Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control

❇✐❧✐♥❡❛r ❧♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② t♦ t❤❡ tr❛❥❡❝t♦r✐❡s ♦❢ t❤❡

❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ ❧♦❝❛❧✐③❡❞ ❝♦♥tr♦❧

▼✐❝❤❡❧ ❉✉♣r❡③

∗

_{P✐❡rr❡ ▲✐ss②}

†

❙❡♣t❡♠❜❡r ✻✱ ✷✵✶✾

❆❜str❛❝t ❚❤✐s ✇♦r❦ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ❝♦♥tr♦❧ ♦❢ t❤❡ ❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥✱ ♣♦s❡❞ ♦♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ♦❢ Rd ✭d > 1✮✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❝♦♥tr♦❧ ✐s t❤❡ ❞r✐❢t ❢♦r❝❡✱ ❧♦❝❛❧✐③❡❞ ♦♥ ❛ s♠❛❧❧ ♦♣❡♥ s✉❜s❡t✳ ❲❡ ♣r♦✈❡ t❤❛t t❤✐s s②st❡♠ ✐s ❧♦❝❛❧❧② ❝♦♥tr♦❧❧❛❜❧❡ t♦ r❡❣✉❧❛r ♥♦♥③❡r♦ tr❛❥❡❝t♦r✐❡s✳ ▼♦r❡♦✈❡r✱ ✉♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥tr♦❧✱ ✇❡ ❡①♣❧❛✐♥ ❤♦✇ t♦ r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧s ❛r♦✉♥❞ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥tr♦❧✳ ❚❤❡ r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ t❤❛♥❦s t♦ ❛ ❧✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ ❛ st❛♥❞❛r❞ ✐♥✈❡rs❡ ♠❛♣♣✐♥❣ ♣r♦❝❡❞✉r❡ ❛♥❞ t❤❡ ✜❝t✐t✐♦✉s ❝♦♥tr♦❧ ♠❡t❤♦❞✳ ❚❤❡ ♠❛✐♥ ♥♦✈❡❧t✐❡s ♦❢ t❤❡ ♣r❡s❡♥t ❛rt✐❝❧❡ ❛r❡ t✇♦❢♦❧❞✳ ❋✐rst❧②✱ ✇❡ ♣r♦♣♦s❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ str❛t❡❣② t♦ t❤❡ st❛♥❞❛r❞ ✜❝t✐t✐♦✉s ❝♦♥tr♦❧ ♠❡t❤♦❞✿ t❤❡ ❛❧❣❡❜r❛✐❝ s♦❧✈❛❜✐❧✐t② ✐s ♣❡r❢♦r♠❡❞ ❛♥❞ ✉s❡❞ ❞✐r❡❝t❧② ♦♥ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✳ ❙❡❝♦♥❞❧②✱ ✇❡ ♣r♦✈❡ ❛ ♥❡✇ ❈❛r❧❡♠❛♥ ✐♥❡q✉❛❧✐t② ❢♦r t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ♦♥❡ ♦r❞❡r s♣❛❝❡✲✈❛r②✐♥❣ ❝♦❡✣❝✐❡♥ts✿ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ s♦❧✉t✐♦♥ ❧♦❝❛❧✐③❡❞ ♦♥ ❛ s✉❜s❡t ✭r❛t❤❡r t❤❛♥ t❤❡ s♦❧✉t✐♦♥ ✐ts❡❧❢✮✱ ❛♥❞ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❝❛♥ ❝♦♥t❛✐♥ ❛r❜✐tr❛r② ❤✐❣❤ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ s♦❧✉t✐♦♥✳ ❑❡②✇♦r❞s✿❈♦♥tr♦❧❧❛❜✐❧✐t②✱ P❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s✱ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡s✱ ❋✐❝t✐t✐♦✉s ❝♦♥tr♦❧ ♠❡t❤♦❞✱ ❆❧❣❡❜r❛✐❝ s♦❧✈❛❜✐❧✐t②✳ ✷✵✶✵ ▼❙❈✿ ✾✸❇✵✺✱ ✾✸❇✵✼✱ ✾✸❇✷✺✱ ✾✸❈✶✵✱ ✸✺❑✹✵✳

❈♦♥t❡♥ts

✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠❛✐♥ r❡s✉❧ts ✷ ✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ▼❛✐♥s r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳✶ ❈♦♥tr♦❧s ✇✐t❤ ❞ ❝♦♠♣♦♥❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳✷ ❈♦♥tr♦❧❧❛❜✐❧✐t② ❛❝t✐♥❣ t❤r♦✉❣❤ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷ ◆✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ❧✐♥❡❛r✐③❡❞ s②st❡♠ ✻ ✷✳✶ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❆❧❣❡❜r❛✐❝ r❡s♦❧✉❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❘❡❣✉❧❛r ❝♦♥tr♦❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸ ❈♦♥tr♦❧❧❛❜✐❧✐t② t♦ t❤❡ tr❛❥❡❝t♦r✐❡s ✷✷ ∗_{❈❊❘❊▼❆❉❊✱ ❯♥✐✈❡rs✐té P❛r✐s✲❉❛✉♣❤✐♥❡ ✫ ❈◆❘❙ ❯▼❘ ✼✺✸✹✱ ❯♥✐✈❡rs✐té P❙▲✱ ✼✺✵✶✻ P❛r✐s✱ ❋r❛♥❝❡✳ ❊✲♠❛✐❧✿} ❞✉♣r❡③❅❝❡r❡♠❛❞❡✳❞❛✉♣❤✐♥❡✳❢r✳ †_{❈❊❘❊▼❆❉❊✱ ❯♥✐✈❡rs✐té P❛r✐s✲❉❛✉♣❤✐♥❡ ✫ ❈◆❘❙ ❯▼❘ ✼✺✸✹✱ ❯♥✐✈❡rs✐té P❙▲✱ ✼✺✵✶✻ P❛r✐s✱ ❋r❛♥❝❡✳ ❊✲♠❛✐❧✿} ❧✐ss②❅❝❡r❡♠❛❞❡✳❞❛✉♣❤✐♥❡✳❢r✳ ✶

✹ ❊①❛♠♣❧❡ ♦❢ ❛ ♥♦♥✲❝♦♥tr♦❧❧❛❜❧❡ tr❛❥❡❝t♦r② ✇✐t❤ ❛ r❡❞✉❝❡❞ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧s ✷✺

✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠❛✐♥ r❡s✉❧ts

✶✳✶ ■♥tr♦❞✉❝t✐♦♥

▲❡t T > 0 ❛♥❞ ❧❡t Ω ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ Rd _{✭d ∈ N}∗_{✮✱ r❡❣✉❧❛r ❡♥♦✉❣❤ ✭❢♦r ❡①❛♠♣❧❡ ♦❢ ❝❧❛ss} C∞_{✮✳ ❉❡♥♦t❡ ❜② Q} T := (0, T ) × Ω❛♥❞ ΣT := (0, T ) × ∂Ω✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠    ∂ty = ∆y + div(uy) ✐♥ QT, y = 0 ♦♥ ΣT, y(0, ·) = y0 _{✐♥ Ω,} ✭✶✳✶✮ ✇❤❡r❡ y0_{∈ L}2_{(Ω)✐s t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ❛♥❞ u = (u} 1, ..., ud) ∈ L∞((0, T ) × Ω)d ✐s t❤❡ ❝♦♥tr♦❧✳ ■t ✐s ✇❡❧❧✲❦♥♦✇♥ ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❬✷✸✱ ❚❤❡♦r❡♠ ❛♥❞ Pr♦♣♦s✐t✐♦♥ ✸✳✶❪✮ t❤❛t ❢♦r ❡✈❡r② ✐♥✐t✐❛❧ ❞❛t❛ y0_{∈ L}2_{(Ω) ❛♥❞ ❡✈❡r② ❝♦♥tr♦❧ u ∈ L}∞_{((0, T ) × Ω)}d_{✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ y t♦ ❙②st❡♠ ✭✶✳✶✮} ✐♥ t❤❡ s♣❛❝❡ W (0, T )✱ ✇❤❡r❡ W (0, T ) := L2((0, T ), H01(Ω)) ∩ H1((0, T ), H−1(Ω)) ֒→ C0([0, T ]; L2(Ω)). ❊q✉❛t✐♦♥ ✭✶✳✶✮✱ ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✸✵❪✱ ✐s ❝❛❧❧❡❞ t❤❡ ❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥ ✐s ♣♦s❡❞ ♦♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡ Rd_{✱ ✐t ✐s str♦♥❣❧② r❡❧❛t❡❞ t♦ t❤❡ st♦❝❤❛st✐❝} ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭❙❉❊✮ ( dXt =Pdi=1ui(Xt)dt + dWt ✐♥ (0, T ) × Rd, X(0, ·) = X0 _{✐♥ R}d_, ✭✶✳✷✮ ✇❤❡r❡ Wt✐s t❤❡ st❛♥❞❛r❞ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ st❛rt✐♥❣ ❢r♦♠ 0✳ ❙②st❡♠ ✭✶✳✷✮ ❞❡s❝r✐❜❡s t❤❡ ♠♦✈❡♠❡♥t ♦❢ ❛ ♣❛rt✐❝✉❧❡ ♦❢ ♥❡❣❧✐❣✐❜❧❡ ♠❛ss✱ ✇✐t❤ ❝♦♥st❛♥t ❛♥❞ ✐s♦tr♦♣✐❝ ❞✐✛✉s✐♦♥✱ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ ❛ ❢♦r❝❡ ✜❡❧❞ u = (u1, . . . , ud)✳ ❯♥❞❡r s♦♠❡ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❞r✐❢t t❡r♠ U✱ ✐t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t✱ ❜② t❤❡ ■tô ▲❡♠♠❛✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ p ❛ss♦❝✐❛t❡❞ t♦ ✭✶✳✷✮ ✈❡r✐✜❡s  ∂tp =1₂∆p + div(up) ✐♥ (0, T ) × Rd, p(0, ·) = p0 _{✐♥ R}d_, ✭✶✳✸✮ ✇❤❡r❡ p0 _{✐s s♦♠❡ ✐♥✐t✐❛❧ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭s❡❡ ❡✳❣✳ ❬✹✶✱ ❙❡❝t✐♦♥ ✺✳✸❪✮✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ ❛} ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ ✇❡ ❤❛✈❡ p0 _{> 0 ❛✳❡✳ ❛♥❞}R Rdp 0 _{= 1✳ ■t ✐s t❤❡♥ ✈❡r② ❡❛s② t♦ ♣r♦✈❡ t❤❛t t❤❡s❡} ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❞✉r✐♥❣ t✐♠❡✿ ❛♥② s♦❧✉t✐♦♥ p ♦❢ ❙②st❡♠ ✭✶✳✸✮ ✈❡r✐✜❡s ❛❧s♦ p(t, ·) > 0 ❛✳❡✳ ❛♥❞ R Rdp(t, ·) = 1✱ ❢♦r ❛♥② t ∈ [0, T ] ❛♥❞ ❤❡♥❝❡ r❡♠❛✐♥s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✳ ❲❡ r❡❢❡r t♦ ❬✹✷❪ ❢♦r ♠♦r❡ ❡①♣❧❛♥❛t✐♦♥s ♦♥ t❤❡ ❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥✱ ♥♦t❛❜❧② ✐♥ t❤❡ ❝❛s❡ ♦❢ ♥♦♥❧✐♥❡❛r ❞r✐❢t t❡r♠s ♦r ♥♦♥✲❝♦♥st❛♥t ❛♥❞ ❛♥✐s♦tr♦♣✐❝ ❞✐✛✉s✐♦♥✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ ✇❡ ✐♠♣♦s❡ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛s ✐♥ ✭✶✳✶✮✱ t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥ ❢r♦♠ ❛ ❙❉❊ ✐s ♠♦r❡ ❞✐✣❝✉❧t✿ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❤❛s t♦ ❜❡ r❡♣❧❛❝❡❞ ❜② ❛♥ ✏❛❜s♦r❜❡❞✑ ♦r ✏❦✐❧❧❡❞✑ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ s❡❡ ❡✳❣✳ ❬✶✶✱ ♣♣✳ ✸✶✲✻✵❪✳ ▼♦r❡♦✈❡r✱ t❤❡ t♦t❛❧ ♠❛ss ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ♥♦t ❝♦♥s❡r✈❡❞ ❛♥②♠♦r❡✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ r❡♠❛✐♥✐♥❣ ✐♥s✐❞❡ Ω ❞❡❝r❡❛s❡s ✐♥ t✐♠❡✱ ❛♥❞ t❤❡ s♦❧✉t✐♦♥ t♦ ✭✶✳✶✮ ✐s ♥♦t ❛ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ❛♥②♠♦r❡✳ ❲❡ r❡❢❡r t♦ ❬✷✸✱ ❙❡❝t✐♦♥ ✷❪ ❢♦r ❛ ❞✐s❝✉ss✐♦♥ ♦♥ t❤❡ r❡❧❡✈❛♥❝❡ ♦❢ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ t❤✐s ❝♦♥t❡①t✳ ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭t❤❛t ✇♦✉❧❞ r❡st♦r❡ t❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ♠❛ss✮ s❡❡♠ t♦ ❜❡ ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ t❤❡ ♣r❡s❡♥t ❛rt✐❝❧❡✳ ✷

