❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
❘ô♠❡❧ ❞❛ ❘♦s❛ ❞❛ ❙✐❧✈❛
❖ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ s✐st❡♠❛s q✉❛s❡✲❧✐♥❡❛r❡s ❤✐♣❡r❜ó❧✐❝♦s
é ❜❡♠ ♣♦st♦ ❡♠ ❡s♣❛ç♦s ❞❡ ❍ö❧❞❡r
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
❘ô♠❡❧ ❞❛ ❘♦s❛ ❞❛ ❙✐❧✈❛
❖ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ s✐st❡♠❛s q✉❛s❡✲❧✐♥❡❛r❡s ❤✐♣❡r❜ó❧✐❝♦s
é ❜❡♠ ♣♦st♦ ❡♠ ❡s♣❛ç♦s ❞❡ ❍ö❧❞❡r
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛çã♦✿ Pr♦❢✳ ❉r✳ ❏♦sé ❘✉✐❞✐✈❛❧ ❙♦❛r❡s ❞♦s ❙❛♥t♦s ❋✐✲ ❧❤♦✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar
S586pc
Silva, Rômel da Rosa da.
O problema de Cauchy para sistemas quase-lineares hiperbólicos é bem posto em espaços de Hölder / Rômel da Rosa da Silva. -- São Carlos : UFSCar, 2012.
75 f.
Tese (Doutorado) -- Universidade Federal de São Carlos, 2012.
1. Análise matemática. 2. Cauchy, Problemas de. 3. Espaços de Hölder. I. Título.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s✳
❆ ♠❡✉s ♣❛✐s✱ ✐r♠ã♦s ❡ ❛ ❡s♣♦s❛ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ❛♣♦✐♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❛♠♦r✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❘✉✐❞✐✈❛❧✱ ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✱ ♣❡❧❛ ❛♠✐③❛❞❡✱ ❝♦♠♣r❡❡♥sã♦✱✳✳✳ ✳
❆♦s ♣r♦❢❡ss♦r❡s✱ ❆❞❛❧❜❡rt♦✱ ❈❡③❛r✱ ▼❛r❝❡❧♦ ❡ ❏♦r❣❡ ♣♦r ❢❛③❡r❡♠ ♣❛rt❡ ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✳
❆ t♦❞♦s ♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡ ❛té ❛❣♦r❛✳
❆♦s ❛♠✐❣♦s ❞♦ ❉▼✳
❘❡s✉♠♦
◆ós ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ s✐st❡♠❛s q✉❛s❡✲❧✐♥❡❛r❡s
∂tu+a(u)∂xu = 0
u(0, x) = ✉0(x),
❝♦♠u= (u1, . . . , uN)❡a(u) = (ajk(u))Nj,k=1 ♠❛tr✐③ r❡❛❧N×N✱ ❝♦♠ ❡♥tr❛❞❛sC∞✱ t❛❧ q✉❡
a(0) t❡♠ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s✱ ♦✉ s❡❥❛ ♦ s✐st❡♠❛ é ❤✐♣❡r❜ó❧✐❝♦ ❡♠ u= 0✳
❉❡♠♦♥str❛♠♦s q✉❡ ❝❡rt♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈ sã♦ ♣r❡s❡r✈❛❞♦s ♣❡❧♦ ✢✉①♦ ❞❛ s♦❧✉çã♦✱ ♣❡rt♦ ❞❛ s♦❧✉çã♦ ♥✉❧❛✳
❆❜str❛❝t
❲❡ ❝♦♥s✐❞❡r t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t❤❡ q✉❛s✐✲❧✐♥❡❛r s②st❡♠s
∂tu+a(u)∂xu = 0
u(0, x) = ✉0(x),
✇✐t❤u= (u1, . . . , uN) ❛♥❞a(u) = (ajk(u))Nj,k=1 r❡❛❧ ♠❛tr✐① N×N✱ ✇✐t❤ ❡♥tr✐❡sC∞✱ s✉❝❤ t❤❛t t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ a(0) ❛r❡ r❡❛❧ ❛♥❞ ❞✐st✐♥❝t✱ t❤❛t ✐s✱ t❤❡ s②st❡♠ ✐s ❤②♣❡r❜♦❧✐❝ ❛t
u= 0✳ ❲❡ s❤♦✇ t❤❛t ❝❡rt❛✐♥ ❇❡s♦✈ s♣❛❝❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ✢♦✇ ♦❢ t❤❡ s♦❧✉t✐♦♥✱ ♥❡❛r t❤❡
♥✉❧❧ s♦❧✉t✐♦♥✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✼
✶ Pr❡❧✐♠✐♥❛r❡s ✶✵
✶✳✶ ❊s♣❛ç♦s ❞❡ ❇❡s♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ▼❛t❡♠át✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ ❍✐♣❡r❜♦❧✐❝✐❞❛❞❡ ♣❛r❛ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❞♦ P❧❛♥♦ ✶✺
✷ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✶✼ ✷✳✶ ❖ ❈❛s♦ C∞
b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❖ ❈❛s♦ Ck ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸ ❖ ❈❛s♦ Cρ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✸ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ✭✵✳✵✳✶✮ é ❜❡♠ ♣♦st♦ ❡♠ ❇ρ
∞,r ✺✹
✸✳✶ ❯♠❛ ❱❡rsã♦ P❛r❛❞✐❢❡r❡♥❝✐❛❧ ❞♦ ❚❡♦r❡♠❛ ✵✳✵✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✷ ❖ ❈❛s♦ ❇ρ
∞,r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
■♥tr♦❞✉çã♦
▲❛rs ❍ör♠❛♥❞❡r ✐♥✐❝✐❛ ♦ ❝❛♣ít✉❧♦ ■❱ ❞❡ ❬✹❪ ❝♦♠ ✉♠❛ ❞✐s❝✉ssã♦ ❛ r❡s♣❡✐t♦ ❞❡ s♦❧✉✲ çõ❡s ❝❧áss✐❝❛s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ s✐st❡♠❛s q✉❛s❡✲❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❈♦♥s✐❞❡r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②
∂tu+a(u)∂xu = 0
u(0, x) = ✉0(x), ✭✵✳✵✳✶✮
❝♦♠u= (u1, . . . , uN)❡a(u) = (ajk(u)) N
j,k=1 ♠❛tr✐③ r❡❛❧N×N✱ ❝♦♠ ❡♥tr❛❞❛sC∞✱ t❛❧ q✉❡
a(0) t❡♠ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s✳ ❙♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s ✐♠♣♦st❛s ❛♦ ❞❛❞♦
✐♥✐❝✐❛❧ ♦ ♣r♦❜❧❡♠❛ ✭✵✳✵✳✶✮ é ❜❡♠ ♣♦st♦ ❡♠Ck✱k ≥1✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♥❛ s❡çã♦ ✹✳✷ ❞❡ ❬✹❪ ❞❡♠♦♥str❛✲s❡ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿
❚❡♦r❡♠❛ ✵✳✵✳✶ ❙❡ ✉0 ∈Ck(R)✱ k ≥1✱ t❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠≤k ❧✐♠✐t❛❞❛s ❡ ❛s ❞❡ ♦r❞❡♠ ≤1 sã♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛s✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ✭✵✳✵✳✶✮ t❡♠
✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ u∈Ck✱ ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ 0≤t≤T✱ ❞❡s❞❡ q✉❡
T sup|✉′
0| ≤c,
♦♥❞❡cé ✉♠❛ ❝♦♥st❛♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡a✳ ▼❛✐s ❛✐♥❞❛✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡sC1, . . . , Ck✱ ✭❞❡♣❡♥❞❡♥❞♦ só ❞❛ ♥♦r♠❛ C1 ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧✮ ♣❛r❛ ❛s q✉❛✐s ✈❛❧❡♠ ❛s ❡st✐♠❛t✐✈❛s
kukCl ≤Clk✉0kCl. ✭✵✳✵✳✷✮
❖❜s❡r✈❛♠♦s q✉❡ t❛❧ r❡s✉❧t❛❞♦ ♥ã♦ ♣♦❞❡ s❡r s✉❜st❛♥❝✐❛❧♠❡♥t❡ ♠❡❧❤♦r❛❞♦✱ ♣❛r❛ t❛❧ ✈❡❥❛♠♦s ♦ q✉❡ s❡ ♣❛ss❛ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❡ ❇✉r❣❡rs❀ ♦✉ s❡❥❛ q✉❛♥❞♦ N = 1 ❡ a(u) = u✳
◆❡st❡ ❝❛s♦✱ ✈✐❛ ♠ét♦❞♦ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ✈❡r✐✜❝❛✲s❡✿ P❛r❛ ✉0 ∈ C1✱ ❛ s♦❧✉çã♦ é ❞❛❞❛ ♣♦r u(t, x) = ✉0(y)❀ x = y+t✉0(y). ❈♦♠♦ xy = 1 +t✉0′(y) s❡ ✉0 ❡ ✉′0 sã♦ ❧✐♠✐t❛❞❛s✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ x= y+t✉0(y) ❞❡t❡r♠✐♥❛ ✐♠♣❧✐❝✐t❛♠❡♥t❡ y ❡♠ t❡r♠♦s ❞❡ (t, x) q✉❛♥❞♦
0≤t ≤T ❝♦♠
1
T = sup{−✉ ′
0}.
