Một số tính chất định tính của phương trình navier stokes

❱■❊❚◆❆▼ ❆❈❆❉❊▼❨ ❖❋ ❙❈■❊◆❈❊ ❆◆❉ ❚❊❈❍◆❖▲❖●❨
■◆❙❚■❚❯❚❊ ❖❋ ▼❆❚❍❊▼❆❚■❈❙

❉❆❖ ◗❯❆◆● ❑❍❆■

❙❖▼❊ ◗❯❆▲■❚❆❚■❱❊ P❘❖P❊❘❚■❊❙ ❖❋ ❙❖▲❯❚■❖◆❙
❚❖ ◆❆❱■❊❘✲❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙

❉❖❈❚❖❘❆▲ ❉■❙❙❊❘❚❆❚■❖◆ ■◆ ▼❆❚❍❊▼❆❚■❈❙

❍❆◆❖■ ✷✵✶✼


❱■❊❚◆❆▼ ❆❈❆❉❊▼❨ ❖❋ ❙❈■❊◆❈❊ ❆◆❉ ❚❊❈❍◆❖▲❖●❨
■◆❙❚■❚❯❚❊ ❖❋ ▼❆❚❍❊▼❆❚■❈❙

❉❆❖ ◗❯❆◆● ❑❍❆■

❙❖▼❊ ◗❯❆▲■❚❆❚■❱❊ P❘❖P❊❘❚■❊❙ ❖❋ ❙❖▲❯❚■❖◆❙
❚❖ ◆❆❱■❊❘✲❙❚❖❑❊❙ ❊◗❯❆❚■❖◆❙
❙♣❡❝✐❛❧✐t②✿ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s
❙♣❡❝✐❛❧✐t② ❝♦❞❡✿ ✻✷ ✹✻ ✵✶ ✵✸