❲❤✐❧❡ t❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s❝❛❧❛r ❧✐♥❡❛r ❤❡❛t ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✐♥t❡r♥❛❧ ❝♦♥tr♦❧ ❛♥❞ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❛r❡ ♥♦✇ ✇❡❧❧✲✉♥❞❡rst♦♦❞ ✭s❡❡ ♥♦t❛❜❧② ❬✸✷❪ ❛♥❞ ❬✷✹❪✮✱ ❜✐❧✐♥❡❛r ❝♦♥tr♦❧❧❛❜✐❧✐t② s❡❡♠s t♦ ❤❛✈❡ ❜❡❡♥ ❧❡ss ❡①♣❧♦r❡❞✳ ❊q✉❛t✐♦♥ ✭✶✳✶✮ ❤❛s ❜❡❡♥ st✉❞✐❡❞ ✐♥ ❬✽❪✱ ✐♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡ ❛♥❞ ✇✐t❤ ❝♦♥tr♦❧s ❧♦❝❛❧✐③❡❞ ❡✈❡r②✇❤❡r❡ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡✳ ❈♦♥❝❡r♥✐♥❣ ❜✐❧✐♥❡❛r ❝♦♥tr♦❧ ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❜✐❧✐♥❡❛r t❡r♠ div(uy) ✐s r❡♣❧❛❝❡❞ ❜② uy ✇✐t❤ u ∈ L∞_{((0, T ) × Ω)✱ ✇❡ r❡❢❡r t♦} ❬✾✱ ✶✵✱ ✷✼✱ ✷✽✱ ✷✻✱ ✷✾✱ ✸✹✱ ✹✵✱ ✹✹✱ ✹✺❪✳ ▲❡t ✉s ♠❡♥t✐♦♥ t❤❛t ❜✐❧✐♥❡❛r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❤❛s ♣r❡✈✐♦✉s❧② ❜❡❡♥ st✉❞✐❡❞✳ ❆ ✜rst r❡s✉❧t ✇❛s ♣r♦✈❡❞ ✐♥ ❬✶❪✱ ✇❤❡r❡ ❛ ❝❧♦s❡ ❢♦rt❤✲♦r❞❡r ✐♥ t✐♠❡ ♠♦❞❡❧ ✐s ✐♥✈❡st✐❣❛t❡❞✱ ✇✐t❤ ❝♦♥tr♦❧s ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ t✐♠❡✳ ❚❤✐s r❡s✉❧t ❤❛s ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ s❡❝♦♥❞✲♦r❞❡r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ✜rst❧② ✐♥ ❬✹❪ ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ t❤❡♥ ✐♥ ❬✺❪ ✐♥ t❤❡ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ st✐❧❧ ❢♦r t✐♠❡✲✈❛r②✐♥❣ ❝♦♥tr♦❧s✳ ❋♦r ❡q✉❛t✐♦♥ ✭✶✳✶✮ ✭✐♥ ❛ s❧✐❣❤t❧② ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r♠✮✱ t❤❡ ❝❛s❡ ♦❢ s♣❛❝❡ ❛♥❞ t✐♠❡✲✈❛r②✐♥❣ ❝♦♥tr♦❧s ✐s tr❡❛t❡❞ ✐♥ ❬✷✸❪✳ ◆♦t❛❜❧②✱ ❢♦r ❛ ❞r✐❢t t❡r♠ t❤❛t ✐s ❛✣♥❡ ✐♥ t❤❡ ❝♦♥tr♦❧✱ t❤❡ ❛✉t❤♦rs ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ❢♦r ❣❡♥❡r❛❧ ❝♦st ❢✉♥❝t✐♦♥❛❧s✱ ❛♥❞ ❞❡r✐✈❡ ✜rst✲♦r❞❡r ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♥ ❛❞❥♦✐♥t st❛t❡✳ ❚❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✱ ✐✳❡✳ ❙②st❡♠ ✭✶✳✶✮ ✇✐t❤♦✉t ❞✐✛✉s✐♦♥✱ ❤❛s ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✶✾✱ ✷✵❪✳ ❚❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❛rt✐❝❧❡ ✐s ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✶✳✷✱ ✇❡ ❣✐✈❡ t❤❡ ♠❛✐♥s r❡s✉❧ts ♦❢ t❤❡ ❛rt✐❝❧❡ ✭❚❤❡♦r❡♠ ✶✳✶✱ r❡s♣✳ ❚❤❡♦r❡♠ ✶✳✷✱ ✇❤✐❝❤ ❣✐✈❡s ❛ r❡s✉❧t ♦❢ ❧♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② t♦ t❤❡ tr❛❥❡❝t♦r✐❡s ✇✐t❤ d❝♦♠♣♦♥❡♥ts✱ r❡s♣✳ ❛ r❡❞✉❝❡❞ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧s ❛r♦✉♥❞ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥tr♦❧✮ ❛♥❞ s♦♠❡ r❡♠❛r❦s✳ ❙❡❝t✐♦♥ ✷ ✐s ❞❡✈♦t❡❞ t♦ st✉❞②✐♥❣ ❛ ❧✐♥❡❛r✐③❡❞ ✈❡rs✐♦♥ ♦❢ ✭✶✳✶✮✳ ■♥ ❙❡❝t✐♦♥ ✷✳✶✱ ✇❡ ♣r♦✈❡ ❛ ♥❡✇ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✭Pr♦♣♦s✐t✐♦♥ ✷✳✶✮ ❢♦r s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❜❛❝❦✇❛r❞ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ♦♥❡✲♦r❞❡r t❡r♠s✳ ❚❤❡ ♠❛✐♥ ♥♦✈❡❧t② ✐s t❤❛t t❤❡ ❧♦❝❛❧ ♦❜s❡r✈❛t✐♦♥ t❡r♠ ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✭✷✳✹✮✳ ❚❤✐s ❤❛s ❛❧r❡❛❞② ❜❡❡♥ ♣r♦✈❡❞ ✐♥ ❬✶✼❪ ❢♦r ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts✳ ▼♦r❡♦✈❡r✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ♣✉t ❛s ♠❛♥② ❞❡r✐✈❛t✐✈❡s ❛s ✇❡ ✇❛♥t ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ ♦✉r ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ♥❡❡❞❡❞ ❢♦r t❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢✳ ■♥ ❙❡❝t✐♦♥ ✷✳✷✱ ✇❡ ❡①♣❧❛✐♥ ❤♦✇ t♦ r❡♠♦✈❡ s♦♠❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❣r❛❞✐❡♥t ✐♥ t❤❡ ❈❛r❧❡♠❛♥ ✐♥❡q✉❛❧✐t②✳ ❚❤✐s ✐s ♣❡r❢♦r♠❡❞ ❜② ✉s✐♥❣ ✇❤❛t ✇❡ ❝❛❧❧ ❛♥ ❛r❣✉♠❡♥t ♦❢ ✏❛❧❣❡❜r❛✐❝ s♦❧✈❛❜✐❧✐t②✑ ✭❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✶✷❪ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ st❛❜✐❧✐③❛t✐♦♥ ♦❢ ❖❉❊s ❛♥❞ ✐♥ ❬✶✻❪ ❢♦r t❤❡ st✉❞② ♦❢ ❝♦✉♣❧❡❞ s②st❡♠s ♦❢ P❉❊s✮✱ ❜❛s❡❞ ♦♥ ✐❞❡❛s ❞❡✈❡❧♦♣❡❞ ❜② ●r♦♠♦✈ ✐♥ ❬✷✺✱ ❙❡❝t✐♦♥ ✷✳✸✳✽❪✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ❤❛s ❛❧r❡❛❞② ❜❡❡♥ ✉s❡❞ s✉❝❝❡ss❢✉❧❧② ✐♥ ❬✷✱ ✶✼✱ ✶✽✱ ✶✺✱ ✸✸✱ ✹✸❪✳ ❚❤❡ ♠❛✐♥ ♥♦✈❡❧t② ❝♦♠♣❛r❡❞ t♦ t❤❡ ❡①✐st✐♥❣ ❧✐t❡r❛t✉r❡ ✐s t❤❛t t❤❡ ❛❧❣❡❜r❛✐❝ s♦❧✈❛❜✐❧✐t② ✐s ♣❡r❢♦r♠❡❞ ❞✐r❡❝t❧② ♦♥ t❤❡ ❞✉❛❧ ♣r♦❜❧❡♠✳ ▼♦r❡♦✈❡r✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❣❡t r✐❞ ♦❢ t❤❡ ❤✐❣❤ ♦r❞❡r ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ r✐❣❤t ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ✜♥❛❧ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✭✷✳✸✹✮✳ ■♥ ❙❡❝t✐♦♥ ✷✳✸✱ ✇❡ ✉s❡ s♦♠❡ ❛r❣✉♠❡♥ts ❝♦♠✐♥❣ ❢r♦♠ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❤❡♦r② ✐♥ ♦r❞❡r t♦ ❞❡r✐✈❡ ❢r♦♠ ♦✉r ♦❜s❡r✈❛❜✐❧✐t② ✐♥❡q✉❛❧✐t② t❤❡ ❡①✐st❡♥❝❡ ♦❢ r❡❣✉❧❛r ❡♥♦✉❣❤ ❝♦♥tr♦❧s✱ ✇✐t❤ ❛ s♣❡❝✐❛❧ ❢♦r♠✱ ✐♥ ❛♣♣r♦♣r✐❛t❡ ✇❡✐❣❤t❡❞ s♣❛❝❡s✳ ■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ♥♦♥❧✐♥❡❛r ♣r♦❜❧❡♠ ❜② ✉s✐♥❣ ❛ st❛♥❞❛r❞ str❛t❡❣② ❝♦♠✐♥❣ ❢r♦♠ ❬✸✼❪ t♦❣❡t❤❡r ✇✐t❤ s♦♠❡ ❛❞❛♣t❡❞ ✐♥✈❡rs❡ ♠❛♣♣✐♥❣ ❚❤❡♦r❡♠✳ ❚♦ ✜♥✐s❤✱ ✐♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ ❣✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ tr❛❥❡❝t♦r② ❢♦r ✇❤✐❝❤ t❤❡ ❧♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ ✇✐t❤ ❛ r❡❞✉❝❡❞ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧s✳

✶✳✷ ▼❛✐♥s r❡s✉❧ts

▲❡t (y, u) ❜❡ ❛ tr❛❥❡❝t♦r② ♦❢ ✭✶✳✶✮✱ ✐✳❡✳ ✈❡r✐❢②✐♥❣    ∂ty = ∆y + div(uy) ✐♥ QT, y = 0 ♦♥ ΣT, y(0, ·) = y0∈ L2_{(Ω) \ {0} ✐♥ Ω.} ✭✶✳✹✮ ✸