◆♦t❡♠♦s ❛✐♥❞❛ q✉❡ s❡ −✉′
0(y) ❛t✐♥❣❡ ♦ ♠á①✐♠♦ ♣♦s✐t✐✈♦ ❡♠ y✱ ❡♥tã♦ ux = x−y1✉′0(y) ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ✐♥✜♥✐t♦ q✉❛♥❞♦t →T✱ ❛ss✐♠ ❛ ❢❛✐①❛ ❡♠ t q✉❡ ❛ s♦❧✉çã♦ ♣❡r♠❛♥❡❝❡C1 é
0≤t < T✳
❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❡①t❡♥sõ❡s ❞♦ ❚❡♦r❡♠❛ ✵✳✵✳✶✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥♦s ❝♦♥❝❡♥tr❛♠♦s ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❞❛❞♦ ✐♥✐❝✐❛❧ ♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ ❞❡ ❍ö❧❞❡r
Cρ✱ ❝♦♠ ρ > 1✳ ❊①t❡♥sõ❡s ❞♦ r❡❢❡r✐❞♦ t❡♦r❡♠❛ t❡♠ s✐❞♦ ❡①❛✉st✐✈❛♠❡♥t❡ ❡st✉❞❛❞♦s ♥♦ â♠❜✐t♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈✱ ✈❡❥❛ ❬✷❪ ❡ ❬✻❪✳
■♥tr♦❞✉çã♦ ✽
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❞❡♠♦♥str❛♠♦s q✉❡ ❛ ❤✐♣ót❡s❡ ❞❡ ♣❡q✉❡♥❡③ ♥❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧✱ ✐♠♣♦st❛ ♣♦r ❍ör♠❛♥❞❡r✱ ♥ã♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ✭✵✳✵✳✶✮ s❡❥❛ ❧♦❝❛❧♠❡♥t❡ ❜❡♠ ♣♦st♦ ❡♠Ck✱ k ≥1✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❞❡♠♦♥str❛♠♦s ♦✿ ❚❡♦r❡♠❛ ✵✳✵✳✷ ❙❡ ✉0 ∈Ck✱ k ≥1✱ t❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❧✐♠✐t❛❞❛s ❡k✉0k∞ é s✉✜❝✐❡♥✲
t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ✭✵✳✵✳✶✮ ❛❞♠✐t❡ s♦❧✉çã♦k✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧✱
❡ ♠❛✐s✱ ❡①✐st❡♠ T > 0 ❡ ❝♦♥st❛♥t❡s C0,C˜0, C1,C˜1, . . . , Ck,C˜k ♣❛r❛ ❛s q✉❛✐s ❛ s♦❧✉çã♦ u s❛t✐s❢❛③
k∂l
xu(t,·)k∞ ≤Clk✉( l)
0 k∞exp(tC˜l), ∀ t∈[0, T].
❊♠ s❡❣✉✐❞❛✱ ❛✐♥❞❛ s❡♠ ❛ ❤✐♣ót❡s❡ ❞❡ ♣❡q✉❡♥❡③ ♥❛ ❞❡r✐✈❛❞❛ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧✱ ✈❡r✐✜❝❛♠♦s q✉❡ ♦ ♣r♦❜❧❡♠❛ é ❧♦❝❛❧♠❡♥t❡ ❜❡♠ ♣♦st♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❍ö❧❞❡r Cρ✱ ❝♦♠ ρ > 1✳ ■st♦ é✱ ❞❡♠♦♥str❛♠♦s ♦✿
❚❡♦r❡♠❛ ✵✳✵✳✸ ❙❡❥❛ρ >1 ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲✐♥t❡✐r♦✳ ❙❡ ✉0 ∈Cρ ❡ k✉0k∞ é s✉✜❝✐❡♥✲
t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ✭✵✳✵✳✶✮ ❛❞♠✐t❡ s♦❧✉çã♦u∈Cρ([0, T]×R)✭♣❛r❛ ❛❧❣✉♠
T >0✮ ❡ ♠❛✐s✱
kukCρ ≤Ck✉0kCρ.
◆♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ❛ r❡❣✉❧❛r✐❞❛❞❡ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧ ❛✉♠❡♥t❛✱ ♥ã♦ ❝♦♥s❡❣✉✐♠♦s r❡❧❛❝✐♦♥❛r ♦ ✈❛❧♦r ❞❡ T ❝♦♠ ❛ ♣❡q✉❡♥❡③ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧✳ ◆♦ ✜♥❛❧ ❞♦
❈❛♣ít✉❧♦ ✷✱ ♠❛♥t❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ ♣❡q✉❡♥❡③ ❞❛ ❞❡r✐✈❛❞❛✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❣❡♥❡r❛❧✐③❛ ♦ ❚❡♦r❡♠❛ ✵✳✵✳✶ ♣❛r❛ ♦ ❝❛s♦ Cρ✱ ρ > 1✱ ❡ ♠❛♥t❡♠ ❛ r❡❧❛çã♦ ❡♥tr❡ T ♦ ❛ ♣❡q✉❡♥❡③ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧✳ ❆ s❛❜❡r✿
❚❡♦r❡♠❛ ✵✳✵✳✹ ❙❡❥❛ ✉0 ∈Cρ✱ ρ >1♥ã♦✲✐♥t❡✐r♦✳ ❙❡ ✉0 t❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠
≤ [ρ] ❧✐♠✐t❛❞❛s ❡ ❛s ❞❡ ♦r❞❡♠ ≤ 1 sã♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛s✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡
❈❛✉❝❤② ✭✵✳✵✳✶✮ t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦u∈Cρ✱ ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ 0≤t≤T✱ ❞❡s❞❡ q✉❡
T sup|✉′
0| ≤c,
♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡ a✳ ▼❛✐s ❛✐♥❞❛✱ ❡①✐st❡ ❝♦♥st❛♥t❡ C ♣❛r❛ ❛
q✉❛❧ ✈❛❧❡
kukCρ ≤Ck✉0kCρ. ✭✵✳✵✳✸✮
◆♦ ❈❛♣ít✉❧♦ ✸✱ ❡♠ ✉♠ ♣r✐♠❡✐r♦ ♠♦♠❡♥t♦✱ ❞❡♠♦♥str❛♠♦s✱ ✉t✐❧✐③❛♥❞♦ ❛ t❡♦r✐❛ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡②✱ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✵✳✵✳✶ q✉❡ é ❞❡ ❢❛t♦ ♠❛✐s ❢r❛❝❛ q✉❡ ❛ ❥á ❛♣r❡s❡♥t❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ♥♦ ❡♥t❛♥t♦✱ t❛❧ ❞❡♠♦♥str❛çã♦ s❡r✈❡ ❞❡ ✐♥s♣✐r❛çã♦ ♣❛r❛✱ ♥✉♠❛ s❡❣✉♥❞❛ ❡t❛♣❛✱ tr❛❜❛❧❤❛r ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ♥✉♠❛ ❝❧❛ss❡ ♠❛✐s ❛♠♣❧❛ ❞❡ ❢✉♥çõ❡s✱ ♦ q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛ ❞❡♠♦♥str❛r ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❚❡♦r❡♠❛ ✵✳✵✳✶ ♣❛r❛ ♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈ ❇ρ
∞,r ❝♦♠ ρ >2♥ã♦✲✐♥t❡✐r♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❝♦♠ t❛❧ té❝♥✐❝❛ ❞❡♠♦♥str❛♠♦s ♦✿
❚❡♦r❡♠❛ ✵✳✵✳✺ ❙❡❥❛ ✉0 ∈❇ρ∞,r✱ ρ >2♥ã♦✲✐♥t❡✐r♦ ❡ 1≤r≤ ∞✳ ❙❡ ✉0 t❡♠ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ ≤[ρ] ❧✐♠✐t❛❞❛s ❡ ❞❡ ♦r❞❡♠ ≤1 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛s✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡
❈❛✉❝❤② ✭✵✳✵✳✶✮ t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦u∈C[ρ]([0, T]×R) ❞❡s❞❡ q✉❡
T sup|✉′0| ≤c,
♦♥❞❡cé ✉♠❛ ❝♦♥st❛♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡a✳ ▼❛✐s ❛✐♥❞❛✱ ♣❛r❛ ❝❛❞❛t∈[0, T], u(t,·)∈
❇ρ
∞,r ❡ s❛t✐s❢❛③
ku(t,·)k❇ρ
∞,r ≤Ck✉0k❇∞ρ,r ∀t∈[0, T],
■♥tr♦❞✉çã♦ ✾
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
❈♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ♥❛ ✐♥tr♦❞✉çã♦✱ ❡st❡ é ✉♠ ❝❛♣ít✉❧♦ ✐♥❢♦r♠❛t✐✈♦✱ ♥❡❧❡ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s✳ ❱❛♠♦s s❡r ❜❡♠ s✉❝✐♥t♦s ✉♠❛ ✈❡③ q✉❡ t❛❧ ❝❛♣ít✉❧♦ é✱ ❜❛s✐❝❛♠❡♥t❡✱ ✉♠ ❛♣❛♥❤❛❞♦ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉❡ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❢♦r♥❡❝❡r ✉♠❛ ❝♦♥s✉❧t❛ rá♣✐❞❛✱ ❛✜♠ ❞❡ s❛♥❛r ❡✈❡♥t✉❛✐s ❞ú✈✐❞❛s✱ s♦❜r❡ ❢❛t♦s ♦✉ ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ P❛r❛ ♠❛✐♦r❡s ✐♥❢♦r♠❛çõ❡s ❛❝❡r❝❛ ❞♦s ❝♦♥t❡ú❞♦s ❡ r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❡st❡ ❝❛♣ít✉❧♦ ✈❡r ❬✷❪✱ ♣❛r❛ ❛s s❡çõ❡s ✶✳✶ ❡ ✶✳✷✱ ❡ ♣❛r❛ ❛s s❡çõ❡s ✶✳✸ ❡ ✶✳✹ ✈❡r ❬✶❪✱ ❬✹❪✱ ❬✺❪ ❡ ❬✼❪✳ ❖❜s❡r✈♦ q✉❡ ♥❡♠ t♦❞❛s ❛s ❞❡✜♥✐çõ❡s ♦✉ ♦s r❡s✉❧t❛❞♦s q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s ❡stã♦ ❛q✉✐ ❝♦❧❡❝✐♦♥❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❡ ✢✉①♦ sã♦ ❛ss✉♠✐❞❛s✳ ❆❧❣✉♥s ❢❛t♦s ❜ás✐❝♦s ❞❡ ❝á❧❝✉❧♦ sã♦ ✉s❛❞♦s s❡♠ q✉❛❧q✉❡r ♠❡♥çã♦✳ ❊st❛♠♦s ❡s♣❡❝✐❛❧♠❡♥t❡ ✐♥t❡r❡ss❛❞♦s ❡♠ ❛♣r❡s❡♥t❛r ❛q✉✐ ❛ ❞❡✜♥✐çã♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈ ❡ ❛ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥②✳
✶✳✶ ❊s♣❛ç♦s ❞❡ ❇❡s♦✈
❆❧❣✉♠❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❜❡♠ ❝♦♥❤❡❝✐❞❛s ❡ q✉❡ t❡♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥♦ ❡st✉❞♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈ sã♦ ♦ ❝♦♥t❡ú❞♦ ❞♦✿
❚❡♦r❡♠❛ ✶✳✶✳✶ ✭❉❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❇❡r♥st❡✐♥✮ ❙❡❥❛♠ C ✉♠❛ ❝♦r♦❛ ❡ B ✉♠❛ ❜♦❧❛✳
❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s (p, q) s❛t✐s❢❛③❡♥❞♦✱
q≥p≥1✱ λ >0 ❡ u∈Lp t❡♠♦s✿
✐✮ S(ˆu)⊂λB⇒ sup
|α|=k
k∂αuk
Lq(Rd) ≤Ck+1λk+d(
1
p−
1
q)kuk
Lp(Rd);
✐✐✮ S(ˆu)⊂λC ⇒C−k−1λkkuk
Lp(Rd) ≤ sup
|α|=k
k∂αuk
Lp(Rd) ≤Ck+1λkkukLp(Rd).