❉❖❈❚❖❘❆▲ ❉■❙❙❊❘❚❆❚■❖◆ ■◆ ▼❆❚❍❊▼❆❚■❈❙

❙✉♣❡r✈✐s♦r✿ Pr♦❢✳ ❉r✳ ❙❝✳ ◆❣✉②❡♥ ▼✐♥❤ ❚r✐

❍❆◆❖■ ✷✵✶✼


VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM

VIỆN TOÁN HỌC

ĐÀO QUANG KHẢI

MỘT SỐ TÍNH CHẤT ĐỊNH TÍNH
CỦA NGHIỆM PHƯƠNG TRÌNH NAVIER-STOKES

Chuyên ngành: Phương trình vi phân và tích phân
Mã ngành: 62 46 01 03

LUẬN ÁN TIẾN SĨ

Người hướng dẫn khoa học: GS. TSKH. Nguyễn Minh Trí

HÀ NỘI 2017


❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts
■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ♠② ❛❞✈✐s♦r Pr♦❢❡ss♦r ◆❣✉②❡♥ ▼✐♥❤ ❚r✐✳ ❍❡ s❤❛r❡❞ ❤✐s
t✐♠❡ ❛♥❞ ♣r♦❢♦✉♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ❦♥♦✇❧❡❞❣❡ ✇✐t❤ ♠❡ ❛♥❞ ❣❛✈❡ ♠❡ s♦♠❡ ♥❡❝❡ss❛r②
❜❛❝❦❣r♦✉♥❞ ♦♥ t❤❡ ✜❡❧❞ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ♣r♦❜❧❡♠s✳ ■ ❛❧s♦ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦
❤✐♠ ❢♦r ❝♦rr❡❝t✐♥❣ ♠② ❊♥❣❧✐s❤ ❛♥❞ t❤❡ ♠✐st❛❦❡s ✐♥ ✇r✐t✐♥❣ t❤❡ ♣❛♣❡rs ❛♥❞ ❞✐ss❡rt❛✲
t✐♦♥✳
■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ Pr♦❢❡ss♦rs ❍❛ ❚✐❡♥ ◆❣♦❛♥ ❛♥❞ ❉✐♥❤ ◆❤♦ ❍❛♦ ❢♦r t❤❡✐r
❝❛r❡❢✉❧ r❡❛❞✐♥❣ ♦❢ t❤❡ ♠❛♥✉s❝r✐♣t ♦❢ ♠② ❞✐ss❡rt❛t✐♦♥ ❛♥❞ ❢♦r t❤❡✐r ❝♦♥str✉❝t✐✈❡
❝♦♠♠❡♥ts ❛♥❞ ✈❛❧✉❛❜❧❡ s✉❣❣❡st✐♦♥s✳
■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ♠② ✐♥st✐t✉t✐♦♥✱ t❤❡ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❢♦r
♣r♦✈✐❞✐♥❣ ♠❡ ❡♥❝♦✉r❛❣❡♠❡♥t ❛♥❞ ✜♥❛♥❝✐❛❧ s✉♣♣♦rt t❤r♦✉❣❤♦✉t ♠② P❤✳ ❉✳ st✉❞✲
✐❡s✳ ❚❤✐s ❞✐ss❡rt❛t✐♦♥ ✇♦✉❧❞ ♥❡✈❡r ❤❛✈❡ ❜❡❡♥ ❝♦♠♣❧❡t❡❞ ✇✐t❤♦✉t t❤❡✐r ❣✉✐❞❛♥❝❡ ❛♥❞
❡♥❞❧❡ss s✉♣♣♦rt✳

■ ✇♦✉❧❞ ♥♦t ❤❛✈❡ ❜❡❝♦♠❡ t❤❡ ♦♥❡ ■ ❛♠ t♦❞❛② ✇✐t❤♦✉t t❤❡ ❤❡❧♣ ❛♥❞ ❣✉✐❞❛♥❝❡ ♦❢ ♠②
❢❛♠✐❧②✳ ❚❤❛♥❦ ②♦✉ ▼❛♠♠❛ ❛♥❞ P❛♣❛ ❢♦r ❛❧✇❛②s ❜❡❧✐❡✈✐♥❣ ✐♥ ♠❡✱ s✉♣♣♦rt✐♥❣ ♠❡✱ ❛♥❞
❡♥❝♦✉r❛❣✐♥❣ ♠❡✳ ▼② s♣❡❝✐❛❧ ❣r❛t✐t✉❞❡ ❣♦❡s t♦ ♠② ✇✐❢❡ ❢♦r ❧♦✈❡ ❛♥❞ ❡♥❝♦✉r❛❣❡♠❡♥t✳


❈♦♥✜r♠❛t✐♦♥
❚❤✐s ✇♦r❦ ❤❛s ❜❡❡♥ ❝♦♠♣❧❡t❡❞ ❛t ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢
❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✉♥❞❡r t❤❡ s✉♣❡r✈✐s✐♦♥ ♦❢ Pr♦❢✳ ❉r✳ ❙❝✳ ◆❣✉②❡♥ ▼✐♥❤ ❚r✐✳ ■
❞❡❝❧❛r❡ ❤❡r❡❜② t❤❛t t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ✐♥ t❤✐s t❤❡s✐s ❛r❡ ♥❡✇ ❛♥❞ ❤❛✈❡ ♥❡✈❡r ❜❡❡♥
♣✉❜❧✐s❤❡❞ ❡❧s❡✇❤❡r❡✳
❆✉r❤♦r✿ ❉❛♦ ◗✉❛♥❣ ❑❤❛✐