✶✳✷✳✶ ❈♦♥tr♦❧s ✇✐t❤ ❞ ❝♦♠♣♦♥❡♥ts ❲❡ ✜rst st❛t❡ ❛ r❡s✉❧t ♦❢ ❧♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② t♦ t❤❡ tr❛❥❡❝t♦r✐❡s t♦ ❙②st❡♠ ✭✶✳✹✮ ✇✐t❤ ❛ ❝♦♥tr♦❧ ❝♦♥t❛✐♥✐♥❣ d ❝♦♠♣♦♥❡♥ts✿ ❚❤❡♦r❡♠ ✶✳✶✳ ▲❡t ω ❜❡ ❛♥② ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t ♦❢ Ω✳ ❆ss✉♠❡ t❤❛t t❤❡ tr❛❥❡❝t♦r② (y, u) ✇✐t❤ u = (u1, ..., ud)♦❢ ❙②st❡♠ ✭✶✳✹✮ ✐s r❡❣✉❧❛r ❡♥♦✉❣❤ ✭❢♦r ❡①❛♠♣❧❡ ♦❢ ❝❧❛ss C∞ ♦♥ (0, T ) × Ω✮✱ ❛♥❞ t❤❛t t❤❡r❡ ❡①✐sts s♦♠❡ ♦♣❡♥ s✉❜s❡t ωu✱ str♦♥❣❧② ✐♥❝❧✉❞❡❞ ✐♥ Ω✱ s✉❝❤ t❤❛t t❤❡ s✉♣♣♦rt ♦❢ u ✐s ✐♥❝❧✉❞❡❞ ✐♥ [0, T ] × ωu✳ ❚❤❡♥✱ ❙②st❡♠ ✭✶✳✶✮ ✐s ❧♦❝❛❧❧② ❝♦♥tr♦❧❧❛❜❧❡ ✇✐t❤ ❧♦❝❛❧✐③❡❞ ❝♦♥tr♦❧s✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❢♦r ❡✈❡r② ε > 0 ❛♥❞ ❡✈❡r② T > 0✱ t❤❡r❡ ❡①✐sts η > 0 s✉❝❤ t❤❛t ❢♦r ❛♥② y0_{∈ L}2_{(Ω)✈❡r✐❢②✐♥❣} ||y0− y(0)||L2_(Ω)6η, ✭✶✳✺✮ t❤❡r❡ ❡①✐sts ❛ tr❛❥❡❝t♦r② (y, u) t♦ ❙②st❡♠ ✭✶✳✶✮ s✉❝❤ t❤❛t              y(T ) = y(T ), u = u + v ❢♦r s♦♠❡ v ∈ L∞_{((0, T ) × Ω)}d_, Supp (v) ⊂ (0, T ) × ω, ||v||L∞_{((0,T )×Ω)}d 6ε, ||y − y||W (0,T ) 6ε. ❘❡♠❛r❦ ✶✳ • ❚❤❡ r❡❣✉❧❛r✐t② ❛ss✉♠♣t✐♦♥s ♦♥ (y, u) ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞✱ ♥♦t❛❜❧② ✐t ✐s ❡♥♦✉❣❤ t❤❛t t❤❡ r❡❢❡r❡♥❝❡ tr❛❥❡❝t♦r② ✐s Cr _{❢♦r s♦♠❡ r ∈ N}∗_{❧❛r❣❡ ❡♥♦✉❣❤✱ ♦♥ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ (0, T ) × ω} 0✳ • ■❢ y0_{= 0✱ t❤❡ ♦♥❧② s♦❧✉t✐♦♥ t♦ ✭✶✳✶✮ ✐s y ≡ 0✱ ✇❤❛t❡✈❡r u ✐s✱ s♦ t❤❛t t❤❡ ♦♥❧② r❡❛❝❤❛❜❧❡ st❛t❡ ❛t} t✐♠❡ T ✐s 0✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ η > 0 ❤❛s ♥♦t❛❜❧② t♦ ❜❡ ❝❤♦s❡♥ s♠❛❧❧ ❡♥♦✉❣❤ s✉❝❤ t❤❛t y0_{6= 0✳} • ❋r♦♠ t❤❡ r❡s✉❧ts ❣✐✈❡♥ ✐♥ ❬✼❪✱ ❛s s♦♦♥ ❛s y0 _{>0✱ t❤❡♥ ❛♥② tr❛❥❡❝t♦r② t♦ ❙②st❡♠ ✭✶✳✶✮ r❡♠❛✐♥s} ♥♦♥✲♥❡❣❛t✐✈❡ ✭s❡❡ ❛❧s♦ ❬✷✸❪✮✳ ❚❤✐s ❢❛❝t ❞✐✛❡rs ❢r♦♠ t❤❡ ✉s✉❛❧ ❧✐♥❡❛r ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ✐♥t❡r♥❛❧ ❝♦♥tr♦❧ ✭s❡❡ ❬✸✽❪✮✳ ✶✳✷✳✷ ❈♦♥tr♦❧❧❛❜✐❧✐t② ❛❝t✐♥❣ t❤r♦✉❣❤ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❣✐✈❡ ❛ r❡s✉❧t ♦❢ ❧♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② t♦ t❤❡ tr❛❥❡❝t♦r✐❡s t♦ ❙②st❡♠ ✭✶✳✹✮ ✇✐t❤ ❛ ❝♦♥tr♦❧ ❛❝t✐♥❣ t❤r♦✉❣❤ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r B ∈ Md,m(R)✇✐t❤ m ∈ N∗ s✉❝❤ t❤❛t m 6 d✳ ❲❡ ✜rst ✐♥tr♦❞✉❝❡ s♦♠❡ ♥♦t❛t✐♦♥s✳ ▲❡t q ∈ N ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t E(m, q) = {(α1, . . . , αm) ∈ Nm| 0 < α1+ . . . + αm6q}, ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t E(m, q) = ∅ ✐❢ q = 0✳ ◆♦t❡ t❤❛t ❜② ❛♥ ❡❧❡♠❡♥t❛r② ❝♦♠♣✉t❛t✐♦♥✱ #E(m, q) = (q + m)! m!q! − 1 =: N (m, q). ❋♦r v ∈ Rm_{✱ ✇❡ ✇r✐t❡ Bv = (B} 1v, . . . , Bdv) ∈ Rd✳ ❋♦r j ∈ {1, ..., m}✱ ✇❡ ✇r✐t❡ (Bm∗.∇) : ψ ∈ C∞(Rd) 7→ Bm∗(∇ψ) ∈ C∞(Rd). ❋♦r (α1, . . . αm) ∈ E(m, q)✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t♦r✿ (B∗.∇)α1,...αm _{: ψ ∈ C}∞_(Rd_{) 7→ (B}∗ 1.∇) . . . (B1∗.∇) | {z } α1 t✐♠❡s . . . (Bm∗.∇) . . . (Bm∗.∇) | {z } αm t✐♠❡s ψ ∈ C∞(Rd). ✹

❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐①✿ Mq(u) =              B∗ 1 ✳✳✳ B∗ m (B∗_.∇)1,0,...,0_u 1 . . . (B∗.∇)1,0,...,0ud (B∗_.∇)0,1,...,0_u 1 . . . (B∗.∇)0,1,...,0ud ✳✳✳ ✳✳✳ ✳✳✳ (B∗_.∇)0,...,0,q_u 1 . . . (B∗.∇)0,...,0,qud              ∈ MN (m,q)+m,d(R). ✭✶✳✻✮ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥tr♦❧❧❛❜✐❧✐t② r❡s✉❧t✳ ❚❤❡♦r❡♠ ✶✳✷✳ ▲❡t m ∈ N∗ _{✭✇✐t❤ ♣♦ss✐❜❧② m < d✮✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s ♦❢ ❚❤❡♦r❡♠ ✶✳✶✱ ❛ss✉♠❡} t❤❛t t❤❡r❡ ❡①✐sts q ∈ N ❛♥❞ s♦♠❡ (t0, x0) ∈ (0, T ) × ω s✉❝❤ t❤❛t ❘❛♥❦(Mq(u)(t0, x0)) = d. ✭✶✳✼✮ ❚❤❡♥✱ ❙②st❡♠ ✭✶✳✶✮ ✐s ❧♦❝❛❧❧② ❝♦♥tr♦❧❧❛❜❧❡ ✇✐t❤ ❧♦❝❛❧✐③❡❞ ❝♦♥tr♦❧s✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❢♦r ❡✈❡r② ε > 0 ❛♥❞ ❡✈❡r② T > 0✱ t❤❡r❡ ❡①✐sts η > 0 s✉❝❤ t❤❛t ❢♦r ❛♥② y0_{∈ L}2_{(Ω)✈❡r✐❢②✐♥❣} ||y0− y(0)||L2_(Ω)6η, t❤❡r❡ ❡①✐sts ❛ tr❛❥❡❝t♦r② (y, u) t♦ ❙②st❡♠ ✭✶✳✶✮ s✉❝❤ t❤❛t              y(T ) = y(T ), u = u + Bv ❢♦r s♦♠❡ v ∈ L∞_{((0, T ) × Ω)}m_, Supp (v) ⊂ (0, T ) × ω, ||v||L∞_{((0,T )×Ω)}m 6ε, ||y − y||W (0,T ) 6ε. ❘❡♠❛r❦ ✷✳ • ❘❡♠❛r❦ t❤❛t ✐❢ B = Id ✭✐✳❡✳ ✇❡ ❝♦♥tr♦❧ ❡✈❡r② ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ u✮✱ ❝♦♥❞✐t✐♦♥ ✭✶✳✼✮ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ✈❡r✐✜❡❞ ❢♦r q = 0✱ ✇❤❛t❡✈❡r u ✐s✳ ❍❡♥❝❡ ❚❤❡♦r❡♠ ✶✳✷ ❝♦♥t❛✐♥s t❤❡ r❡s✉❧t ❣✐✈❡♥ ✐♥ ❚❤❡♦r❡♠ ✶✳✶✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦♥❧② ❣✐✈❡ ❛ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✷✳ • ❈♦♥❞✐t✐♦♥ ✭✶✳✼✮ ♥♦t❛❜❧② ✐♠♣❧✐❡s t❤❛t q ❤❛s t♦ ❜❡ ❝❤♦s❡♥ ❧❛r❣❡ ❡♥♦✉❣❤ s✉❝❤ t❤❛t N(m, q) > d−m✳ • ❆ss✉♠♣t✐♦♥ ✭✶✳✼✮ ✐s ❣❡♥❡r✐❝✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ✐❢ C∞_{((0, T ) × ω)}2 _{✐s ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ C}q t♦♣♦❧♦❣②✱ t❤❡ s❡ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥s (y, u) ∈ C∞_{((0, T ) × ω)}2_{✈❡r✐❢②✐♥❣ ✭✶✳✼✮ ✐s ❛♥ ❞❡♥s❡ ♦♣❡♥ s❡t✳} • ■♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ ❣✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ tr❛❥❡❝t♦r② ✇❤✐❝❤ ❞♦❡s ♥♦t s❛t✐s❢② ❝♦♥❞✐t✐♦♥ ✭✶✳✼✮ ❛♥❞ ❢♦r ✇❤✐❝❤ t❤❡ ❧♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② t♦ t❤❡ tr❛❥❡❝t♦r✐❡s ❞♦❡s ♥♦t ❤♦❧❞✳ ■t ❤✐❣❤❧✐❣❤ts t❤❛t ❈♦♥❞✐t✐♦♥ ✭✶✳✼✮ ✐s ♥♦t ❛rt✐✜❝✐❛❧✳ ❊✈❡♥ ✐❢ t❤❡ ❛✉t❤♦rs t❤✐♥❦ t❤❛t ❈♦♥❞✐t✐♦♥ ✶✳✼ ✐s ♦♣t✐♠❛❧✱ ✜♥❞ ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ r❡♠❛✐♥s ♦♥ ♦♣❡♥ ♣r♦❜❧❡♠✳ ❊①❡♠♣❧❡ ✶✳✶✳ ❲❡ ❣✐✈❡ ❛♥ ❡①♣❧✐❝✐t ❡①❛♠♣❧❡✱ ✐♥ ♦r❞❡r t♦ ❡①♣❧❛✐♥ ❜❡tt❡r ❝♦♥❞✐t✐♦♥ ✭✶✳✼✮✳ ▲❡t ✉s ❛ss✉♠❡ ✺

t❤❛t ✇❡ ✇❛♥t t♦ ❝♦♥tr♦❧ ♦♥❧② t❤❡ m(< n) ✜rst ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❣r❛❞✐❡♥t✱ ✐✳❡✳ B =               1 0 . . . 0 0 1 ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ 0 0 . . . 0 1 0 . . . 0 ✳✳✳ ✳✳✳ 0 . . . 0               ∈ Mn,m(R). ❚❤❡♥ ❢♦r ❛♥② q ∈ N s✉❝❤ t❤❛t N(m, q) > d − m✱ ✇❡ ❤❛✈❡ Mq(u) =                      1 0 . . . 0 0 1 0 0 ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ 0 . . . 0 1 0 . . . 0 ∂x1u1 ∂x1u2 . . . ∂x1um+1 . . . ∂x1ud ✳✳✳ ✳✳✳ ✳✳✳ ∂xmu1 . . . ∂xmum+1 . . . ∂xmud ∂2 x2 1u1 . . . ∂ 2 x2 1um+1 . . . ∂ 2 x2 1ud ✳✳✳ ✳✳✳ ✳✳✳ ∂p_xq mu1 . . . ∂ q xqmum+1 . . . ∂ q xqmum+1                      ∈ MN (m,q)+m,d(R). ❲❡ ♦❜s❡r✈❡ t❤❛t Mq(u)✐s ♦❢ ♠❛①✐♠❛❧ r❛♥❦ d ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐①✿ f Mq(u) =            ∂x1um+1 . . . ∂x1ud ✳✳✳ ✳✳✳ ∂xmum+1 . . . ∂xmud ∂_x22 1um+1 . . . ∂ 2 x2 1ud ✳✳✳ ✳✳✳ ∂_xqq mum+1 . . . ∂ q xq mum+1            ∈ MN (m,q),d−m(R), ✐s ♦❢ ♠❛①✐♠❛❧ r❛♥❦ d − m✳

✷ ◆✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ❧✐♥❡❛r✐③❡❞ s②st❡♠

■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ ❛❧✇❛②s ❛ss✉♠❡ t❤❛t t❤❡ tr❛❥❡❝t♦r② (y, u) ♦❢ ✭✶✳✹✮ ✈❡r✐✜❡s t❤❡ ❤②♣♦t❤❡s✐s ♦❢ ❚❤❡♦r❡♠ ✶✳✶✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ s②st❡♠   

∂ty = ∆y + div(uy) + div(θu) ✐♥ QT,

y = 0 ♦♥ ΣT,

y(0, ·) = y0 _{✐♥ Ω,}

✭✷✳✶✮

✇❤❡r❡ y0_{∈ L}2_{(Ω)❛♥❞ θ ∈ C}∞_{(Ω) ✐s s✉❝❤ t❤❛t}    Supp(θ) ⊆ ω, θ ≡ 1 in ω0, 0 6θ 6 1 in Ω, ✭✷✳✷✮ ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ♦♣❡♥ s✉❜s❡t ω0 ✇❤✐❝❤ ✐s str♦♥❣❧② ✐♥❝❧✉❞❡❞ ✐♥ ω✳ ❚❤❡ ❣♦❛❧ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ ♣r♦✈❡ t❤❡ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ ❙②st❡♠ ✭✷✳✶✮✱ ✇✐t❤ ❧❡ss ❝♦♥tr♦❧s t❤❛♥ ❡q✉❛t✐♦♥s ❛♥❞ r❡❣✉❧❛r ❡♥♦✉❣❤ ❝♦♥tr♦❧s ✐♥ ❛ s♣❡❝✐❛❧ ❢♦r♠✳ ❘❡♠❛r❦ ✸✳ ◆♦t❡ t❤❛t t❤❡ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ ✭✷✳✶✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ✏r❡❛❧✑ ❧✐♥❡❛r✐③❡❞ ✈❡rs✐♦♥ ♦❢ ✭✶✳✶✮ ❛r♦✉♥❞ (y, u) ❣✐✈❡♥ ❜②   