P❛r❛ q✉❡ ♣♦ss❛♠♦s ❛♣r❡s❡♥t❛r ❛ ❞❡✜♥✐çã♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈✱ ♣r❡❝✐s❛♠♦s ❝♦♥s✐❞❡r❛r ❛❧❣✉♠❛s ❢✉♥çõ❡s ❡s♣❡❝✐❛s✱ t❛✐s ❢✉♥çõ❡s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛✿
Pr♦♣♦s✐çã♦ ✶✳✶✳✶ ❙❡❥❛♠ C = C(0,34,83) ❡ B = B(0,43)✳ ❊①✐st❡♠ ❢✉♥çõ❡s r❛❞✐❛✐s χ ∈
C∞
c (B,[0,1]) ❡ ϕ ∈Cc∞(C,[0,1]) s❛t✐s❢❛③❡♥❞♦✿
∀ ξ∈Rn, χ(ξ) +X
j≥0
ϕ(2−jξ) = 1;
∀ ξ ∈Rn\ {0}, X
j∈Z
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✶
|j−j′| ≥2⇒S(ϕ(2−j·))∩S(ϕ(2−j′·)) =∅;
j ≥1⇒S(χ)∩S(ϕ(2−j·)) = ∅. ❆❧é♠ ❞✐st♦✱ s❡ C˜=B(0,2
3) +C✱ ❡♥tã♦
|j−j′| ≥5⇒2j′˜
C ∩2jC =∅;
∀ ξ ∈Rn, 1
3 ≤χ
2(ξ) +X
j≥0
ϕ2 2−jξ≤1;
∀ ξ∈Rn\ {0}, 1
2 ≤
X
j∈Z
ϕ2(2−jξ)≤1.
P❛r❛ ϕ ❡ χ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ ❛ss♦❝✐❛✲s❡ ♦s s❡❣✉✐♥t❡s
♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ❋♦✉r✐❡r✿
∆ju= 0, ∀ j <−1;
∆−1u= (χuˆˇ);
∆ju= (ϕ(2−j·)ˆu)ˇ,∀ j ≥0;
Sju= (χ(2−j·)ˆu)ˇ, ∀ j ∈Z. ✭✶✳✶✳✶✮ ◆♦ ❝❛s♦ ❞❡ u ∈ Lp, 1 ≤ p ≤ ∞, ♦s ♦♣❡r❛❞♦r❡s ❛❝✐♠❛ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ✉t✐❧✐③❛♥❞♦ ❝♦♥✈♦❧✉çã♦✱ ❛ s❛❜❡r
∆−1u= 2−nχˇ(2−1·)∗u,
∆ju= 2jnϕˇ(2j·)∗u, ∀j ≥0,
Sju= 2jnχˇ(2j·)∗u, ∀j ≥ −1.
❖❜s❡r✈❡♠♦s ❛✐♥❞❛ q✉❡ ♦s ♦♣❡r❛❞♦r❡s ∆j ❝♦♠✉t❛♠ ❝♦♠ ❛s ❞❡r✐✈❛❞❛s✱ ✐st♦ é✱
∆j∂αu=∂α∆ju; ∀α∈Nn, ∀u∈ S′,
♥♦t❡♠♦s✱ ❛❧é♠ ❞✐st♦✱ q✉❡∆j ❡Sj ♠❛♣❡✐❛♠Lp ❡♠ Lp ❡ ❡①✐st❡ ❝♦♥st❛♥t❡C ✭ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡j ∈Z ✮✱ t❛❧ q✉❡
k∆jukLp ≤CkukLp ❡
kSjukLp ≤CkukLp.
❱❡r✐✜❝❛✲s❡ t❛♠❜é♠✱ ♣❛r❛ t♦❞❛u∈ S′✱ q✉❡
Sju=
X
j′≤j−1
∆j′u
❡ q✉❡
u= lim
j→∞Sju ♥♦ ❡s♣❛ç♦ S ′(Rn
).
▲♦❣♦ t❡♠ s❡♥t✐❞♦ ❡s❝r❡✈❡r
u=X
j∈Z
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✷
❉❡✜♥✐çã♦ ✶✳✶✳✶ ✭❉❡❝♦♠♣♦s✐çã♦ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡②✮ ❖ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✶✳✶✳✷✮ é ❝❤❛♠❛❞♦ ❞❡❝♦♠♣♦s✐çã♦ ♥ã♦✲❤♦♠♦❣ê♥❡❛ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❞❛ u✳
P♦❞❡♠♦s ❛❣♦r❛ ❛♣r❡s❡♥t❛r ❛ ❞❡✜♥✐çã♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈ ♥ã♦✲❤♦♠♦❣ê♥❡♦s✳
❉❡✜♥✐çã♦ ✶✳✶✳✷ ❙❡❥❛♠ (p, r)∈[1,∞]2 ❡ s ✉♠ ♥ú♠❡r♦ r❡❛❧✳ P❛r❛ u∈ S′✱ ❝♦♥s✐❞❡r❡♠♦s
kuk❇s p,r
.
=k(2sjk∆jukLp)j∈Zklr(Z).
❖ ❡s♣❛ç♦ ❞❡ ❇❡s♦✈ ❇s
p,r é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s u t❛✐s q✉❡
kuk❇s
p,r <∞✳
❖❜s❡r✈❛♠♦s ❛q✉✐ q✉❡ ♦s ❡s♣❛ç♦s ❇s
p,r ♥ã♦ ❞❡♣❡♥❞❡♠ ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣❛r (χ, ϕ)✳ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ♠✉✐t♦ út❡✐s ❞♦s ♦♣❡r❛❞♦r❡s ∆q sã♦ ❞❛❞❛s ♥❛✿
Pr♦♣♦s✐çã♦ ✶✳✶✳✷ P❛r❛ q✉❛✐sq✉❡r u ❡ v ∈ S′(Rd)✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
∆q∆pu≡0 s❡ |q−p| ≥2;
∆q(Sp−1u∆pv)≡0 s❡ |q−p| ≥5.
❆❣♦r❛ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❍ö❧❞❡r ✈✐❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡②✳ Pr✐♠❡✐r❛♠❡♥t❡ ❧❡♠❜r❡♠♦s ❛✿
❉❡✜♥✐çã♦ ✶✳✶✳✸ ✭❊s♣❛ç♦s ❞❡ ❍ö❧❞❡r✮ ❙❡❥❛ ρ ∈(0,1)✳ ❉❡♥♦t❛♠♦s ♣♦r Cρ(Rd) ♦ ❝♦♥✲ ❥✉♥t♦ ❞❛s ❢✉♥çõ❡s u: Rd →R ❝♦♥tí♥✉❛s ❡ ❧✐♠✐t❛❞❛s s❛t✐s❢❛③❡♥❞♦✱ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡
C✱
|u(x)−u(y)| ≤C|x−y|ρ, ∀x ❡ y ∈Rd.
▼❛✐s ❣❡r❛❧♠❡♥t❡✱ s❡ρ >0✱ ❡ ♥ã♦ é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❞❡♥♦t❛♠♦s ♣♦rCρ(Rd)♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s u✱ [ρ] ✈❡③❡s✶ ❞✐❢❡r❡♥❝✐á✈❡✐s t❛✐s q✉❡ ∂αu∈Cρ−[ρ](Rd)✱ ♣❛r❛ t♦❞♦ |α| = [ρ]✳ ❖ ❝♦♥❥✉♥t♦ Cρ(Rd) ♠✉♥✐❞♦ ❝♦♠ ❛ ♥♦r♠❛
kukCρ =.
X
|α|≤[ρ]
k∂αuk∞+ X
|α|=[ρ]
sup
x6=y
|∂αu(x)−∂αu(y)|
|x−y|ρ−[ρ] .
é ♦ ❡s♣❛ç♦ ❞❡ ❍ö❧❞❡r Cρ(Rd)✳
❚❡♠♦s ❛ s❡❣✉✐♥t❡ ❝❛r❛❝t❡r✐③❛çã♦✱ ✈✐❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡②✱ ♣❛r❛ ♦s ❡s♣❛✲ ç♦s ❞❡ ❍ö❧❞❡r✳
❚❡♦r❡♠❛ ✶✳✶✳✷ ❙❡ρé ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ♥ã♦✲✐♥t❡✐r♦✱ ❡♥tã♦ ♦s ❡s♣❛ç♦s ❇ρ∞,∞ ❡ Cρ
sã♦ ✐❣✉❛✐s ❡✱ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C✱ ✈❛❧❡
C−1kuk❇ρ
∞,∞ ≤ kukCρ ≤Ckuk❇ρ∞,∞.
❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ❝♦♥té♠ ✉♠❛ ❧✐st❛ ❞❡ ✐♠♣♦rt❛♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❡s♣❛ç♦s ❞❡ ❇❡s♦✈✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✸
✭✐✮ Pr♦♣r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛✿ ❇s
p,r é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♥t✐♥✉❛♠❡♥t❡ ♠❡r❣✉❧❤❛❞♦ ❡♠
S′✳
✭✐✐✮ ❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s s✉❛✈❡s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ é ❞❡♥s♦ ❡♠ ❇s
p,r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p ❡ r sã♦ ✜♥✐t♦s✳
✭✐✐✐✮ ❙❡ 1≤p ❡ r <∞✱ ❡♥tã♦ ❇s
p,r é s❡♣❛rá✈❡❧✳ ✭✐✈✮ ❆❧❣✉♥s ♠❡r❣✉❧❤♦s✿
✭❛✮ ❇s
p,r ֒→❇˜ s
p,r˜ s❡♠♣r❡ q✉❡ s < s˜ ♦✉ s˜=s ❡ r ≤˜r❀ ✭❜✮ ❇s
p,r(Rd)֒→❇
s−d(1p−1 ˜
p)
˜
p,r (Rd) s❡♠♣r❡ q✉❡ p≤p˜❀
✭✈✮ Pr♦♣r✐❡❞❛❞❡ ❞❡ ❋❛t♦✉✿ s❡ (un)n∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ ❇
s
p,r q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ u ❡♠ S′✱ ❡♥tã♦ u∈❇s
p,r ❡
kuk❇s
p,r ≤lim infkunk❇sp,r.
