐♠♣•② t❤❛t R ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ♣❡rt✉r❜❡❞ ◆❛✈✐❡r✲❙t♦❦❡s s②st❡♠ ✭✼✷✮    ∂ t R + R • ∇R -∆R + R • ∇N S(u 0 ) + N S(u 0 ) • ∇R = -∇p div R = 0 R |t=0 = v 0 . ▲❡t ε 0 > 0✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ t✐♠❡ T 0 ❞❡✜♥❡❞ ❜② T 0 := sup 0 < T < T * (u 0 ) | R(t) t♦ ❚❤❡♦r❡♠ ✻✳✶✱ ✇❡ ✐♥❢❡r t❤❛t ❢♦r ❛♥② T T 0 ✈❡ ❡①♣r❡ss✐♦♥ ❣✐✈❡s t❤❡ s♠❛••♥❡ss ❝♦♥❞✐t✐♦♥ ♦♥ v 0 Ḣs ✳ ■♥❞❡❡❞✱ s✉♣♣♦s❡ t❤❛t v 0 s❛t✐✜❡s ✭✼✹✮ t❤❡ ❡rr♦r t❡r♠ R✱ ❦❡❡♣s ♦♥ •✐✈✐♥❣ ✉♥t✐• t❤❡ t✐♠❡ T * (u 0 ) -ε✱ ❢♦r ❛♥② ε > 0✳ ❚❤✐s ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥✳ |❡❢❡r❡♥❝❡s ❬✶( ❍✳ ❇❛❤♦✉r✐✱ ❏✳✲(✳ ❈❤❡♠✐♥✱ |✳ ❉❛♥❝❤✐♥✿ ❋♦✉r✐❡r ❆♥❛•②s✐s ❛♥❞ ◆♦♥•✐♥❡❛r P❛rt✐❛• ❉✐✛❡r❡♥t✐❛• ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ✸✹✸✱ ✷✵✶✶✳ ❬✷( ❍✳ ❇❛❤♦✉r✐✱ ❆✳ ❈♦❤❡♥✱ •✳ ❑♦❝❤✿ ❆ ❣❡♥❡r❛• ✇❛✈❡•❡t✲❜❛s❡❞ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❤❡ ❝r✐t✐❝❛• ❡♠❜❡❞❞✐♥❣ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡s✱ ❈♦♥✢✉❡♥t❡s ▼❛t❤❡♠❛t✐❝✐✱ ✸✱ ✷✵✶✶✱ ♣❛❣❡s ✶✲✷✺✳ ❬✸( ❍✳ ❇❛❤♦✉r✐ ❛♥❞ ■✳ •❛••❛❣❤❡r✿ ❖♥ t❤❡ st❛❜✐•✐t② ✐♥ ✇❡❛❦ t♦♣♦•♦❣② ♦❢ t❤❡ s❡t ♦❢ ❣•♦❜❛• s♦•✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ❆r❝❤✐✈❡ ❢♦r |❛t✐♦♥❛• ▼❡❝❤❛♥✐❝s ❛♥❞ ❆♥❛•②s✐s✱ ✷✵✾✱ ✷✵✶✸✱ ♣❛❣❡s ✺✻✾✲✻✷✾✳ ❬✹( ❍✳ ❇❛❤♦✉r✐✱ P✳ •ér❛r❞✿ ❍✐❣❤ ❢r❡q✉❡♥❝② ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s♦•✉t✐♦♥s t♦ ❝r✐t✐❝❛• ♥♦♥•✐♥❡❛r ✇❛✈❡ ❡q✉❛t✐♦♥s✱ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛• ♦❢ ▼❛t❤✱ ✶✷✶✱ ✶✾✾✾✱ ♣❛❣❡s ✶✸✶✲✶✼✺✳ ❬✺( ❍✳ ❇❛❤♦✉r✐✱ ▼✳ ▼❛•❞♦✉❜ ❛♥❞ ◆✳ ▼❛s♠♦✉❞✐✿ ▲❛❝❦ ♦❢ ❝♦♠♣❛❝t♥❡ss ✐♥ t❤❡ ✷❉ ❝r✐t✐❝❛• ❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣✱ t❤❡ ❣❡♥❡r❛• ❝❛s❡✱ t♦ ❛♣♣❡❛r ✐♥ ❏♦✉r♥❛• ❞❡ ▼❛t❤é♠❛t✐q✉❡s P✉r❡s ❡t ❆♣♣•✐q✉é❡s✳ ❬✻( ❏✳ ❇♦✉r❣❛✐♥✿ |❡✜♥❡♠❡♥ts ♦❢ ❙tr✐❝❤❛rt③✬ ✐♥❡q✉❛•✐t② ❛♥❞ ❛♣♣•✐❝❛t✐♦♥s t♦ 2 ❉✲◆▲❙ ✇✐t❤ ❝r✐t✐❝❛• ♥♦♥•✐♥❡❛r✐t②✱ ■♥t❡r♥❛t✐♥❛• ▼❛t❤❡♠❛t✐❝❛• |❡s❡❛r❝❤ ◆♦t✐❝❡s✱ ✺✱ ✶✾✾✽✱ ✷✺✸✲✷✽✸✳ ❬✼( ❍✳ ❇ré③✐s ❛♥❞ ❏✳✲▼✳ ❈♦r♦♥✿ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ s♦•✉t✐♦♥s ♦❢ ❍✲❙②st❡♠s ♦r ❤♦✇ t♦ ❜•♦✇ ❜✉❜❜•❡s✱ ❆r❝❤✐✈❡ ❢♦r |❛t✐♦♥❛• ▼❡❝❤❛♥✐❝s ❛♥❞ ❆♥❛•②s✐s✱ ✽✾✱ ✶✾✽✺✱ ♣❛❣❡s ✷✶✲✽✻✳ ❬✽( ❏✳✲(✳ ❈❤❡♠✐♥✿ |❡♠❛rq✉❡s s✉r •✬❡①✐st❡♥❝❡ ❣•♦❜❛•❡ ♣♦✉r •❡ s②stè♠❡ ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s ✐♥❝♦♠♣r❡ss✐❜•❡✱ ❙■❆▼✱ ❏♦✉r♥❛• ♦♥ ▼❛t❤❡♠❛t✐❝❛• ❆♥❛•②s✐s✱ ✷✸✱ ✶✾✾✷✱ ♣❛❣❡s ✷✵✲✷✽✳ ❬✾( ❏✳✲(✳ ❈❤❡♠✐♥✿ ❚❤é♦rè♠❡s ❞✬✉♥✐❝✐té ♣♦✉r •❡ s②stè♠❡ ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s tr✐❞✐♠❡♥s✐♦♥♥❡•✱ ❏♦✉r♥❛• ❞✬❆♥❛•②s❡ ▼❛t❤é♠❛✲ t✐q✉❡✱ ✼✼✱ ✶✾✾✾✱ ♣❛❣❡s ✷✼✲✺✵✳ ❬✶✵( ❏✳✲(✳ ❈❤❡♠✐♥ ❛♥❞ ■✳ •❛••❛❣❤❡r✿ ▲❛r❣❡✱ ❣•♦❜❛• s♦•✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ s•♦✇•② ✈❛r②✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥✱ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛• ❙♦❝✐❡t②✱ ✸✻✷✱ ✷✵✶✵✱ ♣❛❣❡s ✷✽✺✾✲✷✽✼✸✳ ❬✶✶( ❏✳✲(✳ ❈❤❡♠✐♥✱ ■✳ •❛••❛❣❤❡r✿ ❲❡••♣♦s❡❞♥❡ss ❛♥❞ st❛❜✐•✐t② r❡s✉•ts ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐♥ ■| 3 ✱ ❆♥♥✳ ■✳ ❍✳ P♦✐♥❝❛ré ✲ ❆◆✱ ✷✻✱ ✷✵✵✾✱ ♣❛❣❡s ✺✾✾✲✻✷✹✳ ❬✶✷( ❏✳✲(✳ ❈❤❡♠✐♥ ❛♥❞ ◆✳ ▲❡r♥❡r✿ ❋•♦t ❞❡ ❝❤❛♠♣s ❞❡ ✈❡❝t❡✉rs ♥♦♥ •✐♣s❝❤✐t③✐❡♥s ❡t éq✉❛t✐♦♥s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ ❏♦✉r♥❛• ♦❢ ❉✐✛❡r❡♥t✐❛• ❊q✉❛t✐♦♥s✱ ✶✷✶✱ ✶✾✾✺✱ ♣❛❣❡s ✸✶✹✲✸✷✽✳ ❬✶✸( ❍✳ ❋✉•✐t❛✱ ❚✳ ❑❛t♦✿ ❖♥ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ✐♥✐t✐❛• ✈❛•✉❡ ♣r♦❜•❡♠ ■✱ ❆r❝❤✐✈❡ ❢♦r |❛t✐♦♥❛• ▼❡❝❤❛♥✐❝s ❛♥❞ ❆♥❛•②s✐s✱ ✶✻✱ ✶✾✻✹✱ ♣❛❣❡s ✷✻✾✲✸✶✺✳ ❬✶✹( ■✳ •❛••❛❣❤❡r✿ Pr♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❢♦r s♦•✉t✐♦♥s ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ❇✉••✳ ❙♦❝✳ ▼❛t❤✳ ❋r❛♥❝❡✱ ✶✷✾ ✭✷✮✱ ✷✵✵✶✱ ♣❛❣❡s ✷✽✺✲✸✶✻✳ ❬✶✺( ■✳ •❛••❛❣❤❡r✱ ❉✳ ■❢t✐♠✐❡ ❛♥❞ ❋✳ P•❛♥❝❤♦♥✿ ❆s②♠♣t♦t✐❝s ❛♥❞ st❛❜✐•✐t② ❢♦r ❣•♦❜❛• s♦•✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ❆♥♥❛•❡s ❞❡ •✬■♥st✐t✉t ❋♦✉r✐❡r✱ ✺✸✱ ✷✵✵✸✱ ♣❛❣❡s ✶✸✽✼✲✶✹✷✹✳ ❬✶✻( ■✳ •❛••❛❣❤❡r✱ •✳ ❑♦❝❤✱ ❋✳ P•❛♥❝❤♦♥✿ ❆ ♣r♦✜•❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♣♣r♦❛❝❤ t♦ t❤❡ L ∞ t (L 3 x ) ◆❛✈✐❡r✲❙t♦❦❡s r❡❣✉•❛r✐t② ❝r✐t❡r✐♦♥✱ t♦ ❛♣♣❡❛r✱ ▼❛t❤❡♠❛t✐s❝❤❡ ❆♥♥❛•❡♥✱ ✷✵✶✶✳ ❬✶✼( P✳ •ér❛r❞✿ ❉❡s❝r✐♣t✐♦♥ ❞✉ ❞é❢❛✉t ❞❡ ❝♦♠♣❛❝✐té ❞❡ •✬✐♥•❡❝t✐♦♥ ❞❡ ❙♦❜♦•❡✈✱ ❊❙❆■▼ ❈♦♥trô•❡ ❖♣t✐♠❛• ❡t ❈❛•❝✉• ❞❡s ❱❛r✐❛t✐♦♥s✱ ✈♦•✳ ✸✱ ▼❛✐ ✶✾✾✽✱ ♣❛❣❡s ✷✶✸✲✷✸✸✳ ❬✶✽( ❙✳ ❏❛✛❛r❞✱ ❆♥❛•②s✐s ♦❢ t❤❡ •❛❝❦ ♦❢ ❝♦♠♣❛❝t♥❡ss ✐♥ t❤❡ ❝r✐t✐❝❛• ❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣s✱ ❏♦✉r♥❛• ♦❢ ❋✉♥❝t✐♦♥❛• ❆♥❛•②s✐s✱ ✶✻✶✱ ✶✾✾✾✱ ♣❛❣❡s ✸✽✹✲✸✾✻✳

2 )✈❡r✱ ✐❢ u 0 Ḣ 1 2 r Λ λ,x 0 1 2 < s < 3 2 ♠ ♣❛❣❡ 146 t