∂ty = ∆y + div(uy) + div(y ˜u) ✐♥ QT,

y = 0 ♦♥ ΣT, y(0, ·) = y0 _{✐♥ Ω.} ✭✷✳✸✮ ■♥❞❡❡❞✱ ❜② ✉♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ y ♦❢ ✭✶✳✹✮✱ ❛s s♦♦♥ ❛s y0 _{6= 0✱ s✐♥❝❡ y ❝❛♥♥♦t ✈❛♥✐s❤} ♦♥ ❛ s✉❜s❡t ♦❢ (0, T ) × Ω ♦❢ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ ✭s❡❡ ❬✻❪✮ ❛♥❞ y ✐s ✐♥ C∞_{((0, T ) × Ω)✱ t❤❡r❡ ❡①✐sts s♦♠❡} s✉❜s❡t (T1, T2) × ˜ω0 ♦❢ (0, T ) × ω0 s✉❝❤ t❤❛t |y| > C > 0 ♦♥ (T1, T2) × ω0✱ t❤❛t ✇❡ ❝❛♥ ❛ss✉♠❡ t♦ ❜❡ ❡①❛❝t❧② (0, T ) × ω0✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✳ ❍❡♥❝❡✱ ❢♦r ❛♥② i ∈ {1, . . . , d}✱ ♦♥❡ ❝❛♥ s♦❧✈❡ ✭✐♥ ˜ui✮ t❤❡ ❡q✉❛t✐♦♥ θui= y ˜ui ❜② ♣♦s✐♥❣ ˜ ui = θui y . ❘❡♠❛r❦ t❤❛t ˜ui ❡♥❥♦②s t❤❡ s❛♠❡ r❡❣✉❧❛r✐t② ♣r♦♣❡rt✐❡s ❛s ui✳

✷✳✶ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡s

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❞❥♦✐♥t s②st❡♠ ❛ss♦❝✐❛t❡❞ t♦ ❙②st❡♠ ✭✷✳✶✮    −∂tψ = ∆ψ + u · ∇ψ ✐♥ QT, ψ = 0 ♦♥ ΣT, ψ(T, ·) = ψ0 _{✐♥ Ω.} ✭✷✳✹✮ ❋✐rst ♦❢ ❛❧❧✱ ✇❡ ✇✐❧❧ ✐♥tr♦❞✉❝❡ s♦♠❡ ♥♦t❛t✐♦♥s✳ ❲❡ ❞❡♥♦t❡ ❜② | · | t❤❡ ❡✉❝❧✐❞❡❛♥ ♥♦r♠ ♦♥ RM_✱ ✇❤❛t❡✈❡r M ∈ N∗ _{✐s✳ ❋♦r s, λ > 0 ❛♥❞ p > 1✱ ❧❡t ✉s ❞❡✜♥❡ t❤❡ t✇♦ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿} α(t, x) := exp((2p + 2)λkη 0_k ∞) − exp[λ(2pkη0k∞+ η0(x))] tp_{(T − t)}p ✭✷✳✺✮ ❛♥❞ ξ(t, x) := exp[λ(2pkη 0_k ∞+ η0(x))] tp_{(T − t)}p . ✭✷✳✻✮ ❍❡r❡✱ η0_{∈ C}∞_{(Ω)✐s ❛ ❢✉♥❝t✐♦♥ s❛t✐s❢②✐♥❣} |∇η0| > κ in Ω\ω1, η0> 0 in Ω and η0= 0 on ∂Ω, ✇✐t❤ κ > 0 ❛♥❞ ω1 s♦♠❡ ♦♣❡♥ s✉❜s❡t ✈❡r✐❢②✐♥❣ ω1 ⊂⊂ ω0✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ η0 _{❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✹✱ ▲❡♠♠❛ ✶✳✶✱ ❈❤❛♣✳ ✶❪ ✭s❡❡ ❛❧s♦ ❬✶✸✱ ▲❡♠♠❛ ✷✳✻✽✱ ❈❤❛♣✳ ✷❪✮✳ ❲❡ ✇✐❧❧} ✉s❡ t❤❡ t✇♦ ♥♦t❛t✐♦♥s α∗(t) := max x∈Ω α(t, x) and ξ∗(t) := min x∈Ω ξ(t, x), ✭✷✳✼✮ ✼

❢♦r ❛❧❧ t ∈ (0, T )✳ ◆♦t❡ t❤❛t t❤❡s❡ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ❛r❡ r❡❛❝❤❡❞ ❛t t❤❡ ❜♦✉♥❞❛r② ∂Ω✳ ❋♦r s, λ > 0✱ ❧❡t ✉s ❞❡✜♥❡ I(s, λ; u) := s3λ4 ZZ QT e−2sαξ3u2dxdt + sλ2 ZZ QT e−2sαξ|∇u|2dxdt. ✭✷✳✽✮ ▲❡t ✉s ♥♦✇ ❣✐✈❡ s♦♠❡ ✉s❡❢✉❧ ❛✉①✐❧✐❛r② r❡s✉❧ts t❤❛t ✇❡ ✇✐❧❧ ♥❡❡❞ ✐♥ ♦✉r ♣r♦♦❢s✳ ❚❤❡ ✜rst ♦♥❡ ✐s ❛ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✇❤✐❝❤ ❤♦❧❞s ❢♦r s♦❧✉t✐♦♥s ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ▲❡♠♠❛ ✷✳✶✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C > 0 s✉❝❤ t❤❛t ❢♦r ❛♥② u0 _{∈ L}2_{(Ω)✱ f} 1 ∈ L2(QT) ❛♥❞ f2∈ L2(ΣT)✳✱ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ s②st❡♠    −∂tu − ∆u = f1 in QT, ∂u ∂n = f2 on ΣT, u(T, ·) = u0 _{in Ω} s❛t✐s✜❡s I(s, λ; u) 6 C s3λ4 ZZ (0,T )×ω1 e−2sαξ3u2dxdt + sλ ZZ ΣT e−2sα∗ξ∗f22dσdt + ZZ QT e−2sαf12dxdt  , ❢♦r ❛❧❧ λ > C ❛♥❞ s > C(Tp_{+ T}2p_)✳ ▲❡♠♠❛ ✷✳✶ ✐s ♣r♦✈❡❞ ✐♥ ❬✷✷✱ ❚❤❡♦r❡♠ ✶❪ ✐♥ t❤❡ ❝❛s❡ p = 1✳ ❍♦✇❡✈❡r✱ ❢♦❧❧♦✇✐♥❣ t❤❡ st❡♣s ♦❢ t❤❡ ♣r♦♦❢ ❣✐✈❡♥ ✐♥ ❬✷✷❪✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ✐♥❡q✉❛❧✐t② ❢♦r ❛♥② p ∈ N∗_✳ ❋r♦♠ ▲❡♠♠❛ ✷✳✶✱ ♦♥❡ ❝❛♥ ❞❡❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿ ▲❡♠♠❛ ✷✳✷✳ ▲❡t f ∈ L2_(Σ T)✱ G = (g1, . . . gd) ∈ L∞(QT)d ❛♥❞ h ∈ L2(QT)✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C > 0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ϕT _{∈ L}2_{(Ω)✱ t❤❡ s♦❧✉t✐♦♥ ϕ t♦ t❤❡ s②st❡♠}    −∂tϕ = ∆ϕ + G · ∇ϕ + h in QT, ∂ϕ ∂n = f on ΣT, ϕ(T, ·) = ϕT _{in Ω} s❛t✐s✜❡s I(s, λ; ϕ) 6 C s3_λ4 ZZ (0,T )×ω1 e−2sα_ξ3_ϕ2_{dxdt + sλ} ZZ ΣT e−2sα∗ ξ∗_f2_{dσdt +} ZZ QT e−2sα_h2_dxdt ! , ❢♦r ❡✈❡r② λ > C ❛♥❞ s > s0= C(Tp+ T2p)✳ ❚❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✷✳✷ ✐s st❛♥❞❛r❞ ❛♥❞ ✐s ❧❡❢t t♦ t❤❡ r❡❛❞❡r ✭♦♥❡ ❥✉st ❤❛s t♦ ❛♣♣❧② ▲❡♠♠❛ ✷✳✶ ❛♥❞ ❛❜s♦r❜ t❤❡ r❡♠❛✐♥✐♥❣ ❧♦✇❡r✲♦r❞❡r t❡r♠s t❤❛♥❦s t♦ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✮✳ ❲❡ ✇✐❧❧ ❛❧s♦ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s✳ ▲❡♠♠❛ ✷✳✸✳ ▲❡t r ∈ R✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts C := C(r, ω1, Ω) > 0 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② T > 0 ❛♥❞ ❡✈❡r② u ∈ L2_{((0, T ), H}1_(Ω))✱ sr+2_λr+2 ZZ QT e−2sαξr+2u2dxdt 6 C  srλr ZZ QT e−2sαξr|∇u|2_dxdt +sr+2_λr+2 ZZ (0,T )×ω1 e−2sαξr+2u2dxdt ! , ❢♦r ❡✈❡r② λ > C ❛♥❞ s > C(T2p_)✳ ✽

❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s ❧❡♠♠❛ ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r ❡①❛♠♣❧❡ ✐♥ ❬✶✹✱ ▲❡♠♠❛ ✸❪ ✐♥ t❤❡ ❝❛s❡ p = 9✳ ❍♦✇❡✈❡r✱ ❢♦❧❧♦✇✐♥❣ t❤❡ st❡♣s ♦❢ t❤❡ ♣r♦♦❢ ❣✐✈❡♥ ✐♥ ❬✶✹❪✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ✐♥❡q✉❛❧✐t② ❢♦r ❛♥② p ∈ N∗_✳ ■♥ ♦r❞❡r t♦ ❞❡❛❧ ✇✐t❤ ♠♦r❡ r❡❣✉❧❛r s♦❧✉t✐♦♥s✱ ♦♥❡ ♥❡❡❞s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳ ▲❡♠♠❛ ✷✳✹✳ ▲❡t z0∈ H01(Ω)✱ G ∈ C∞(QT)d ❛♥❞ f ∈ L2(QT)m✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② R := −∆ − G · ∇ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ s♦❧✉t✐♦♥ z t♦ t❤❡ s②st❡♠    ∂tz = ∆z + G · ∇z + f in QT, z = 0 on ΣT, z(0, ·) = z0 in Ω. ▲❡t n ∈ N✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t z0 ∈ H2n+1(Ω)✱ f ∈ L2((0, T ), H2n(Ω)) ∩ Hn((0, T ), L2(Ω)) ❛♥❞ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥s✿          g0:= z0∈ H01(Ω), g1:= f (0, ·) − Rg0∈ H01(Ω), ✳✳✳ gd:= ∂tn−1f (0, ·) − Rgd−1∈ H01(Ω). ✭✷✳✾✮ ❚❤❡♥ z ∈ L2_{((0, T ), H}2n+2_{(Ω)) ∩ H}n+1_{((0, T ), L}2_{(Ω)) ❛♥❞ ✇❡ ❤❛✈❡ t❤❡ ❡st✐♠❛t❡} kzkL2_{((0,T ),H}2n+2_(Ω))∩Hn+1_{((0,T ),L}2_(Ω))6C(kf k_L2_{((0,T ),H}2n_(Ω))∩Hn_{((0,T ),L}2_(Ω))+ kz₀k_H2n+1_(Ω)). ■t ✐s ❛ ❝❧❛ss✐❝❛❧ r❡s✉❧t t❤❛t ❝❛♥ ❜❡ ❡❛s✐❧② ❞❡❞✉❝❡❞ ❢♦r ❡①❛♠♣❧❡ ❢r♦♠ ❬✷✶✱ ❚❤✳ ✻✱ ♣✳ ✸✻✺❪✳ ❲❡ ❛r❡ ♥♦✇ ❛❜❧❡ t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝r✉❝✐❛❧ ✐♥❡q✉❛❧✐t②✿ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳ ▲❡t N ∈ N ✇✐t❤ N > 3 ✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts C > 0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ψ0_{∈ L}2_{(Ω)✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥ ψ t♦ ❙②st❡♠ ✭✷✳✹✮ s❛t✐s✜❡s} λ2 ZZ QT e−2sα−2µsα∗(sξ)|∇N +1ψ|2dxdt + . . . + λ2N +2 ZZ QT e−2sα−2µsα∗(sξ)2N +1|∇ψ|2_dxdt +λ2N +2 ZZ QT e−2sα∗−2µsα∗(sξ∗)2N +1|ψ|2_dxdt 6Cλ2N +2 ZZ (0,T )×ω0 e−2sα−2µsα∗(sξ)2N +1|∇ψ|2_dxdt ✭✷✳✶✵✮ ❢♦r ❡✈❡r② λ > C ❛♥❞ s > s0= C(Tp+ T2p)✳ ❙✉❝❤ ❛ ❈❛r❧❡♠❛♥ ✐♥❡q✉❛❧✐t② s❡❡♠s ♥❡✇ t♦ t❤❡ ❛✉t❤♦rs ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♥♦♥✲❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ✭✐t ✇❛s ♣r♦✈❡❞ ✐♥ ❬✶✼❪ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts✮✳ ❚❤❡ ♠❛✐♥ ✐♠♣r♦✈❡♠❡♥t ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ♦❜s❡r✈❛t✐♦♥ ✐s ❛ ❣r❛❞✐❡♥t ♦❢ t❤❡ s♦❧✉t✐♦♥ ψ ♦♥ ω0 ✭❛♥❞ ♥♦t t❤❡ s♦❧✉t✐♦♥ ✐ts❡❧❢✮✳ ❲❡ ❛r❡ ❛❧s♦ ❛❜❧❡ t♦ ✐♥tr♦❞✉❝❡ ❛s ♠❛♥② ❞❡r✐✈❛t✐✈❡s ♦❢ ψ ❛s ✇❡ ✇❛♥t ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✱ ❛s s♦♦♥ ❛s ui ✐s r❡❣✉❧❛r ❡♥♦✉❣❤✳ ❘❡♠❛r❦ ✹✳ • ◆♦t❡ t❤❛t t❤❡ ♣r♦♦❢ ♣r♦♣♦s❡❞ ❤❡r❡ r❡❧✐❡s ♦♥ t❤❡ ❢❛❝t t❤❛t t❤❡ ❧♦✇❡r✲♦r❞❡r t❡r♠s ✐♥ ❡q✉❛t✐♦♥ ✭✷✳✹✮ ❛r❡ ♦❢ ♦r❞❡r 1✱ ❛♥❞ ✇♦✉❧❞ ❢❛✐❧ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❧♦✇❡r✲♦r❞❡r t❡r♠s ♦❢ ♦r❞❡r 0✳ ■♥❞❡❡❞✱ ✐♥ t❤❡ ✜rst st❡♣ ♦❢ ♦✉r ♣r♦♦❢ ✭✐♥❡q✉❛❧✐t② ✭✷✳✶✸✮✮✱ s♦♠❡ t❡r♠ t❤❛t ❝❛♥♥♦t ❜❡ ❛❜s♦r❜❡❞ ✇✐❧❧ ❛♣♣❡❛r✳ • ◆♦t❡ t❤❛t ✐♥❡q✉❛❧✐t② ✭✷✳✶✵✮ ❛✉t♦♠❛t✐❝❛❧❧② ✐♠♣❧✐❡s t❤❛t ❛♥② s♦❧✉t✐♦♥ ψ ♦❢ ✭✷✳✹✮ ❧✐✈❡s ✐♥ ❤✐❣❤ ♦r❞❡r ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❚❤✐s ✐s ♥♦t ❛ s✉r♣r✐s❡ s✐♥❝❡ ✇❡ ❦♥♦✇ t❤❛t ❛✇❛② ❢r♦♠ t❤❡ ✜♥❛❧ t✐♠❡ t = T✱ ❛♥② s♦❧✉t✐♦♥ ♦❢ ✭✷✳✹✮ ✐s r❡❣✉❧❛r✳ ✾