❉❡st❛❝❛♠♦s ❛✐♥❞❛ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♠✉t❛❞♦r ❞❡ ♦♣❡r❛❞♦r❡s✳
▲❡♠❛ ✶✳✶✳✶ ❊①✐st❡ ❝♦♥st❛♥t❡C t❛❧ q✉❡ ♣❛r❛ t♦❞❛f ❡♠Lp✱ g ❢✉♥çã♦ ❧✐♣s❝❤✐t③ ❡ p∈[1,∞] ✈❛❧❡✿
k[∆q, g]fkLp ≤C2−qk∇gk∞kfkLp.
✶✳✷ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥②
❆ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥② é ✉♠❛ ❢❡rr❛♠❡♥t❛ ✐♠♣♦rt❛♥t❡ ❡ ♠✉✐t♦ út✐❧ ♣❛r❛ ❡st✉❞❛r ♦ ♣r♦❞✉t♦ ❞❡ ❢✉♥çõ❡s ❡ ❞❡ ❞✐str✐❜✉✐çõ❡s✳ P❛r❛ ❞✉❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛su ❡ v t❡♠♦s ❛
s❡❣✉✐♥t❡ ❞❡❝♦♠♣♦s✐çã♦✿
uv = X
q,q′∈Z
∆qu∆q′v.
❆ ♣ró①✐♠❛ ❞❡✜♥✐çã♦ ✐♥tr♦❞✉③ ❛ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥② q✉❡ é ✉♠❛ ❜♦❛ ❛❜♦r❞❛❣❡♠ ❛♦ ♣r♦❞✉t♦ ❞❡ ❞✐str✐❜✉✐çõ❡s✳
❉❡✜♥✐çã♦ ✶✳✷✳✶ ❙❡❥❛♠ u ❡ v ❞✉❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s✳ ❈♦♥s✐❞❡r❡♠♦s
Tuv =.
X
q∈Z
Sq−1u∆qv
❡
R(u, v)=. X
|q−q′|≤1
q,q′∈Z
∆qu∆q′v.
P❡❧♦ ♠❡♥♦s ✐♥❢♦r♠❛❧♠❡♥t❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
uv =Tuv+Tvu+R(u, v) ✭❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥②✮. ✭✶✳✷✳✶✮
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✹
❚❡♦r❡♠❛ ✶✳✷✳✶ P❛r❛ q✉❛❧q✉❡r s ∈ R ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r
(p, r)∈[1,+∞]2✱ t❡♠♦s
kTuvk❇s
p,r ≤CkukL∞kvk❇sp,r, ∀(u, v)∈L
∞×❇s
p,r.
❚❡♦r❡♠❛ ✶✳✷✳✷ ❙❡ (s1, s2) ∈ R2 sã♦ t❛✐s q✉❡ s1+s2 >0✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡
C t❛❧ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r (p1, p2, r1, r2)∈[1,∞]4 s❛t✐s❢❛③❡♥❞♦
1
p .
= 1
p1
+ 1
p2
≤1 ❡ 1
r .
= 1
r1
+ 1
r2
≤1,
t❡♠♦s
kR(u, v)k❇s1+s2
p,r ≤Ckuk❇ s1
p1,r1kvk❇ s2
p2,r2, ∀(u, v)∈❇
s1
p1,r1 ×❇
s2
p2,r2.
❈♦♠❜✐♥❛❞♦s✱ ♦s ❞♦✐s t❡♦r❡♠❛s ❛❝✐♠❛ ✐♠♣❧✐❝❛♠ ♦✿
❈♦r♦❧ár✐♦ ✶✳✷✳✶ P❛r❛ q✉❛❧q✉❡r r❡❛❧ ♣♦s✐t✐✈♦ s✱ ♦ ❡s♣❛ç♦ L∞∩❇s
p,r é ✉♠❛ á❧❣❡❜r❛✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡C t❛❧ q✉❡
kuvk❇s
p,r ≤C(kukL∞kvk❇sp,r +kuk❇sp,rkvkL∞), ∀u, v ∈L
∞∩❇s p,r.
✶✳✸ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ▼❛t❡♠át✐❝❛✳
❆♣r❡s❡♥t❛♠♦s ❛q✉✐ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ❝✐t❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ■♥✐✲ ❝✐❛♠♦s ❝♦♠ ✉♠ r❡s✉❧t❛❞♦ ♠✉✐t♦ út✐❧ q✉❛♥❞♦ ♣r❡❝✐s❛♠♦s ❡①♣❧✐❝✐t❛r ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ♣❛r❛ ❝♦♠♣♦st❛ ❞❡ ❢✉♥çõ❡s✳
❚❡♦r❡♠❛ ✶✳✸✳✶ ❙❡❥❛♠ U ⊂ Rm ❡ V ⊂ Rn ❛❜❡rt♦s✱ f : U → Rn ✉♠❛ ❛♣❧✐❝❛çã♦ i ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x ∈ U✱ ❝♦♠ f(U) ⊂ V✱ ❡ g : V → Rp ✉♠❛ ❛♣❧✐❝❛çã♦ i ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ y = f(x)✳ P❛r❛ ❝❛❞❛ ♣❛rt✐çã♦ i = i1 +. . .+ik ✭❞❡ i ❝♦♠♦ s♦♠❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✮ ❡①✐st❡ ✉♠ n(i1, . . . , ik)∈Q t❛❧ q✉❡ ❛ i✲és✐♠❛ ❞❡r✐✈❛❞❛ ❞❛ ❛♣❧✐❝❛çã♦
g◦f :U →Rp t❡♠ ❛ ❡①♣r❡ssã♦
(g◦f)(i)= i
X
k=1
n(i1, . . . , ik)g(k)◦f· f(i1), . . . , f(ik)
.
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ r❡❧❛❝✐♦♥❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ❝♦♠ ❛ ❝♦♥✈❡r❣ê♥✲ ❝✐❛ ❞❛ s❡q✉ê♥❝✐❛ ❞❛s ❞❡r✐✈❛❞❛s✳
❚❡♦r❡♠❛ ✶✳✸✳✷ ✭❉❡r✐✈❛çã♦ ❚❡r♠♦ ❛ ❚❡r♠♦✮ ❙❡❥❛ U ⊂ Rd ✉♠ ❛❜❡rt♦ ❝♦♥❡①♦✳ ❙❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s fk : U → Rn ❝♦♥✈❡r❣❡ ♥✉♠ ♣♦♥t♦ c ∈ U ❡ ❛ s❡q✉ê♥❝✐❛ ❞❛s ❞❡r✐✈❛❞❛s f′
k :U → L(Rd;Rn) ❝♦♥✈❡r❣❡ ❞❡ ♠♦❞♦ ❧♦❝❛❧♠❡♥t❡ ✉♥✐❢♦r♠❡ ♣❛r❛ ✉♠❛ ❛♣❧✐❝❛çã♦ g : U → L(Rd;Rn)✱ ❡♥tã♦ (f
k)k ❝♦♥✈❡r❣❡ ❞❡ ♠♦❞♦ ❧♦❝❛❧♠❡♥t❡ ✉♥✐❢♦r♠❡ ♣❛r❛ ✉♠❛ ❛♣❧✐❝❛çã♦ f :U →Rn✱ ❛ q✉❛❧ é ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❝♦♠ f′ =g✳
❚❛♠❜é♠ r❡❧❛❝✐♦♥❛❞♦ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s t❡♠♦s ♦s✿
❚❡♦r❡♠❛ ✶✳✸✳✸ ✭❚❡♦r❡♠❛ ❞❡ ❆r③❡❧á✮ ❙❡❥❛♠ ✭❳✱❞✮ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❛❝t♦ ❡ F
✉♠❛ ❢❛♠í❧✐❛ ❡q✉✐❝♦♥tí♥✉❛ ❞❡ ❢✉♥çõ❡s ϕ:X →R✳ ❙❡ F é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛✱ ❡♥tã♦
t♦❞❛ s❡q✉ê♥❝✐❛ (ϕn)n∈N ❞❡ ❡❧❡♠❡♥t♦s ❞❡ F t❡♠ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ (ϕnk)nk∈N ✉♥✐❢♦r♠❡✲
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✺
❚❡♦r❡♠❛ ✶✳✸✳✹ ❙❡❥❛ (ϕn)n∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❡q✉✐❝♦♥tí♥✉❛ ❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❞❡
❢✉♥çõ❡s r❡❛✐s ❡ ❝♦♥tí♥✉❛s ♥✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❛❝t♦ X✳ ❙✉♣♦♥❤❛♠♦s q✉❡ t♦❞❛ ❛
s✉❜s❡q✉ê♥❝✐❛ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡ ❞❡st❛ s❡q✉ê♥❝✐❛ t❡♠ ♦ ♠❡s♠♦ ❧✐♠✐t❡ ϕ✱ ❡♥tã♦
(ϕn)n∈N é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ϕ✳
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥t❛ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ✐♥t❡r♣♦❧❛çã♦ q✉❡ é ✉♠ ❝❛s♦ ♣❛r✲ t✐❝✉❧❛r ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●❛❣❧✐❛r❞♦✲◆✐r❡♥❜❡r❣✳
❚❡♦r❡♠❛ ✶✳✸✳✺ ❙❡v ∈Ck(R) t❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❧✐♠✐t❛❞❛s✱ ❡♥tã♦
kv(j)k
∞≤2
j(k−j) 2 kv(k)k
j k
∞kvkk−kj
∞
♣❛r❛ t♦❞♦0< j < k✳
❯♠ r❡s✉❧t❛❞♦ ♠❛✐s ❝♦♠✉♠ ❡♠ ❧✐✈r♦s q✉❡ tr❛t❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛s ♠❛s q✉❡ t❛♠❜é♠ ❝♦❧♦❝❛♠♦s ❛q✉✐ é✿
▲❡♠❛ ✶✳✸✳✶ ✭▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧✮ ❙❡ u ❡ v sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♥ã♦ ♥❡❣❛t✐✈❛s ❡♠
[a, b] ❡ t❛✐s q✉❡ ♣❛r❛ ❛❧❣✉♠ r ≥0 s❛t✐s❢❛③❡♠
u(t)≤r+
Z t
a
v(s)u(s)ds, ∀t∈[a, b],
❡♥tã♦
u(t)≤rexp(
Z t
a
v(s)ds), ∀t∈[a, b].