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳ ❚❤❡ ♣r♦♦❢ ✐s ✐♥s♣✐r❡❞ ❜② ❬✶✹❪ ❛♥❞ ✐s q✉✐t❡ s✐♠✐❧❛r t♦ ❬✶✼❪✳ ▲❡t µ > 0✳ ■♥ ❛❧❧ ✇❤❛t ❢♦❧❧♦✇s✱ C > 0 ✐s ❛ ❝♦♥st❛♥t t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ s ♦r λ ✭❜✉t t❤❛t ♠✐❣❤t ❞❡♣❡♥❞ ♦♥ t❤❡ ♦t❤❡r ♣❛r❛♠❡t❡rs✱ ♥♦t❛❜❧② p✱ N✱ η✱ T ✱ µ✮ ❛♥❞ t❤❛t ♠✐❣❤t ❝❤❛♥❣❡ ❢r♦♠ ✐♥❡q✉❛❧✐t② t♦ ✐♥❡q✉❛❧✐t②✳ ❲❡ ❛ss✉♠❡ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② t❤❛t N ✐s ♦❞❞ ✭t❤❡ ❝❛s❡ N ❡✈❡♥ ❝❛♥ ❜❡ tr❡❛t❡❞ s✐♠✐❧❛r❧②✮✳ ▲❡t ψ t❤❡ s♦❧✉t✐♦♥ t♦ ❙②st❡♠ ✭✷✳✹✮✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛✉①✐❧✐❛r② ❢✉♥❝t✐♦♥s✿ ρ∗0:= e−µsα ∗ , ψ1:= ρ∗0ψ. ✭✷✳✶✶✮ ❚❤❡♥ ψ1✐s s♦❧✉t✐♦♥ ♦❢    −∂tψ1 = ∆ψ1+ u · ∇ψ1− ∂tρ∗0ψ ✐♥ QT, ψ1 = 0 ♦♥ ΣT, ψ1(T, ·) = 0 ✐♥ Ω. ✭✷✳✶✷✮ ❲❡ r❡♠❛r❦ t❤❛t φ := ∇N_ψ 1 ✭t❤❡ ♦♣❡r❛t♦r ∇ ❛♣♣❧✐❡❞ N t✐♠❡s✱ ♦r ✐♥ ♦t❤❡r ✇♦r❞s✱ ❛❧❧ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ♦r❞❡r N ♦❢ ψ1✱ ♦r❞❡r❡❞ ❢♦r ❡①❛♠♣❧❡ ❧❡①✐❝♦❣r❛♣❤✐❝❛❧❧②✮ s❛t✐s✜❡s t❤❡ s②st❡♠        −∂tφ = ∆φ + N P i=1 Gi· ∇iψ1+ u · ∇φ − ∂tρ∗0∇Nψ ✐♥ QT, ∂φ ∂n = ∂φ ∂n ♦♥ ΣT, φ(T, ·) = 0 ✐♥ Ω, ✇❤❡r❡✱ ❢♦r ❛♥② i ∈ {1, ..., N}✱ Gi ✐s ❛♥ ❡ss❡♥t✐❛❧❧② ❜♦✉♥❞❡❞ t❡♥s♦r ♦❢ ❛♣♣r♦♣r✐❛t❡❞ s✐③❡✱ ✇❤♦s❡ ❝♦❡✣✲ ❝✐❡♥ts ❛r❡ ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ ui ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡s ✐♥ s♣❛❝❡ ✉♣ t♦ t❤❡ ♦r❞❡r i✳ ❆♣♣❧②✐♥❣ ▲❡♠♠❛ ✷✳✷ t♦ t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts ♦❢ φ✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ I(s, λ; φ) 6 C s3λ4 ZZ (0,T )×ω1 e−2sαξ3|φ|2_{dxdt + sλ} ZZ ΣT e−2sα∗ξ∗     ∂φ ∂n     2 dσdt + ZZ QT e−2sα N X i=1 |∇i_ψ 1|2dxdt + ZZ QT e−2sα|∂tρ∗0∇Nψ|2dxdt ! . ✭✷✳✶✸✮ ❚❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ❢♦✉r st❡♣s✿ • ■♥ ❛ ✜rst st❡♣✱ ✇❡ ✇✐❧❧ ❡st✐♠❛t❡ t❤❡ ❜♦✉♥❞❛r② t❡r♠ ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✷✳✶✸✮ ❜② s♦♠❡ ❣❧♦❜❛❧ ✐♥t❡r✐♦r t❡r♠ ✐♥✈♦❧✈✐♥❣ ψ1✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❛❜s♦r❜ ❧❛t❡r ♦♥ ✭✐♥ t❤❡ ❧❛st st❡♣✮✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❛❜s♦r❜ t❤❡ ❧❛st t❡r♠ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✉♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥ ♦♥ p✳ • ■♥ ❛ s❡❝♦♥❞ st❡♣✱ ✇❡ ✇✐❧❧ ❡st✐♠❛t❡ t❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✷✳✶✸✮ ❜② s♦♠❡ ❧♦❝❛❧ t❡r♠s ✐♥✈♦❧✈✐♥❣ ∇ψ1 ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡s ♦♥ ω1✱ ❛♥❞ ❣❡t r✐❞ ♦❢ t❤❡ t❤✐r❞ t❡r♠ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ • ■♥ ❛ t❤✐r❞ st❡♣✱ ✇❡ ✇✐❧❧ ❡st✐♠❛t❡ t❤❡ ❤✐❣❤✲♦r❞❡r ❧♦❝❛❧ t❡r♠s ❝r❡❛t❡❞ ❛t t❤❡ ♣r❡✈✐♦✉s st❡♣ ❜② s♦♠❡ ❧♦❝❛❧ t❡r♠s ✐♥✈♦❧✈✐♥❣ ♦♥❧② ∇ψ1 ♦♥ ω0✳ • ■♥ ❛ ❧❛st st❡♣✱ ✇❡ ✇✐❧❧ ✉s❡ s♦♠❡ P♦✐♥❝❛ré✲❧✐❦❡ ✐♥❡q✉❛❧✐t② ✐♥ ♦r❞❡r t♦ r❡❝♦✈❡r t❤❡ ✈❛r✐❛❜❧❡ ψ ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❛♥❞ ❜♦✉♥❞ t❤❡ ❣❧♦❜❛❧ ✐♥t❡r✐♦r t❡r♠ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐♥✈♦❧✈✐♥❣ ψ1 ❜② ❛♥ ✐♥t❡r✐♦r t❡r♠ ✐♥✈♦❧✈✐♥❣ ∇ψ✳ ❲❡ ✇✐❧❧ ❝♦♥❝❧✉❞❡ ❜② ❝♦♠✐♥❣ ❜❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ ✈❛r✐❛❜❧❡ ψ✱ ✐♥ ♦r❞❡r t♦ ❡st❛❜❧✐s❤ ✭✷✳✶✵✮✳ ✶✵

❙t❡♣ ✶✿ ▲❡t ˜θ ∈ C2_{(Ω)❛ ❢✉♥❝t✐♦♥ s❛t✐s❢②✐♥❣} ∂ ˜θ ∂n = ˜θ = 1 on ∂Ω. ❆♥ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ♦❢ t❤❡ ❜♦✉♥❞❛r② t❡r♠ ❧❡❛❞s t♦ sλ Z T 0 e−2sα∗ξ∗ Z ∂Ω    ∂φ_∂n     2 dσdt = sλ Z T 0 e−2sα∗ξ∗ Z ∂Ω ∂φ ∂n∇φ · ∇˜θdσdt = sλ Z T 0 e−2sα∗ξ∗ Z Ω ∆φ∇φ · ∇˜θdxdt + sλ Z T 0 e−2sα∗ξ∗ Z Ω ∇(∇˜θ · ∇φ) · ∇φdxdt. ❍❡♥❝❡ sλ Z T 0 e−2sα∗ξ∗ Z ∂Ω     ∂φ ∂n     2 dσdt 6 Cλ Z T 0 e−2sα∗sξ∗kψ1kHN+2_(Ω)kψ₁k_HN+1_(Ω)dt. ❯s✐♥❣ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ✐♥❡q✉❛❧✐t② kψ1kHN+2_(Ω)6Ckψ₁k1/2 HN+1_(Ω)kψ1k 1/2 HN+3_(Ω) ❛♥❞ ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ab 6 aq q + bq′ q′ ✭ 1 q + 1 q′ = 1✮ ❢♦r a, b > 0 ❛♥❞ q = 4✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ❢♦r ❛♥② c ∈ R✱ ✇❡ ❤❛✈❡ λ Z T 0 e−2sα∗sξ∗ Z ∂Ω    ∂φ_∂n     2 dσdt 6 Cλ Z T 0 e−2sα∗(sξ∗)ckψ1k1/2_HN+3_(Ω)(sξ ∗₎(1−c)_kψ 1k3/2_HN+1_(Ω)dt 6 Cλ Z T 0 e−2sα∗(sξ∗)4ckψ1k2HN+3_(Ω)dt +Cλ Z T 0 e−2sα∗(sξ∗)4(1−c)3 kψ 1k2HN+1_(Ω)dt. ✭✷✳✶✹✮ ❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ ψ2:= ρ∗1ψ1✱ ✇❤❡r❡ ρ∗1:= (sξ∗) 2(1−c) 3 e−sα ∗ . ✭✷✳✶✺✮ ❚❤❡ ❢✉♥❝t✐♦♥ ψ2 ✐s s♦❧✉t✐♦♥ t♦ t❤❡ s②st❡♠    −∂tψ2 = ∆ψ2+ u · ∇ψ2− ∂t(ρ∗1)ψ1− ρ∗1∂t(ρ∗0)ψ ✐♥ QT, ψ2 = 0 ♦♥ ΣT, ψ2(T, ·) = 0 ✐♥ Ω. ❯s✐♥❣ ▲❡♠♠❛ ✷✳✹ ❢♦r ψ2✭r❡♠❛r❦ t❤❛t t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ✭✷✳✾✮ ❛r❡ ✈❡r✐✜❡❞✱ s✐♥❝❡ ψ2(T, ·) = 0 ❛♥❞ u ❤❛s s♣❛t✐❛❧ s✉♣♣♦rt str♦♥❣❧② ✐♥❝❧✉❞❡❞ ✐♥ Ω✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t kψ2kL2_{((0,T ),H}2n+2_(Ω))∩Hn+1_{((0,T ),L}2_(Ω))6Ck∂_t(ρ∗₁)ψ₁+ ρ∗₁∂_t(ρ∗₀)ψk_L2_{((0,T ),H}2n_(Ω))∩Hn_{((0,T ),L}2_(Ω)), ✭✷✳✶✻✮ ❢♦r n = 1, 2, . . . , (N + 1)/2✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ξ∗ _{❛♥❞ α}∗ _{❣✐✈❡♥ ✐♥ ✭✷✳✼✮✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ρ}∗ 0 ❣✐✈❡♥ ✐♥ ✶✶

✭✷✳✶✶✮✱ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ρ∗ 1 ❣✐✈❡♥ ✐♥ ✭✷✳✶✺✮ ❧❡❛❞ t♦                        |∂tρ∗0| 6C(sξ∗)1+ 1 pe−µsα ∗ , ✳✳✳ |∂ N+3 2 t ρ∗0| 6 C(sξ∗) N+3 2 + N+3 2p e−µsα ∗ , |∂tρ∗1| 6C(sξ∗) 2(1−c) 3 +1+ 1 pe−sα∗, ✳✳✳ |∂ N+3 2 t ρ∗1| 6 C(sξ∗) 2(1−c) 3 + N+3 2 + N+3 2p e−sα ∗ . ✭✷✳✶✼✮ ❘❡♠❛r❦ t❤❛t ❢♦r ❛♥② k 6 l✱ ✇❡ ❤❛✈❡ |∂k tρ∗0| 6 C|∂tlρ∗0|. ✭✷✳✶✽✮ ❈♦♠❜✐♥✐♥❣ ✭✷✳✶✻✮ ❢♦r n = (N − 1)/2✱ ✭✷✳✶✼✮✱ ✭✷✳✶✽✮ ❛♥❞ t❤❡ ❡q✉❛t✐♦♥s s❛t✐s✜❡❞ ❜② ψ ❛♥❞ ψ1✱ ✇❡ ♦❜t❛✐♥ λ Z T 0 e−2sα∗(sξ∗)4(1−c)3 kψ 1k2HN+1_(Ω)dt 6 Cλ Z T 0 e−2sα∗(sξ∗)4(1−c)3 +N +1+ N+1 p ||ψ 1||2L2_(Ω)dt + Z T 0 e−2(1+µ)sα∗(sξ∗)4(1−c)3 +N +1+ N+1 p ||ψ||2 L2_(Ω)dt + Z T 0 e−2sα∗ (sξ∗₎4(1−c)3 +2+ 2 p||ψ 1||2HN −1_(Ω)dt + Z T 0 e−2(1+µ)sα∗ (sξ∗₎4(1−c)3 +2+ 2 p||ψ||2 HN −1_(Ω)dt ! . ✭✷✳✶✾✮ ■♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✷✳✶✾✮✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡st✐♠❛t❡ t❤❡ t❡r♠s Z T 0 e−2sα∗(sξ∗)4(1−c)3 +2+ 2 p||ψ 1||2HN −1_(Ω)dt❛♥❞ Z T 0 e−2(1+µ)sα∗(sξ∗)4(1−c)3 +2+ 2 p||ψ||2 HN −1_(Ω)dt. ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ ❡①❛❝t❧② t❤❡ s❛♠❡ ♣r♦❝❡ss✉s ❜② ✐♥tr♦❞✉❝✐♥❣ s♦♠❡ ❛♣♣r♦♣r✐❛t❡ ❛✉①✐❧✐❛r② ✇❡✐❣❤t t❤❛t ♠✉❧t✐♣❧✐❡s ψ ♦r ψ1 ❛s ✐♥ ✭✷✳✶✺✮✱ ✉s✐♥❣ ▲❡♠♠❛ ✷✳✹ s✉❝❝❡ss✐✈❡❧② ❢♦r n = (N − 1)/2, . . . , 0✱ ✭✷✳✶✼✮ ❛♥❞ ✭✷✳✶✽✮✳ ❆t t❤❡ ❡♥❞✱ ❜② ❣❛t❤❡r✐♥❣ ❛❧❧ t❤❡ ✐♥❡q✉❛❧✐t✐❡s✱ ✇❡ ♦❜t❛✐♥ λ Z T 0 e−2sα∗(sξ∗)4(1−c)3 kψ 1k2HN+1_(Ω)dt 6Cλ Z T 0 (sξ∗)4(1−c)3 +N +1+ N+1 p ||ψ 1||2L2_(Ω)dt + Z T 0 e−2(1+µ)sα∗(sξ∗)4(1−c)3 +N +1+ N+1 p ||ψ||2 L2_(Ω)dt ! . ✭✷✳✷✵✮ ❆♣♣❧②✐♥❣ t❤❡ s❛♠❡ t❡❝❤♥✐q✉❡ ❛❧s♦ ❧❡❛❞s t♦ λ Z T 0 e−2sα∗(sξ∗)4ckψ1k2HN+3_(Ω)dt 6Cλ Z T 0 (sξ∗)4c+N +3+N+3p ||ψ 1||2L2_(Ω)dt + Z T 0 e−2(1+µ)sα∗(sξ∗)4c+N +3+N+3p ||ψ||2 L2_(Ω)dt ! . ✭✷✳✷✶✮ ✶✷

❋r♦♠ ✭✷✳✶✹✮✱ ✭✷✳✷✵✮ ❛♥❞ ✭✷✳✷✶✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t λ Z T 0 e−2sα∗ sξ∗ Z ∂Ω     ∂φ ∂n     2 dσdt 6Cλ Z T 0 (sξ∗)4(1−c)3 +N +1+ N+1 p ||ψ₁||2 L2_(Ω)dt + Z T 0 e−2(1+µ)sα∗(sξ∗)4(1−c)3 +N +1+ N+1 p ||ψ||2 L2_(Ω)dt + Z T 0 (sξ∗)4c+N +3+N+3p ||ψ 1||2L2_(Ω)dt + Z T 0 e−2(1+µ)sα∗(sξ∗)4c+N +3+N+3p ||ψ||2 L2_(Ω)dt ! . ✭✷✳✷✷✮ ❙✐♥❝❡ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t❤❡ ♣♦✇❡rs ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ t♦ ❜❡ ❡q✉❛❧✱ ✐t ✐s ♥❛t✉r❛❧ t♦ ✐♠♣♦s❡ t❤❛t 4c + N + 3 +N + 3 p = 4(1 − c) 3 + N + 1 + N + 1 p , ✐✳❡✳ c = −3 − p 8p . ✭✷✳✷✸✮ ❚❤✉s✱ ✉s✐♥❣ ✭✷✳✷✷✮ ❛♥❞ ✭✷✳✷✸✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t λ Z T 0 e−2sα∗sξ∗ Z ∂Ω    ∂φ_∂n     2 dσdt 6Cλ Z T 0 e−2sα∗(sξ∗)2N (p+1)+5p+32p ||ψ 1||2L2_(Ω)dt + Z T 0 e−2(1+µ)sα∗(sξ∗)2N (p+1)+5p+32p ||ψ||2 L2_(Ω)dt ! . ✭✷✳✷✹✮ ❋r♦♠ ✭✷✳✶✸✮✱ ✭✷✳✷✹✮✱ t❤❡ ✜rst ❧✐♥❡ ♦❢ ✭✷✳✶✼✮ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ψ1❣✐✈❡♥ ✐♥ ✭✷✳✶✶✮✱ ✇❡ ❛❧r❡❛❞② ❞❡❞✉❝❡ t❤❛t I(s, λ; φ) 6 C s3_λ4 ZZ (0,T )×ω1 e−2sαξ3|∇N_ψ 1|2dxdt + λ ZZ QT e−2sα∗(sξ∗)2N (p+1)+5p+32p |ψ 1|2dxdt + ZZ QT e−2sα N X i=1 |∇i_ψ 1|2dxdt + ZZ QT e−2sα(sξ∗)2+2p|∇N_ψ 1|2dxdt ! . ❇② ❞❡✜♥✐t✐♦♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ξ∗ _{❣✐✈❡♥ ✐♥ ✭✷✳✼✮✱ ✐t ✐s ❝❧❡❛r t❤❛t ξ}∗_{6ξ✳ ❍❡♥❝❡✱ t❛❦✐♥❣ p ❧❛r❣❡ ❡♥♦✉❣❤} s✉❝❤ t❤❛t 2 +2 p 63✭✐✳❡✳ p > 2✮✱ s, λ ❧❛r❣❡ ❡♥♦✉❣❤ ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ I(s, λ; φ) ❣✐✈❡♥ ✐♥ ✭✷✳✽✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ✇❡ ❝❛♥ ❛❜s♦r❜ t❤❡ ❧❛st t❡r♠ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞✲s✐❞❡✱ s♦ t❤❛t ✇❡ ♦❜t❛✐♥ I(s, λ; φ) 6 C s3_λ4 ZZ (0,T )×ω1 e−2sαξ3|∇Nψ1|2dxdt + λ ZZ QT e−2sα∗(sξ∗)2N (p+1)+5p+32p |ψ 1|2dxdt + ZZ QT e−2sα N X i=1 |∇iψ1|2dxdt ! . ✭✷✳✷✺✮ ❙t❡♣ ✷✿ ❲❡ ❛♣♣❧② ▲❡♠♠❛ ✷✳✸ s✉❝❝❡ss✐✈❡❧② ✇✐t❤ (u, r) = (∇N −1ψ1, 3), . . . , (u, r) = (∇ψ1, 2N − 1). ✶✸

❲❡ ♦❜t❛✐♥ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥❡q✉❛❧✐t✐❡s ♦❢ t❤❡ ❢♦r♠ s5_λ6 ZZ QT e−2sαξ5|∇N −1_ψ 1|2dxdt 6 C  s3λ4 ZZ QT e−2sαξ3|∇N_ψ 1|2dxdt +s5_λ6 ZZ (0,T )×ω1 e−2sα_ξ5_|∇N −1_ψ 1|2dxdt ! , . . . s2N +1_λ2N +2 ZZ QT e−2sα_ξ2N +1_|∇ψ 1|2dxdt 6 C  s2N −1_λ2N ZZ QT e−2sα_ξ2N −1_|∇2_ψ 1|2dxdt +s2N +1_λ2N +2 ZZ (0,T )×ω1 e−2sαξ2N +1|∇ψ1|2dxdt ! . ❲❡ ❞❡❞✉❝❡ ❜② st❛rt✐♥❣ ❢r♦♠ t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ❛♥❞ ✉s✐♥❣ ✐♥ ❝❛s❝❛❞❡ t❤❡ ♦t❤❡r ♦♥❡s t❤❛t s5_λ6 ZZ QT e−2sα_ξ5_|∇N −1_ψ 1|2dxdt + . . . + s2N +1λ2N +2 ZZ QT e−2sα_ξ2N +1_|∇ψ 1|2dxdt 6C s3_λ4 ZZ QT e−2sαξ3|∇Nψ1|2dxdt + s5λ6 ZZ (0,T )×ω1 e−2sαξ5|∇N −1ψ1|2dxdt + . . . + s2N +1_λ2N +2 ZZ (0,T )×ω1 e−2sαξ2N +1|∇ψ1|2dxdt ! . ✭✷✳✷✻✮ ❈♦♠❜✐♥✐♥❣ ✭✷✳✷✺✮✱ ✭✷✳✷✻✮ ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ I(s, λ, φ) ❣✐✈❡♥ ✐♥ ✭✷✳✽✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ✇❡ ❝❛♥ ❛❜s♦r❜ t❤❡ ✜rst t❡r♠ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✷✳✷✻✮ ❛♥❞ ♦❜t❛✐♥ sλ2 ZZ QT e−2sαξ|∇N +1ψ1|2dxdt + . . . + s2N +1λ2N +2 ZZ QT e−2sαξ2N +1|∇ψ1|2dxdt 6C λ ZZ QT e−2sα∗(sξ∗)2N (p+1)+5p+32p |ψ 1|2dxdt + ZZ QT e−2sα N −1_X i=1 |∇i_ψ 1|2dxdt +s3_λ4 ZZ (0,T )×ω1 e−2sαξ3|∇Nψ1|2dxdt + s5λ6 ZZ (0,T )×ω1 e−2sαξ5|∇Nψ1|2dxdt + . . . +s2N +1_λ2N +2 ZZ (0,T )×ω1 e−2sα_ξ2N +1_|∇ψ 1|2dxdt ! . ❆❜s♦r❜✐♥❣ t❤❡ s❡❝♦♥❞ t❡r♠ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ❢♦r s, λ ❧❛r❣❡ ❡♥♦✉❣❤✱ ✇❡ ❤❛✈❡ sλ2 ZZ QT e−2sαξ|∇N +1ψ1|2dxdt + . . . + s2N +1λ2N +2 ZZ QT e−2sαξ2N +1|∇ψ1|2dxdt 6C  λ ZZ QT e−2sα∗(sξ∗)2N (p+1)+5p+32p _|ψ 1|2dxdt +s3_λ4 ZZ (0,T )×ω1 e−2sαξ3|∇N_ψ 1|2dxdt + . . . + s2N +1λ2N ZZ (0,T )×ω1 e−2sαξ2N +1|∇ψ1|2dxdt ! . ✭✷✳✷✼✮ ❙t❡♣ ✸✿ ◆♦✇✱ ✇❡ ❝♦♥s✐❞❡r s♦♠❡ ♦♣❡♥ s✉❜s❡t ω2 s✉❝❤ t❤❛t ω1⊂⊂ ω2⊂⊂ ω0✳ ❲❡ ❝♦♥s✐❞❡r s♦♠❡ ❢✉♥❝t✐♦♥ ˜θ ∈ C∞_{(Ω, R)s✉❝❤ t❤❛t✿} • ❙✉♣♣(˜θ) ⊂ ω2✱ • ˜θ = 1♦♥ ω1✱ • ˜θ ∈ [0, 1]✳ ✶✹