❈♦❧♦❝❛♠♦s ❛✐♥❞❛ ❛q✉✐ ♦✿
❚❡♦r❡♠❛ ✶✳✸✳✻ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✮ ❙✉♣♦♥❤❛1≤p, q, r≤ ∞❡ q✉❡p−1+q−1 =
r−1+ 1✳ ❙❡ f ∈Lp ❡ g ∈Lq✱ ❡♥tã♦ f∗g ∈Lr ❡ kf ∗gk
Lr ≤ kfkLpkgkLq✳
✶✳✹ ❍✐♣❡r❜♦❧✐❝✐❞❛❞❡ ♣❛r❛ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❞❡ ❊q✉❛✲
çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❞♦ P❧❛♥♦
❉❡✜♥✐çã♦ ✶✳✹✳✶ ❯♠ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ✉♠ ♦♣❡r❛❞♦r ❞❛ ❢♦r♠❛
L=S(t, x)∂t+A(t, x)∂x+B(t, x),
♦♥❞❡ S, A ❡ B sã♦ ♠❛tr✐③❡s C∞✱ N ×N✳
❉❡✜♥✐çã♦ ✶✳✹✳✷ ✭❍✐♣❡r❜♦❧✐❝✐❞❛❞❡✮ ❖ s✐st❡♠❛ Lé ❤✐♣❡r❜ó❧✐❝♦ s❡ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s✱ λi✱ ❞❡ S−1A sã♦ r❡❛✐s✳ ❊st❡s ❛✉t♦✈❛❧♦r❡s sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈❡❧♦❝✐❞❛❞❡s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ♦♣❡r❛❞♦r L✳ ❙❡ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s sã♦ r❡❛✐s ❡ ❞✐st✐♥t♦s ♦ s✐st❡♠❛ é ❞✐t♦ ❡str✐t❛♠❡♥t❡
❤✐♣❡r❜ó❧✐❝♦✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✻
❉❡✜♥✐çã♦ ✶✳✹✳✹ ❯♠ ❞♦♠í♥✐♦ ❢❡❝❤❛❞♦ D⊂[0,∞)×R ❝♦♠ ❜❛s❡
ω =D∩ {t = 0}
é ✉♠ ❞♦♠í♥✐♦ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ❞❡ω ♣❛r❛ ✉♠ ♦♣❡r❛❞♦rLs❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦m= (t0, x0)∈
D✱ ❡ t♦❞♦i✱ ❛ ❝✉r✈❛ ✐✲❝❛r❛❝t❡ríst✐❝❛ ♣❛r❛ trás ✭✐st♦ é t≤t0✮ q✉❡ s❛✐ ❞❡ m✱ ♣❡r♠❛♥❡❝❡ ❡♠
D ❛té ❡♥❝♦♥tr❛r ω✳
❚❡♦r❡♠❛ ✶✳✹✳✶ ❙❡❥❛D✉♠ ❞♦♠í♥✐♦ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ❝♦♠♣❛❝t♦ ❝♦♠ ❜❛s❡ω♥♦ ❡✐①♦x♣❛r❛
✉♠ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡str✐t❛♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦ L✳ ❙❡❥❛ Dt = {x : (t, x) ∈ D}✳ ❊♥tã♦ ❡①✐st❡ ❝♦♥st❛♥t❡ C t❛❧ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r U ∈C( ¯D)✱ ✈❛❧❡
max
0≤s≤tkU(s,·)kL ∞(D
s)≤C
kU(0,·)kL∞(ω)+
Z t
0
k(LU)(s,·)kL∞(D
s)ds
.
❈❛♣ít✉❧♦ ✷
❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é
❜❡♠ ♣♦st♦ ❡♠
C
ρ
❱❛♠♦s ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ ♦ s✐st❡♠❛
∂tu+a(u)∂xu= 0, ✭✷✳✵✳✶✮ ♦♥❞❡u= (u1, . . . , uN) ❡a(u) = (ajk(u))Nj,k=1 é ✉♠❛ ♠❛tr✐③ r❡❛❧N×N ❝♦♠ ❡♥tr❛❞❛s C∞✱ t❛❧ q✉❡ a(0) t❡♠ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s✱ ♦✉ s❡❥❛✱ ♦ s✐st❡♠❛ é ❡str✐t❛♠❡♥t❡
❤✐♣❡r❜ó❧✐❝♦✳ ❉❡st❡ ♠♦❞♦ ♣❛r❛u❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ③❡r♦ ❛ ♠❛tr✐③a(u)t❡♠ ❛✉t♦✈❛❧♦r❡s
r❡❛✐s ❡ ❞✐st✐♥t♦s ❞❡♥♦t❛❞♦s ♣♦r
λ1(u)< . . . < λN(u) ✭✷✳✵✳✷✮
❡ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛✉t♦✈❡t♦r❡sr1(u), . . . , rN(u)❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ✉♥✐tár✐❛ ❞❡♣❡♥❞❡♥❞♦ s✉❛✈❡♠❡♥t❡ ❞❡ u✳
❉❛❞♦ ✉0 ∈ Cρ(R) ❝♦♠ k✉0k∞ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ρ > 1✱ ✈❛♠♦s ♠♦str❛r q✉❡
❡①✐st❡ ú♥✐❝❛ u∈Cρ([0, T]×RN)s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛
∂tu+a(u)∂xu = 0
u(0, x) = ✉0(x) ✭✷✳✵✳✸✮ ❛❧é♠ ❞✐st♦✱ ♠♦str❛r❡♠♦s q✉❡ us❛t✐s❢❛③ ❛ ❡st✐♠❛t✐✈❛ kukCρ ≤Cku0kCρ✳
❆♥t❡s ❞❡ ♣❛ss❛r♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦s r❡s✉❧t❛❞♦s✱ ❛♣r❡s❡♥t❛♠♦s três ❡t❛♣❛s ❜❛s❡❛❞❛s ❡♠ ✐❞❡✐❛s ✉t✐❧✐③❛❞❛s ❡♠ ❬✹❪ ♣❛r❛ ❞❡♠♦♥str❛r ♦ ❚❡♦r❡♠❛ ✵✳✵✳✶✳
❊t❛♣❛ ✶✳ Pr♦❝❡ss♦ ■t❡r❛t✐✈♦✳ ❱❛♠♦s s✉♣♦r q✉❡ ♦ ❞❛❞♦ ✐♥✐❝✐❛❧ ✉0 éC∞✱ t❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❧✐♠✐t❛❞❛s ❡ k✉0k∞ é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✳ ❙❡❥❛ (uν)ν≥−1 ❛ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
u−1(t, x) = u0(t, x) =✉0(x) ✭✷✳✵✳✹✮ ❡ ♣❛r❛ν ≥1 ❞❡✜♥✐♠♦s uν ❝♦♠♦ s❡♥❞♦ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛
∂tuν +a(uν−1)∂xuν = 0
uν(0, x) = ✉0(x). ✭✷✳✵✳✺✮ ❆ss✉♠✐♥❞♦ q✉❡ kuν−1k∞ é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❡s❝r❡✈❡♠♦s
uν(t, x) = N
X
i=1
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✶✽
❡
∂k
xuν(t, x) =wνk(t, x) = N
X
i=1
wνki(t, x)ri(uν−1(t, x)). ✭✷✳✵✳✼✮
❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s✉♣♦r kuν−1k∞ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ é ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❛
❜❛s❡ ❞❡ ❛✉t♦✈❡t♦r❡s {r1(uν−1), . . . , rN(uν−1)}✳ ❈♦♠♦ ku−1k∞ = ku0k∞ = k✉0k∞ é ❝❧❛r♦ q✉❡ s❡k✉0k∞ é ♣❡q✉❡♥♦ ❡s❝r❡✈❡r ✭✷✳✵✳✻✮ ❡ ✭✷✳✵✳✼✮ t❡♠ s❡♥t✐❞♦ ♣❛r❛ν = 0,1✳ ▼❛✐s ❛❞✐❛♥t❡✱ ♠♦str❛r❡♠♦s✱ ✈✐❛ ✐♥❞✉çã♦ ❡♠ ν✱ q✉❡ s❡ ❛ ♥♦r♠❛ L∞ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧ é ♣❡q✉❡♥❛✱ ❡♥tã♦
r❡str✐♥❣✐♥❞♦t❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥t❡r✈❛❧♦ [0, T]✱ ❛ ♥♦r♠❛L∞([0, T]×R)❞❡ t♦❞♦ ♦ t❡r♠♦
❞❛ s❡q✉ê♥❝✐❛(uν)ν≥−1 ♣❡r♠❛♥❡❝❡ ♣❡q✉❡♥❛✱ ❞❡ ❢❛t♦ ❝♦♠♣❛rá✈❡❧ ❛k✉0k∞✳ ❆ss✐♠ é ❛❝❡✐tá✈❡❧ ❛ss✉♠✐r♠♦s q✉❡kuν−1k∞ é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❡ ❡s❝r❡✈❡r ✭✷✳✵✳✻✮ ❡ ✭✷✳✵✳✼✮✳
❯s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ uν ❞❛❞❛ ❡♠ ✭✷✳✵✳✻✮ ❡ q✉❡ a(uν−1)rj(uν−1) =
λj(uν−1)rj(uν−1) ♦❜t❡♠♦s✱
a(uν−1)∂xuν = N
X
i=1
∂xuνiλi(uν−1)ri(uν−1) + N
X
i=1
uνia(uν−1)∂x(ri(uν−1))
❡
∂tuν = N
X
i=1
∂tuνiri(uν−1) + N
X
i=1
uνi∂t(ri(uν−1)).
❈♦♠♦∂tuν +a(uν−1)∂xuν = 0✱ s❡❣✉❡ s♦♠❛♥❞♦ ❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ q✉❡ N
X
i=1
(∂tuνi+λi(uν−1)∂xuνi)ri(uν−1) + N
X
i=1
uνi
h
a(uν−1)∂x(ri(uν−1)) +∂t(ri(uν−1))
i
= 0,
✐st♦ é✱ N
X
i=1
(∂tuνi +λi(uν−1)∂xuνi)ri(uν−1)
=−
N
X
i=1
uνi
h
a(uν−1)∂x(ri(uν−1)) +∂t(ri(uν−1))
i
. ✭✷✳✵✳✽✮
❖❧❤❡♠♦s ❛❣♦r❛ ♣❛r❛ ♦ t❡r♠♦ ❡♥tr❡ ❝♦❧❝❤❡t❡s ♥♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✳ ❚❡♠♦s
h
a(uν−1)∂x(ri(uν−1)) +∂t(ri(uν−1))
i
=a(uν−1) (ri′(uν−1)∂xuν−1) +r′i(uν−1)∂tuν−1,
❝♦♠♦uν−1 s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❡♠ ✭✷✳✵✳✺✮✱ s❡❣✉❡ q✉❡ ∂tuν−1 =−a(uν−2)∂xuν−1✱ ❛ss✐♠
h
a(uν−1)∂x(ri(uν−1)) +∂t(ri(uν−1))
i
=a(uν−1) (r′i(uν−1)∂xuν−1)
−r′i(uν−1) (a(uν−2)∂xuν−1), s❡❣✉❡✱ ❛❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ✭✷✳✵✳✼✮ q✉❡
h
a(uν−1)∂x(ri(uν−1)) +∂t(ri(uν−1))
i
=
N
X
l=1
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✶✾
s✉❜st✐t✉✐♥❞♦ ✐st♦ ❡♠ ✭✷✳✵✳✽✮ ♦❜t❡♠♦s
N
X
i=1
(∂tuνi+λi(uν−1)∂xuνi)ri(uν−1) =
−
N
X
i,l=1
uνiw(ν−1)1l{a(uν−1) [r′i(uν−1)rl(uν−2)]−ri′(uν−1) [a(uν−2)rl(uν−2)]}.
◆♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ❛♣❛r❡❝❡✱ ❡①♣❧✐❝✐t❛♠❡♥t❡✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r ❡s❝r✐t♦ ♥❛ ❜❛s❡ {r1(uν−1), . . . , rN(uν−1)}✳ P❛r❛ t✐r❛r♠♦s ✈❛♥t❛❣❡♠ ❞✐st♦✱ ✈❛♠♦s ❡s❝r❡✈❡r t❛♠❜é♠ ♥❡st❛ ❜❛s❡ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ r❡❢❡r✐❞❛ ✐❣✉❛❧❞❛❞❡✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♣❛r❛ ❝❛❞❛
i, j ∈ {1, . . . , N}❡s❝r❡✈❡♠♦s
− {a(uν−1) [ri′(uν−1)rl(uν−2)]−r′i(uν−1) [a(uν−2)rl(uν−2)]}= N
X
j=1
Φilj(uν−1, uν−2)rj(uν−1),
❛ss✐♠ t❡♠♦s
N
X
i=1
(∂tuνi+λi(uν−1)∂xuνi)ri(uν−1) = N
X
i,l,j=1
uνiw(ν−1)1lΦilj(uν−1, uν−2)rj(uν−1),
❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡
∂tuνj +λj(uν−1)∂xuνj = N
X
i,l=1
uνiw(ν−1)1lΦilj(uν−1, uν−2).
❈♦♥s✐❞❡r❛♥❞♦ ♦ ♦♣❡r❛❞♦r ❞❡ ♦r❞❡♠ ✉♠
Lνj−1v
.
=∂tv+λj(uν−1)∂xv, ✭✷✳✵✳✾✮
♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❝♦♠♦
Lνj−1uνj = N
X
i,l=1
uνiw(ν−1)1lΦilj(uν−1, uν−2). ✭✷✳✵✳✶✵✮
❊t❛♣❛ ✷✳ Pr♦❝❡ss♦ ✐t❡r❛t✐✈♦ ♣❛r❛ ❞❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ ✉♠✳ ◗✉❡r❡♠♦s ❛❣♦r❛ ♦❜t❡r ✉♠❛ ✐❣✉❛❧❞❛❞❡ s✐♠✐❧❛r ❛ ✭✷✳✵✳✶✵✮ ❛ q✉❛❧ ❡♥✈♦❧✈❛ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s wν1j ❞❡
∂xuν✳ P❛r❛ ✐st♦✱ ✐♥✐❝✐❛♠♦s ❞❡r✐✈❛♥❞♦ ❡♠ r❡❧❛çã♦ ❛ x ❛ ♣r✐♠❡✐r❛ ✐❣✉❛❧❞❛❞❡ ❡♠ ✭✷✳✵✳✺✮✱ ❢❛③❡♥❞♦ ✐st♦ ♦❜t❡♠♦s
∂t∂xuν +a(uν−1)∂x∂xuν =−(a′(uν−1)∂xuν−1)∂xuν, ✐st♦ é✱ ✉t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦ ✐♥tr♦❞✉③✐❞❛ ❡♠ ✭✷✳✵✳✼✮✱
∂twν1+a(uν−1)∂xwν1 =− a′(uν−1)w(ν−1)1
wν1. ✭✷✳✵✳✶✶✮ ❯s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ wν1 ❡♠ t❡r♠♦s ❞♦s ❛✉t♦✈❡t♦r❡s✱ ❞❛❞❛ ❡♠ ✭✷✳✵✳✼✮✱ ❡ q✉❡
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✷✵
∂twν1+a(uν−1)∂xwν1 =∂t N
X
i=1
wν1iri(uν−1)
!
+a(uν−1)∂x N
X
i=1
wν1iri(uν−1)
!
=
N
X
i=1
∂twν1iri(uν−1) +wν1iri′(uν−1)
−a(uν−2)w(ν−1)1
+
N
X
i=1
∂xwν1ia(uν−1)ri(uν−1) +wν1ia(uν−1)
r′
i(uν−1)w(ν−1)1
=
N
X
i=1
[∂twν1i+λi(uν−1)∂xwν1i]ri(uν−1)
+
N
X
i,l=1
wν1iw(ν−1)1l{a(uν−1) [r′i(uν−1)rl(uν−2)]−r′i(uν−1) [a(uν−2)rl(uν−2)]},
✐st♦ é✱
∂twν1+a(uν−1)∂xwν1 = N
X
i=1
[∂twν1i+λi(uν−1)∂xwν1i]ri(uν−1)
+
N
X
i,l=1
wν1iw(ν−1)1l{a(uν−1) [r′i(uν−1)rl(uν−2)]−ri′(uν−1) [a(uν−2)rl(uν−2)]}. ✭✷✳✵✳✶✷✮
❚❡♠♦s t❛♠❜é♠✱ ✉s❛♥❞♦ ❛s ❞❡❝♦♠♣♦s✐çõ❡s ❞❡ w(ν−1)1 ❡ wν1✱ q✉❡
a′(uν−1)w(ν−1)1
wν1 = N
X
i,l=1
wν1iw(ν−1)1l[a′(uν−1)rl(uν−2)]ri(uν−1). ✭✷✳✵✳✶✸✮
❉❡ ✭✷✳✵✳✶✶✮✱ ✭✷✳✵✳✶✷✮ ❡ ✭✷✳✵✳✶✸✮ s❡❣✉❡ q✉❡
N
X
i=1
(∂twν1i+λi(uν−1)wν1i)ri(uν−1) =− N
X
i,l=1
wν1iw(ν−1)1l{a(uν−1) [ri′(uν−1)rl(uν−2)]
−r′
i(uν−1) [a(uν−2)rl(uν−2)] + [a′(uν−1)rl(uν−2)]ri(uν−1)}. ❉❡❝♦♠♣♦♥❞♦ ❛❣♦r❛ ❝❛❞❛ ✈❡t♦r
− {a(uν−1) [ri′(uν−1)rl(uν−2)]−r′i(uν−1) [a(uν−2)rl(uν−2)] + [a′(uν−1)rl(uν−2)]ri(uν−1)}
❡♠ t❡r♠♦s ❞❛ ❜❛s❡ {r1(uν−1), . . . , rN(uν−1)} ♦❜t❡♠♦s ♦ ❛♥á❧♦❣♦ ❛ ✭✷✳✵✳✶✵✮ ❞❡s❡❥❛❞♦✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱
Lνj−1wν1j = N
X
i,l=1
wν1iw(ν−1)1iΦ˜ilj(uν−1, uν−2), ✭✷✳✵✳✶✹✮
♦♥❞❡Lν−1
j é ❞❛❞♦ ❡♠ ✭✷✳✵✳✾✮✳
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✷✶
M0ν(t)= sup.
i,x
|uνi(t, x)|
❡ ♣❛r❛k ≥1
Mkν(t)
.
= sup
i,x
|wνki(t, x)|.
❖❜s❡r✈❛çã♦ ✷✳✵✳✶ ➱ ✈á❧✐❞♦ ❛q✉✐ ❢❛③❡r♠♦s ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s ❛ r❡s♣❡✐t♦ ❞❡Mν
k✳ ❆❜❛✐①♦✱ ♣❛r❛ ✉♠ ✈❡t♦r u ∈ R✱ [u]β ❞❡♥♦t❛rá ❛ ♠❛tr✐③ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ u ♥❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ β ={e1, . . . , eN} ❡ [u]βv ❛ ♠❛tr✐③ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ u ♥❛ ❜❛s❡ βv = {r1(v), . . . , rN(v)}✱ ❝♦♥st✐t✉í❞❛ ♣❡❧♦s ❛✉t♦✈❡t♦r❡s ❞❡ a(v)✳ ❚❡♠♦s
[u]β =P(v) [u]βv ✭✷✳✵✳✶✺✮
♦♥❞❡ P(v) é ♠❛tr✐③ ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡ ❛ q✉❛❧ t❡♠ ❝♦♠♦ ✈❡t♦r❡s ❝♦❧✉♥❛ r1(v), . . . , rN(v)✳ ❙❡ |u|β ❡ |u|βv ❞❡♥♦t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ s✉♣r❡♠♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ u ♥❛ ❜❛s❡ β ❡
♥❛ ❜❛s❡ βv✱ ❡♥tã♦ |u|β ❡ |u|βv sã♦ ♥♦r♠❛s ❡♠ R
N✳ ❊s❝r❡✈❡♥❞♦ kP(v)k ♣❛r❛ ❛ ♥♦r♠❛ ❞❛ ♠❛tr✐③ P(v) (s✉♣r❡♠♦ ❞♦ ♠ó❞✉❧♦ ❞❛s ❡♥tr❛❞❛s ❞❡ P(v)) s❡❣✉❡ ❞❡ (2.0.15) q✉❡
|u|β ≤NkP(v)k |u|βv.