❙♦♠❡ ✐♥t❡❣r❛t✐♦♥s ❜② ♣❛rts ❣✐✈❡ s3_λ4 ZZ (0,T )×ω1 e−2sαξ3|∇Nψ1|2dxdt 6 s3λ4 ZZ (0,T )×ω2 θe−2sαξ3|∇Nψ1|2dxdt 6Cs3_λ4 ZZ (0,T )×ω2 |∇(θe−2sα_ξ3_)|.|∇N_ψ 1|.|∇N −1ψ1| + |θe−2sαξ3|.|∇N +1ψ1|.|∇N −1ψ1|
dxdt. ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ξ ❛♥❞ α ❣✐✈❡♥ ✐♥ ✭✷✳✺✮ ❛♥❞ ✭✷✳✻✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t |∇(θe−2sα_ξ3_{)| 6 Csλe}−2sα_ξ4_{. ✭✷✳✷✽✮} ❈♦♠❜✐♥✐♥❣ t❤✐s ❡st✐♠❛t❡ ✇✐t❤ ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t②✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❢♦r ❛♥② ε > 0✱ t❤❡r❡ ❡①✐sts Cε> 0 s✉❝❤ t❤❛t ❢♦r ❛♥② s ❛♥❞ λ ❧❛r❣❡ ❡♥♦✉❣❤✱ ✇❡ ❤❛✈❡ s3_λ4 ZZ (0,T )×ω1 e−2sαξ3|∇N_ψ 1|2dxdt 6 C εs3λ4 ZZ (0,T )×ω2 e−2sαξ3|∇N_ψ 1|2dxdt +εsλ2 ZZ (0,T )×ω2 e−2sαξ|∇N +1ψ1|2dxdt + Cεs5λ6 ZZ (0,T )×ω2 e−2sαξ5|∇N −1_ψ 1|2dxdt ! . ✭✷✳✷✾✮ ❈♦♠❜✐♥✐♥❣ ✭✷✳✷✼✮ ❛♥❞ ✭✷✳✷✾✮✱ ✇❡ ❝❛♥ ❛❜s♦r❜ t❤❡ ❧♦❝❛❧ t❡r♠s ✐♥ |∇N +1_ψ 1|2 ❛♥❞ |∇Nψ1|2 t♦ ❞❡❞✉❝❡ sλ2 ZZ QT e−2sαξ|∇N +1ψ1|2dxdt + . . . + s2N +1λ2N +2 ZZ QT e−2sαξ2N +1|∇ψ1|2dxdt 6C λ ZZ QT e−2sα∗(sξ∗)2N (p+1)+5p+32p _|ψ 1|2dxdt + s5λ6 ZZ (0,T )×ω2 e−2sαξ5|∇N −1ψ1|2dxdt + . . . + s2N +1_λ2N +2 ZZ (0,T )×ω2 e−2sαξ2N +1|∇ψ1|2dxdt ! . ❲❡ ❝❛♥ ♣❡r❢♦r♠ ❡①❛❝t❧② t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡ ♦♥ t❤❡ t❡r♠s s5λ6 ZZ (0,T )×ω2 e−2sαξ5|∇N −1ψ1|2dxdt, . . . , s2N −1λ2N −2 ZZ (0,T )×ω2 e−2sαξ2N −1|∇2ψ1|2dxdt ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡✿ sλ2 ZZ QT e−2sαξ|∇N +1ψ1|2dxdt + . . . + s2N +1λ2N +2 ZZ QT e−2sαξ2N +1|∇ψ1|2dxdt 6C  λ ZZ QT e−2sα∗(sξ∗)2N (r+1)+5r+32r |ψ₁|2dxdt +s2N +1_λ2N +2 ZZ (0,T )×ω0 e−2sα_ξ2N +1_|∇ψ 1|2dxdt ! . ✭✷✳✸✵✮ ❙t❡♣ ✹✿ ❙✐♥❝❡ t❤❡ ✇❡✐❣❤t (sξ∗₎2N −1 _{❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ s♣❛❝❡ ✈❛r✐❛❜❧❡✱ ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ α}∗ ❛♥❞ ξ∗ _{❣✐✈❡♥ ✐♥ ✭✷✳✼✮✱ t❤❡ ❢♦❧❧♦✇✐♥❣ P♦✐♥❝❛ré✬s ✐♥❡q✉❛❧✐t② ❤♦❧❞s✿} λ2N +2 ZZ QT e−2sα∗(sξ∗)2N +1|ψ1|2dxdt 6 Cλ2N +2 ZZ QT e−2sα∗(sξ∗)2N +1|∇ψ1|2dxdt 6Cλ2N +2 ZZ QT e−2sα(sξ)2N +1|∇ψ1|2dxdt. ✭✷✳✸✶✮ ✶✺

❈♦♠❜✐♥✐♥❣ ✭✷✳✸✵✮ ❛♥❞ ✭✷✳✸✶✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ❢♦r s ❧❛r❣❡ ❡♥♦✉❣❤ λ2 ZZ QT e−2sαsξ|∇N +1ψ1|2dxdt + . . . + λ2N +2 ZZ QT e−2sα(sξ)2N +1|∇ψ1|2dxdt +λ2N +2 ZZ QT e−2sα∗(sξ∗)2N +1|ψ1|2dxdt 6C λ ZZ QT e−2sα∗(sξ∗)2N (p+1)+5p+32p _|ψ 1|2dxdt + λ2N +2 ZZ (0,T )×ω0 e−2sα(sξ)2N +1|∇ψ1|2dxdt ! . ✭✷✳✸✷✮ ❲❡ ♥♦✇ ✜① p > 2 ❧❛r❣❡ ❡♥♦✉❣❤ s✉❝❤ t❤❛t 2N (p + 1) + 5p + 3 2p < 2N + 1, ✇❤✐❝❤ ✐s ❝❧❡❛r❧② ♣♦ss✐❜❧❡ s✐♥❝❡2N (p+1)+5p+3 2p → N + 5 2❛s p → ∞ ❛♥❞ N > 3 ✭s♦ t❤❛t N +5/2 < 2N +1✮✳ ❯s✐♥❣ t❤❛t e−2sα∗ (sξ∗)2N (p+1)+5p+32p _6Ce−2sα_(sξ)2N +1✱ ✇❡ ❞❡❞✉❝❡ ❜② ❛❜s♦r❜✐♥❣ t❤❡ ✜rst t❡r♠ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✷✳✸✷✮ t❤❛t λ2 ZZ QT e−2sα(sξ)|∇N +1ψ1|2dxdt + . . . + λ2N +2 ZZ QT e−2sα(sξ)2N +1|∇ψ1|2dxdt +λ2N +2 ZZ QT e−2sα∗(sξ∗)2N +1|ψ1|2dxdt 6 Cλ2N +2 ZZ (0,T )×ω0 e−2sα(sξ)2N +1|∇ψ1|2dxdt. ●♦✐♥❣ ❜❛❝❦ t♦ ψ t❤❛♥❦s t♦ ✭✷✳✶✷✮✱ ✇❡ ❞❡❞✉❝❡ ✭✷✳✶✵✮✳

✷✳✷ ❆❧❣❡❜r❛✐❝ r❡s♦❧✉❜✐❧✐t②

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❞❡r✐✈❡ ❛ ♥❡✇ ❈❛r❧❡♠❛♥ ✐♥❡q✉❛❧✐t②✱ ❛❞❛♣t❡❞ t♦ t❤❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ✇✐t❤ ❧❡ss ❝♦♥tr♦❧s ✇❡ ✇❛♥t t♦ ♣r♦✈❡✳ ❲❡ ❛ss✉♠❡ ❤❡r❡ t❤❛t q ∈ N∗ _{✭✐❢ q = 0✱ ♥❡❝❡ss❛r✐❧②✱ ❜② ❝♦♥❞✐t✐♦♥ ✭✶✳✼✮✱ ✇❡} ❤❛✈❡ m = d ❛♥❞ ✇❡ ❝❛♥ t❛❦❡ M = (B∗₎−1 _{❛♥❞ M} 2= 0✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ▲❡♠♠❛✮✳ ▲❡♠♠❛ ✷✳✺✳ ▲❡t m ∈ N∗ _{s✉❝❤ t❤❛t m 6 d − 1✳ ❆ss✉♠❡ t❤❛t t❤❡ u ✐s r❡❣✉❧❛r ❡♥♦✉❣❤ ✭❢♦r ❡①❛♠♣❧❡ ♦❢} ❝❧❛ss C∞_✮✳ ❈♦♥s✐❞❡r t✇♦ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs L1 : C∞(Rd) → C∞(Rd)m ❛♥❞ L2 : C∞(Rd) → C∞_(Rd_{)❞❡✜♥❡❞ ❢♦r ❡✈❡r② ϕ ∈ C}∞_(Rd_)❜② L1ϕ := B∗(∇ϕ)❛♥❞ L2ϕ := ∂tϕ + ∆ϕ + (u · ∇)ϕ. ❆ss✉♠❡ t❤❛t ✭✶✳✼✮ ❤♦❧❞s✳ ❚❤❡r❡ ❡①✐sts ❛♥ ♦♣❡♥ s✉❜s❡t (t1, t2)×eω♦❢ (0, T )×ω ❛♥❞ t❤❡r❡ ❡①✐st t✇♦ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs M1: C∞(Rd)m→ C∞(Rd)d ✭♦❢ ♦r❞❡r 1 ✐♥ t✐♠❡ ❛♥❞ q + 1 ✐♥ s♣❛❝❡✮ ❛♥❞ M2 : C∞(R) → C∞(Rd)d ✭♦❢ ♦r❞❡r 0 ✐♥ t✐♠❡ ❛♥❞ q ✐♥ s♣❛❝❡✮ s✉❝❤ t❤❛t M1◦ L1+ M2◦ L2= ∇ ✐♥ C∞((t1, t2) × eω). ✭✷✳✸✸✮ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✷✳✺✿ ✶✻