❈♦♠♦ P(v) é ✐♥✈❡rtí✈❡❧✱ t❡♠♦s t❛♠❜é♠
[u]βv = (P(v))−1[u]β,
❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡
|u|βv ≤N
(P(v))−1|u|β.
❙❡ r❡str✐♥❣✐r♠♦s v ❛ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ❞❛ ♦r✐❣❡♠✱ ❡♥tã♦ ❡①✐st❡♠
❝♦♥st❛♥t❡s C0 ❡ C1✱ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ v✱ ❞❡ ♠♦❞♦ q✉❡
C0|u|βv ≤ |u|β ≤C1|u|βv.
❆ss✐♠ s❡ ❣❛r❛♥t✐r♠♦s✱ ♣❛r❛ ❛❧❣✉♠ T >0✱ q✉❡ kuνkL∞([0,T]×R) ♣❡r♠❛♥❡❝❡ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ✭✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ν✮✱ ❡♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s C0 ❡ C1 ❞❡ ♠♦❞♦ q✉❡
C0 sup
i=1,...,N
{|uνi|} ≤ |uν| ≤C1 sup i=1,...,N
{|uνi|},
❧♦❣♦
C0M0ν(t)≤ kuν(t,·)k∞≤C1M0ν(t), ✭✷✳✵✳✶✻✮ ❡✱ ❝❧❛r♦ q✉❡✱ ♣❛r❛ ❛s ♠❡s♠❛s ❝♦♥st❛♥t❡s ✈❛❧❡
C0Mkν(t)≤
∂xkuν(t,·)
∞≤C1M
ν
k(t). ✭✷✳✵✳✶✼✮
✷✳✶ ❖ ❈❛s♦
C
b∞✳
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s✉♣♦♥❞♦ q✉❡ ♦ ❞❛❞♦ ✐♥✐❝✐❛❧ ❡stá ❡♠ C∞
b ✱ ✐st♦ é✱ q✉❡ ♦ ❞❛❞♦ ✐♥✐❝✐❛❧ ❡stá ❡♠C∞ ❡ t❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❧✐♠✐t❛❞❛s✳ ■♥✐❝✐❛♠♦s ♣❡❧♦
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✷✷
▲❡♠❛ ✷✳✶✳✶ ❙❡❥❛♠ ✉0 : R→ RN ♣❡rt❡♥❝❡♥t❡ ❛ Cb∞ ❡ (uν)ν≥−1 ❛ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ❡♠ ✭✷✳✵✳✹✮ ❡ ✭✷✳✵✳✺✮✳ ❙❡ k✉0k∞ é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s T > 0✱
C0 ❡ C1✱ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ν✱ ♣❛r❛ ❛s q✉❛✐s ✈❛❧❡♠✿
Mν
0(t)≤M0ν(0) exp
Z t
0
C0M1ν−1(s)
≤2M0
0(0) ∀ν ≥0, ∀t∈[0, T] ✭✷✳✶✳✶✮ ❡
Mν
1(t)≤M1ν(0) exp
Z t
0
C1M1ν−1(s)
≤2M10(0) ∀ν ≥0, ∀t∈[0, T]. ✭✷✳✶✳✷✮
❉❡♠♦♥str❛çã♦✳ ➱ ✐♠❡❞✐❛t♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ u0 q✉❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♠ ✭✷✳✶✳✶✮ ❡ ✭✷✳✶✳✷✮ ✈❛❧❡♠ ♣❛r❛ ν = 0✳ ❈♦♠♦ ❡st❛♠♦s ❛ss✉♠✐♥❞♦ k✉0k∞ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ✜❝❛ ❝❧❛r♦
q✉❡ ✭✷✳✵✳✻✮ ❡ ✭✷✳✵✳✼✮ ❢❛③❡♠ s❡♥t✐❞♦ ♣❛r❛ ν = 1✱ s❡♥❞♦ ❛ss✐♠✱ t❛♥t♦ ✭✷✳✵✳✶✵✮ ❝♦♠♦ ✭✷✳✵✳✶✹✮
sã♦ ✈á❧✐❞❛s✳ ■♥t❡❣r❛♥❞♦ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ ✭✷✳✵✳✶✵✮ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧✱
γ(t) = (t, γ2(t))✱ ❞♦ ❝❛♠♣♦ L0j✱ ♦❜t❡♠♦s
Z t
0
L0ju1j(γ(s))ds =
Z t
0
(u1j◦γ)′(s)ds
=u1j(t, γ2(t))−u1j(0, γ2(0)). ✭✷✳✶✳✸✮
❏á ♣❛r❛ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✷✳✵✳✶✵✮ t❡♠♦s
Z t
0 N
X
i,l=1
u1iw01lΦilj(u0, u−1)
!
(γ(s))ds
≤
N
X
i,l=1
Z t
0
|u1i(s, γ2(s))w01l(s, γ2(s))| |Φilj(u0, u−1)γ(s)|ds,
❡st✐♠❛♥❞♦|Φilj(u0, u−1)γ(s)|♣♦r ✉♠❛ ❝♦♥st❛♥t❡C✱ ❞❡ ♠♦❞♦ q✉❡ ♣❛r❛ t❛❧ ❝♦♥st❛♥t❡ ✈❛❧❤❛
sup
i,l,j
sup
v,v˜∈RN
|v|,|˜v|≤2M00 (0)
|Φilj(v,v˜)| ≤C, ✭✷✳✶✳✹✮
♣♦❞❡♠♦s ❡s❝r❡✈❡r
Z t
0 N
X
i,l=1
u1iw01lΦilj(u0, u−1)
!
(γ(s))ds
≤C0
Z t
0
M10(s)M01(s)ds, ✭✷✳✶✳✺✮
♣♦✐s✱|u1i(s, γ2(s))| ≤M01(s) ❡ |w01l(s, γ2(s))| ≤M11(s)✳ ❉❡ ✭✷✳✶✳✸✮ ❡ ✭✷✳✶✳✺✮ s❡❣✉❡ q✉❡
u1j(t, γ2(t))≤M01(0) +C0
Z t
0
M10(s)M01(s)ds. ✭✷✳✶✳✻✮
❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❧❤❛♥❞♦ ♣❛r❛ ✲✭✷✳✵✳✶✵✮✱ ♦❜t❡♠♦s
−u1j(t, γ2(t))≤M01(0) +C0
Z t
0
M0
1(s)M01(s)ds.
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✷✸
|u1j(t, γ2(t))| ≤M01(0) +C0
Z t
0
M10(s)M01(s)ds.
❈♦♠♦ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ j ❡ ❞❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ γ✱
♦❜t❡♠♦s t♦♠❛♥❞♦ ♦ s✉♣r❡♠♦ ❡♠γ ❡ j q✉❡
M01(t)≤M01(0) +C0
Z t
0
M10(s)M01(s)ds,
❞♦ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧✱ s❡❣✉❡ q✉❡
M01(t)≤M01(0) exp
Z t
0
C0M10(s)ds
. ✭✷✳✶✳✼✮
❆❣♦r❛✱ ✉s❛♥❞♦ ✭✷✳✵✳✶✹✮ ❛♦ ✐♥✈és ❞❡ ✭✷✳✵✳✶✵✮✱ ✈❛♠♦s ♦❜t❡r✱ ♣r♦❝❡❞❡♥❞♦ ❞❡ ♠♦❞♦ ✐♥t❡✐✲ r❛♠❡♥t❡ ❛♥á❧♦❣♦ ❛♦ ✉s❛❞♦ ♣❛r❛ ♣❛rt✐r ❞❡ ✭✷✳✵✳✶✵✮ ❡ ♦❜t❡r ✭✷✳✶✳✼✮✱ q✉❡
M11(t)≤M11(0) exp
Z t
0
C1M10(s)ds
. ✭✷✳✶✳✽✮
➱ ♣❡rt✐♥❡♥t❡ ♦❜s❡r✈❛r q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡C0 ❡C1 ✈❡♠ ❞❡ q✉❡✱ ♣❛r❛ ♦❜t❡r ✭✷✳✶✳✼✮ ✉s❛♠♦s ✭✷✳✶✳✹✮ ❡ ♣❛r❛ ♦❜t❡r ✭✷✳✶✳✽✮ ✉s❛♠♦s ♦ ❛♥á❧♦❣♦ ❛ ✭✷✳✶✳✹✮
sup
i,l,j
sup
v,v˜∈RN
|v|,|˜v|≤2M00 (0)
|Φ˜ilj(v,v˜)| ≤C. ✭✷✳✶✳✾✮
❖ ✜♥❛❧ ❞❛ ❞❡♠♦♥str❛çã♦ ❝♦♥s✐st❡ ❛❣♦r❛✱ ❜❛s✐❝❛♠❡♥t❡✱ ❡♠ ❡s❝♦❧❤❡r T✳ ❉❡ ❢❛t♦✱ s❡
✜①❛r♠♦sT > 0❞❡ ♠♦❞♦ q✉❡
exp(T max{C0, C1}2M10(0)) <2,
❡♥tã♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♠ ✭✷✳✶✳✶✮ ❡ ✭✷✳✶✳✷✮ s❡❣✉❡♠ ♣❛r❛ν = 1 ❞❡ ✭✷✳✶✳✼✮ ❡ ✭✷✳✶✳✽✮✱ ♣♦✐s✱
M0
1(t) =M10(0) ∀t∈R✳ ❆❣♦r❛ s❡ s✉♣♦r♠♦s ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♠ ✭✷✳✶✳✶✮ ❡ ✭✷✳✶✳✷✮ ✈á❧✐❞❛s ♣❛r❛ t♦❞♦ 0 ≤ ν0 ≤ ν ❡ t ∈ [0, T]✱ t❡r❡♠♦s✱ ❡♠ ❡s♣❡❝✐❛❧✱ q✉❡ ❛s ♥♦t❛çõ❡s ❡♠ ✭✷✳✵✳✻✮ ❡ ✭✷✳✵✳✼✮ ❢❛③❡♠ s❡♥t✐❞♦ ♣❛r❛ν+ 1✱ ❧♦❣♦ sã♦ ✈á❧✐❞❛s ✭✷✳✵✳✶✵✮ ❡ ✭✷✳✵✳✶✹✮✱ ♣❛r❛ν+ 1✳ ❆ss✐♠✱ ♦s
❝á❧❝✉❧♦s ❛❝✐♠❛ r♦❞❛♠ ❝♦♠ν ❡ν+ 1 ♥♦ ❧✉❣❛r ❞❡ 0 ❡1✳ ❊❢❡t✉❛♥❞♦ t❛✐s ❝á❧❝✉❧♦s✱ ♦❜t❡♠♦s
M0ν+1(t)≤M0ν(0) exp
Z t
0
C0M1ν(s)ds
❡
Mν+1
1 (t)≤M1ν(0) exp
Z t
0
C1M1ν(s)ds
.
❈♦♠♦ ♣❛r❛ν t❡♠♦s Mν
1(s)≤2M10(s) ∀s∈[0, T] ✭❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✮✱ s❡❣✉❡ ❞❛ ❡s❝♦❧❤❛ ❞❡T q✉❡
Mν+1
0 (t)≤2M0ν(0) ❡
Mν+1
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✷✹
❖❜s❡r✈❛çã♦ ✷✳✶✳✶ ❉❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✶✳✶ é ❝❧❛r♦ q✉❡ s❡ ✉0 ❡ ✉˜0 sã♦ t❛✐s q✉❡
k✉˜0k∞≤ k✉0k∞ ❡ k✉˜′0k∞≤ k✉′0k∞✱ ❡♥tã♦ s❡ ♦ ▲❡♠❛ ✷✳✶✳✶ ✈❛❧❡ ♣❛r❛ ✉0 ❡♠ ✉♠ ✐♥t❡r✈❛❧♦
[0, T] ❡ ❝♦♥st❛♥t❡s C0 ❡ C1 ❡❧❡✱ ♦ ❧❡♠❛✱ t❛♠❜é♠ ✈❛❧❡rá ♥♦ ♠❡s♠♦ ✐♥t❡r✈❛❧♦ ❡ ❝♦♠ ❛s ♠❡s♠❛s ❝♦♥st❛♥t❡s ♣❛r❛ ✉˜0✳ ■st♦ é✱ ♦ ✐♥t❡r✈❛❧♦ [0, T] ❡ ❛s ❝♦♥st❛♥t❡s C0 ❡ C1 ❞❡♣❡♥❞❡♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞❛ ♥♦r♠❛ ❞♦ ❞❛❞♦ ✐♥✐❝✐❛❧ ❡ ❞❛ ♥♦r♠❛ ❞❡ s✉❛ ❞❡r✐✈❛❞❛✳
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ r❡❧❛❝✐♦♥❛ ❛ ♥♦r♠❛ ❞❛s ❞❡r✐✈❛❞❛s ❞♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛(uν)ν≥−1✱ ❞❡✜♥✐❞❛ ❡♠ ✭✷✳✵✳✹✮ ❡ ✭✷✳✵✳✺✮✱ ❝♦♠ ❛ ♥♦r♠❛ ❞❛s ❞❡r✐✈❛❞❛s ❡♠ r❡❧❛çã♦ ❛ ✈❛r✐á✈❡❧x✳
Pr♦♣♦s✐çã♦ ✷✳✶✳✶ ❙❡❥❛(uν)ν≥−1 ❛ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ❡♠ ✭✷✳✵✳✹✮ ❡ ✭✷✳✵✳✺✮✳ ❙❡ ❛ s❡q✉ê♥✲ ❝✐❛ (kuνk∞)ν≥−1 é ❧✐♠✐t❛❞❛ ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s C1, . . . , Ck ♣♦s✐t✐✈❛s✱ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡
ν✱ ❞❡ ♠♦❞♦ q✉❡
k∂l
xuνk∞≤Clk✉0(l)k∞ ∀ν ≥0,
❡♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s C˜1, . . . ,C˜k ♣♦s✐t✐✈❛s✱ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ν✱ ♣❛r❛ ❛s q✉❛✐s ✈❛❧❡♠
k∂α
uνk∞≤C|˜α|k✉(0|α|)k∞
♣❛r❛ t♦❞♦ ♠✉❧t✐✲í♥❞✐❝❡ α t❛❧ q✉❡ 1≤ |α| ≤k✳
❉❡♠♦♥str❛çã♦✳ ❊st❡ r❡s✉❧t❛❞♦ s❡❣✉❡ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✶✳✸✳✺ ❡ ♦ ❢❛t♦ ❞❡ q✉❡ ❝❛❞❛ uν é s♦❧✉çã♦ ❞❡ ✭✷✳✵✳✺✮✳
Pr♦♣♦s✐çã♦ ✷✳✶✳✷ ❙❡ (uν)ν≥−1 ❡ ✉0 s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞♦ ▲❡♠❛ ✷✳✶✳✶✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ k ∈N ❡①✐st❡♠ Tk ❡ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s Ck ❡ C˜k ❞❡ ♠♦❞♦ q✉❡ ✈❛❧❡
Mν
k(t)≤CkMkν(0) exp(tC˜k) ∀ν≥0 ❡ t∈[0, Tk]. ✭✷✳✶✳✶✵✮ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s C˜˜0, . . . ,C˜˜k ❞❡ ♠♦❞♦ q✉❡
k∂αu
νk∞≤C|˜˜α|k✉(| α|)
0 k∞
♣❛r❛ t♦❞♦ ♠✉❧t✐✲í♥❞✐❝❡ α t❛❧ q✉❡ 1≤ |α| ≤k✳
❉❡♠♦♥str❛çã♦✳ ❖ ▲❡♠❛ ✷✳✶✳✶ ❣❛r❛♥t❡ q✉❡ ❛ ❡st✐♠❛t✐✈❛ ❡♠ ✭✷✳✶✳✶✵✮ ✈❛❧❡ ♣❛r❛ k = 0,1✳
❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ ✭✷✳✶✳✶✵✮ ✈❛❧❡ ♣❛r❛ t♦❞♦0 ≤ k0 ≤ k−1✱ ❝♦♠
k≥2✳ Pr❡❝✐s❛♠♦s ♠♦str❛r q✉❡ ✭✷✳✶✳✶✵✮ ✈❛❧❡ ♣❛r❛k✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❛ s❡q✉ê♥❝✐❛(uν)ν≥−1 t❡♠♦s ♣❛r❛ν ≥1
∂tuν +a(uν−1)∂xuν = 0.
❉✐❢❡r❡♥❝✐❛♥❞♦ t❛❧ ✐❣✉❛❧❞❛❞❡k✲✈❡③❡s ❡♠ r❡❧❛çã♦ ❛ x✱ ♦❜t❡♠♦s✱ ✈✐❛ ❢ór♠✉❧❛ ❞❡ ▲❡✐❜♥✐③ ♣❛r❛
❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦✱ q✉❡
∂t∂xkuν+a(uν−1)∂x∂xkuν =−k a′(uν−1)∂xuν−1
∂xkuν−∂kx a(uν−1)
∂xuν
−
k−1
X
m=2
k m
❈❛♣ít✉❧♦ ✷✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② q✉❛s❡✲❧✐♥❡❛r é ❜❡♠ ♣♦st♦ ❡♠ Cρ ✷✺
♥♦ ❝❛s♦ k = 2 ❛ ú❧t✐♠❛ ♣❛r❝❡❧❛ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ✭♦ s♦♠❛tór✐♦✮ ♥ã♦ ❛♣❛r❡❝❡✱ ❡st❛r❡♠♦s
♣♦rt❛♥t♦ ❝♦♥✈❡♥❝✐♦♥❛♥❞♦ q✉❡ P1
m=2(· · ·) = 0✳ ❯t✐❧✐③❛♥❞♦ ❛❣♦r❛ ♦ ❚❡♦r❡♠❛ ✶✳✸✳✶ t❡♠♦s
∂t∂xkuν +a(uν−1)∂x∂xkuν =−k a′(uν−1)∂xuν−1
∂xkuν
−h
k
X
l=1
n(i1, . . . , il)a(l)(uν−1)
∂i1
x uν−1, . . . , ∂xiluν−1
i
∂xuν
−
k−1
X
m=2
k m
"Xm
τ=1
n(j1, . . . , jτ)a(τ)(uν−1)
∂j1
x uν−1, . . . , ∂xjτuν−1
#
∂xk+1−muν
◆♦t❛♥❞♦ q✉❡ ♣❛r❛l= 1 ❛ ú♥✐❝❛ ♣❛rt✐çã♦ ♣❛r❛k =i1 é ❝♦♠i1 =k ❡ q✉❡ ♥❡st❡ ❝❛s♦ t❡♠♦s
n(i1) = 1✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡♠♦s
∂t∂xkuν +a(uν−1)∂x∂xkuν =−k a′(uν−1)∂xuν−1
∂xkuν − a′(uν−1)∂xkuν−1
∂xuν
−h
k
X
l=2
n(i1, . . . , il)a(l)(uν−1)
∂i1
xuν−1, . . . , ∂xiluν−1
i
∂xuν
−
k−1
X
m=2
k m
"Xm
τ=1
n(j1, . . . , jτ)a(τ)(uν−1)
∂j1
x uν−1, . . . , ∂xjτuν−1
#
∂xk+1−muν.
❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ∂k
xuν−1 ❡ ∂xkuν ❡♠ t❡r♠♦s ❞♦s ❛✉t♦✈❡t♦r❡s ❞❛❞❛ ❡♠ ✭✷✳✵✳✼✮✱ ♦❜t❡♠♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ q✉❡
N
X
i=1
∂twνki+λi(uν−1)∂xwνki
ri(uν−1) =
− N X i=1 wνki h
∂t ri(uν−1)
+a(uν−1)∂x ri(uν−1)
+k a′(u
ν−1)∂xuν−1
ri(uν−1)
i
−
N
X
i=1
w(ν−1)ki a′(uν−1)ri(uν−2)
∂xuν
−h
k
X
l=2
n(i1, . . . , il)a(l)(uν−1)
∂i1
xuν−1, . . . , ∂xiluν−1
i
∂xuν
−
k−1
X
m=2
k m
"Xm
τ=1
n(j1, . . . , jτ)a(τ)(uν−1)
∂j1
x uν−1, . . . , ∂xjτuν−1
#
∂kx+1−muν,
✜♥❛❧♠❡♥t❡✱ ❡s❝r❡✈❡♥❞♦
N
X
i=1
w(ν−1)ki a′(uν−1)ri(uν−2)
∂xuν = N
X
i,l=1
w(ν−1)kiwν1l a′(uν−1)ri(uν−2)
rl(uν−1)