❙✐♥❝❡ ✭✶✳✼✮ ✐s ✈❡r✐✜❡❞✱ t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡♥ s✉❜s❡t (t1, t2) × eω♦❢ (0, T ) × ω ❛♥❞ ❛ ❝♦♥st❛♥t C > 0 s✉❝❤ t❤❛t | det(M)| > C ♦♥ (t1, t2) × eω✳ ▲❡t j ∈ {1, ..., m}✳ ❲❡ ❝❛❧❧ Lj1 t❤❡ j − th ❧✐♥❡ ♦❢ L1✳ ❲❡ r❡♠❛r❦ t❤❛t (B∗ j · ∇)L2ϕ − (∂t+ ∆)Lj1ϕ − (u · ∇)L j 1ϕ = (B∗j· ∇)(u · ∇)ϕ − (u · ∇)(Bj∗.∇)ϕ = (u · ∇)(Bj∗.∇)ϕ + d X k=1 ((B∗j · ∇)uk)∂kϕ − (u · ∇)(Bj∗.∇)ϕ = d X k=1 ((B∗ j · ∇)uk)∂kϕ =: L3. ◆♦✇✱ ❢♦r s♦♠❡ l ∈ {1, ..., m}✱ t❤❡ s❛♠❡ ❝♦♠♣✉t❛t✐♦♥s ❡❛s✐❧② ❣✐✈❡ (Bl∗· ∇)L3ϕ − d X k=1 ((B∗j · ∇)uk)∂kLl1ϕ = d X k=1 ((B∗l · ∇)(B∗j· ∇)uk)∂kϕ =: L4ϕ. ❈♦♥t✐♥✉✐♥❣ t❤✐s ♣r♦❝❡❞✉r❡✱ ✇❡ ❝❛♥ ❡❛s✐❧② ❝r❡❛t❡ t✇♦ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs fM1 ✭♦❢ ♦r❞❡r 1 ✐♥ t✐♠❡ ❛♥❞ q + 1 ✐♥ s♣❛❝❡✮ ❛♥❞ fM2 ✭♦❢ ♦r❞❡r 0 ✐♥ t✐♠❡ ❛♥❞ q ✐♥ s♣❛❝❡✮ s✉❝❤ t❤❛t g M1(L1(ϕ)) + gM2(L2(ϕ)) = Mq(∇ϕ), ✇❤❡r❡ Mq ✐s ❞❡✜♥❡❞ ✐♥ ✭✶✳✻✮✳ ❯♥❞❡r ❝♦♥❞✐t✐♦♥ ✭✶✳✼✮✱ Mq ✐s ♦❢ ♠❛①✐♠❛❧ r❛♥❦ ♦♥ (t1, t2) × eω✱ s♦ t❤❛t ✐t ❛❞♠✐ts ❛ ❧❡❢t ✐♥✈❡rs❡ ❛t ❛♥② ♣♦✐♥t ♦❢ ♦♥ (t1, t2) × eω✳ ❲❡ ❝❛❧❧ Mq(u)−1 ❛♥② ♦❢ ✐ts ❧❡❢t ✐♥✈❡rs❡s✳ ❚❤❡♥✱ ✐s ✐s ❝❧❡❛r t❤❛t M1:= Mq−1Mf1❛♥❞ M2:= Mq−1Mf2 ✈❡r✐❢② ✭✷✳✸✸✮ ❛♥❞ ❤❛✈❡ C∞❝♦❡✣❝✐❡♥ts ♦♥ (t1, t2) × eω✳ ❲❡ ♥♦✇ ❤❛✈❡ ❛❧❧ t❤❡ t♦♦❧s t♦ ❞❡❞✉❝❡ ♦✉r ✜♥❛❧ ❈❛r❧❡♠❛♥ ✐♥❡q✉❛❧✐t②✿ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳ ❆ss✉♠❡ t❤❛t ❈♦♥❞✐t✐♦♥ ✭✶✳✼✮ ❛♥❞ t❤❡ ❤②♣♦t❤❡s❡s ♦❢ Pr♦♣♦s✐t✐♦♥ ✷✳✶ ❤♦❧❞✳ ❚❤❡♥✱ ❢♦r ❛❧❧ η ∈ (0, 1)✱ t❤❡r❡ ❡①✐sts p > 2✱ C > 0 ❛♥❞ K > 0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ψ0 _{∈ L}2_{(Ω)✱ t❤❡} ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥ ψ t♦ ❙②st❡♠ ✭✷✳✹✮ s❛t✐s✜❡s Z Ω ψ(0)2dx + ZZ QT eη(T −t)p−2K {ψ2_{+ |∂} tψ|2+ . . . + |∂ ⌊N+1 2 ⌋ t...t ψ|2+ |∇ψ|2+ . . . + |∇N +1ψ|2}dxdt 6CeK/TpZZ (0,T )×ω0 e(T −t)p−2K |B∗_(∇ψ)|2_dxdt. ✭✷✳✸✹✮ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳ ❲❡ ❛ss✉♠❡ t❤❛t q ∈ N∗_{✳ ▲❡t ω} 1 s♦♠❡ ♦♣❡♥ s✉❜s❡t str♦♥❣❧② ✐♥❝❧✉❞❡❞ ✐♥ ω0✳ ❈♦♠❜✐♥✐♥❣ Pr♦♣♦s✐t✐♦♥ ✷✳✶✱ ▲❡♠♠❛ ✷✳✺ ✭t❤❛t ✐s st✐❧❧ tr✉❡ ❜② r❡♣❧❛❝✐♥❣ ω0 ❜② ω1✮✱ ❛♥❞ t❤❡ ❢❛❝t t❤❛t ❛♥② s♦❧✉t✐♦♥ ψ ♦❢ ✭✷✳✹✮ ✈❡r✐✜❡s ❜② ❞❡✜♥✐t✐♦♥ L2ψ = 0✱ ✇❡ ❞❡❞✉❝❡ t❤❛t✱ ❢♦r ❛♥② ψ0∈ L2(Ω)✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥ ψ t♦ ❙②st❡♠ ✭✷✳✹✮ s❛t✐s✜❡s λ2 ZZ QT e−2sα−2µsα∗(sξ)|∇N +1ψ|2dxdt + . . . + λ2N +2 ZZ QT e−2sα−2µsα∗(sξ)2N +1|∇ψ|2_dxdt +λ2N +2 ZZ QT e−2sα∗−2µsα∗(sξ∗)2N +1|ψ|2_dxdt 6Cλ2N +2 ZZ QT ˜ θe−2sα−2µsα∗(sξ)2N +1|M1B∗(∇ψ)|2dxdt, ✶✼

✇❤❡r❡ M1 ✐s ❛ ❧✐♥❡❛r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ♦❢ ♦r❞❡r 1 ✐♥ t✐♠❡ ❛♥❞ q + 1 ✐♥ s♣❛❝❡✱ ❛♥❞ ˜θ ∈ C∞_{(Ω, R)s✉❝❤ t❤❛t✿} • ˜θ = 1♦♥ ω1✱ • ❙✉♣♣(˜θ) ⊂ ω0✱ • ˜θ ∈ [0, 1]✳ ❲❡ ✜rst r❡♠❛r❦ t❤❛t λ2N +2 ZZ QT θe−2sα−2µsα∗(sξ)2N +1|M1B∗(∇ψ)|2dxdt 6Cλ2N +2 ZZ QT θe−2sα−2µsα∗(sξ)2N +1 q+1 X i=0 |∇iB∗∇ψ|2+ |∂t∇iB∗∇ψ|2
! dxdt. ❯s✐♥❣ t❤❛t ψ ✈❡r✐✜❡s ✭✷✳✹✮✱ ✇❡ ❝❛♥ ❞❡❞✉❝❡ t❤❛t λ2N +2 ZZ QT θe−2sα−2µsα∗ (sξ)2N +1|M1B∗(∇ψ)|2dxdt 6Cλ2N +2 ZZ QT θe−2sα−2µsα∗(sξ)2N +1 q+3 X i=0 |∇i_B∗_∇ψ|2 ! dxdt. ❙♦♠❡ ✐♥t❡❣r❛t✐♦♥s ❜② ♣❛rts ❣✐✈❡ λ2N +2 ZZ QT ˜ θe−2sα−2µsα∗(sξ)2N +1|∇B∗(∇ψ)|2dxdt 6Cλ2N +2 ZZ QT ˜ θe−2sα−2µsα∗(sξ)2N +1|B∗(∇ψ)||∇3ψ|dxdt +Cλ2N +2 ZZ QT |∇(˜θe−2sα−2µsα∗(sξ)2N +1)||B∗(∇ψ)||∇2ψ|dxdt. ▲❡t ε > 0✳ ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ❣✐✈❡s λ2N +2 ZZ QT ˜ θe−2sα−2µsα∗(sξ)2N +1|B∗(∇ψ)||∇3ψ|dxdt 6Cελ2N +6 ZZ (0,T )×ω0 e−2sα−2µsα∗(sξ)2N +5|B∗(∇ψ)|2dxdt +ελ2N −2 ZZ (0,T )×ω0 e−2sα−2µsα∗(sξ)2N −3|∇3_ψ|2_dxdt ❛♥❞ ❛❧s♦✱ ❜② ✭✷✳✷✽✮✱ λ2N +2 ZZ QT |∇(˜θe−2sα−2µsα∗(sξ)2N +1)||B∗(∇ψ)||∇2ψ|dxdt 6Cλ2N +3 ZZ QT ˜ θe−2sα−2µsα∗(sξ)2N +2|B∗(∇ψ)||∇2ψ|dxdt 6Cελ2N +6 ZZ (0,T )×ω0 e−2sα−2µsα∗(sξ)2N +5|B∗(∇ψ)|dxdt +ελ2N ZZ (0,T )×ω0 e−2sα−2µsα∗(sξ)2N −1|∇3_ψ|dxdt. ✶✽

❚❤✉s✱ ❜② t❛❦✐♥❣ ε s♠❛❧❧ ❡♥♦✉❣❤✱ ✇❡ ❞❡❞✉❝❡ t❤❛t λ2 ZZ QT e−2sα−2µsα∗(sξ)|∇N +1ψ|2dxdt + . . . + λ2N +2 ZZ QT e−2sα−2µsα∗(sξ)2N +1|∇ψ|2_dxdt +λ2N +2 ZZ QT e−2sα∗−2µsα∗(sξ∗)2N +1|ψ|2_dxdt 6Cλ2N +6 ZZ (0,T )×ω0 e−2sα−2µsα∗(sξ)2N +5|B∗_(∇ψ)|2_dxdt +Cλ2N +2 ZZ QT ˜ θe−2sα−2µsα∗ (sξ)2N +1 q+3 X i=2 |∇i_B∗_∇ψ|2 ! dxdt. ❇② ✐t❡r❛t✐♥❣ t❤✐s ♣r♦❝❡ss ❢♦r i = 2, . . . , q + 3✱ ✇❡ ❝❛♥ ❣❡t r✐❞ ♦❢ t❤❡ s✉♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❛♥❞ ♦❜t❛✐♥ λ2 ZZ QT e−2sα−2µsα∗(sξ)|∇N +1ψ|2dxdt + . . . + λ2N +2 ZZ QT e−2sα−2µsα∗(sξ)2N +1|∇ψ|2_dxdt +λ2N +2 ZZ QT e−2sα∗−2µsα∗(sξ∗)2N +1|ψ|2_dxdt 6C λ2N +2+4(q+2) ZZ (0,T )×ω0 e−2sα−2µsα∗ (sξ)2N +1+4(q+2)}|B∗_(∇ψ)|2_dxdt ! . ■♥❡q✉❛❧✐t② ✭✷✳✸✹✮ ✐s ❡❛s✐❧② ❞❡❞✉❝❡❞ ❜② r❡♣❧❛❝✐♥❣ t❤❡ s♣❛❝❡✲❞❡♣❡♥❞❡♥t ✇❡✐❣❤ts ❜② t❤❡✐r ✐♥✜♠✉♠ ✐♥ s♣❛❝❡ ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞✲s✐❞❡ ❛♥❞ t❤❡✐r s✉♣r❡♠✉♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✜①✐♥❣ s ❛♥❞ λ ❧❛r❣❡ ❡♥♦✉❣❤✱ t❤❡♥ ❝❤♦♦s✐♥❣ µ ❧❛r❣❡ ❡♥♦✉❣❤ ✭❞❡♣❡♥❞✐♥❣ ♦♥ ||η0||∞( ¯Ω)✮ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣❛r❛♠❡t❡r η ∈ (0, 1)✱ ❛♣♣❧②✐♥❣ ✉s✉❛❧ ❡♥❡r❣② ❡st✐♠❛t❡s ❛♥❞ r❡♠❛r❦✐♥❣ t❤❛t t❤❡ ❢❛❝t t❤❛t ψ ✈❡r✐✜❡s ✭✷✳✹✮ ❡♥❛❜❧❡s ✉s t♦ ❛❞❞ ❛❧❧ t❤❡ ❞❡r✐✈❛t✐✈❡s ✐♥ t✐♠❡ ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✳

✷✳✸ ❘❡❣✉❧❛r ❝♦♥tr♦❧

❖✉r ❣♦❛❧ ✐♥ t❤✐s s❡❝t✐♦♥ ✐s t♦ ❝♦♥str✉❝t r❡❣✉❧❛r ❡♥♦✉❣❤ ❝♦♥tr♦❧s✳ ❘❡♠✐♥❞ t❤❛t θ ✐s ❞❡✜♥❡❞ ✐♥ ✭✷✳✷✮✳ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳ ▲❡t r ∈ N✳ ❆ss✉♠❡ t❤❛t ❈♦♥❞✐t✐♦♥ ✭✶✳✼✮ ❤♦❧❞s✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ♦❢ Pr♦♣♦s✐t✐♦♥ ✷✳✶✱ ❙②st❡♠   

∂ty = ∆y + div(uy) + div(θBv) ✐♥ QT,

y = 0 ♦♥ ΣT, y(0, ·) = y0 _{✐♥ Ω,} ✭✷✳✸✺✮ ✐s ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ❛t t✐♠❡ T ✱ ✐✳❡✳ ❢♦r ❡✈❡r② y0 _{∈ L}2_{(Ω)✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥tr♦❧ v ∈ L}2_(Q T)m s✉❝❤ t❤❛t t❤❡ s♦❧✉t✐♦♥ z t♦ ❙②st❡♠ ✭✷✳✸✺✮ s❛t✐s✜❡s z(T ) ≡ 0 ✐♥ Ω✳ ▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ u ∈ L2_{((0, T ), H}2r+2_(Ω))m_{∩ H}r+1_{((0, T ), L}2_(Ω))m_✇✐t❤ kvkL2_{((0,T ),H}2r+2_(Ω))m_∩Hr+2_{((0,T ),L}2_(Ω))m 6CeK/T p ky0_k L2_(Ω), ✇❤❡r❡ K ✐s t❤❡ ❝♦♥st❛♥t ✐♥ ✭✷✳✸✹✮✳ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳ ▲❡t k ∈ N∗_{❛♥❞ ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠}    minimize Jk(v) := 1 2keρ −1/2_vk2 L2_(QT)m+ k 2 Z Ω |z(T )|2dx, v ∈ U := {w ∈ L2(QT)m: eρ−1/2w ∈ L2(QT)m}, ✭✷✳✸✻✮ ✶✾