Tóm tắt
Trong luận án này, chúng tôi sử dụng những tiến bộ đạt được trong lĩnh vực giải tích
điều hòa trong mười lăm năm gần đây để nghiên cứu phương trình Navier-Stokes.
Chúng tôi muốn nói đến việc sử dụng biến đổi Fourier và các tính chất của nó, phù
hợp hơn cho việc nghiên cứu các bài toán phi tuyến.
Chương 1 được dành cho việc nhắc lại một số kết quả đã biết về giải tích điều hòa.
Trong Chương 2, chúng tôi xây dựng và nghiên cứu các không gian Sobolev sau
(không gian Sobolev trên một không gian Banach bất biến với phép dịch chuyển):
- Không gian Sobolev không thuần nhất và không gian Sobolev thuần nhất trên
các không gian Lebesgue.
- Không gian Sobolev thuần nhất trên các không gian Fourier-Lorentz.
- Không gian Sobolev thuần nhất trên các không gian Lorentz.
- Không gian Sobolev thuần nhất trên các không gian với chuẩn Lorentz hỗn
hợp.
Trong các không gian này, chúng tôi chứng minh một số bất đẳng thức kiểu
Young cho tích chập của hai hàm, một số bất đẳng thức kiểu Holder cho tích thông
thường giữa hai hàm và một số bất đẳng thức kiểu Sobolev. Chúng tôi áp dụng

những bất đẳng thức này để nghiên cứu bài toán Cauchy cho phương trình NavierStokes. Chúng tôi xây dựng nghiệm mềm cho phương trình Navier-Stokes trong
những không gian này bằng nguyên lý ánh xạ co Picard và chỉ ra rằng phương trình
Navier-Stokes được đặt chỉnh trong các không gian này theo nghĩa Hadarmard.
Chúng tôi chứng minh sự tồn tại toàn cục và duy nhất của nghiệm mềm khi giá trị
ban đầu đủ nhỏ và sự tồn tại địa phương của nghiệm mềm đối với giá trị ban đầu
tùy ý. Những kết quả thu được có một quan hệ chặt chẽ giữa thời gian tồn tại và
độ lớn của dữ liệu ban đầu: Thời gian lớn với dữ liệu ban đầu nhỏ hoặc dữ liệu ban
đầu lớn với thời gian nhỏ.
Trong Chương 3, sử dụng phương pháp của Foias-Temam, chúng tôi nghiên cứu
số chiều Hausdorff của tập hợp các điểm kỳ dị theo thời gian của nghiệm yếu của
phương trình Navier-Stokes trên hình xuyến 3 chiều.


❆❜str❛❝t
■♥ t❤✐s t❤❡s✐s✱ ✇❡ ✉s❡ t❤❡ ♣r♦❣r❡ss ❛❝❤✐❡✈❡❞ ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ❢♦r t❤❡
❧❛st ✜❢t❡❡♥ ②❡❛rs t♦ st✉❞② t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ◆❛♠❡❧②✱ ✇❡ ✉s❡ t❤❡ t♦♦❧s ♦❢
❋♦✉r✐❡r ❆♥❛❧②s✐s ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐♥ ♦r❞❡r t♦ st✉❞② t❤❡ ◆❛✈✐❡r✲
❙t♦❦❡s ❡q✉❛t✐♦♥s✳
❈❤❛♣t❡r ✶ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ r❡❝❛❧❧✐♥❣ ♦❢ s♦♠❡ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧ts ♦❢ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s✳
■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥❞ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ ❙♦❜♦❧❡✈ s♣❛❝❡s ✭❙♦❜♦❧❡✈ s♣❛❝❡s
♦✈❡r ❛ s❤✐❢t✲✐♥✈❛r✐❛♥t ❇❛♥❛❝❤ s♣❛❝❡✮✿
✲ ■♥❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s ♦✈❡r t❤❡
▲❡❜❡s❣✉❡ s♣❛❝❡s✳
✲ ❍♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s ♦✈❡r t❤❡ ❋♦✉r✐❡r✲▲♦r❡♥t③ s♣❛❝❡s✳
✲ ❍♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s ♦✈❡r t❤❡ ▲♦r❡♥t③ s♣❛❝❡s✳
✲ ❍♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s ♦✈❡r t❤❡ ♠✐①❡❞✲♥♦r♠ ▲♦r❡♥t③ s♣❛❝❡s✳
■♥ t❤❡s❡ s♣❛❝❡s✱ ✇❡ ♣r♦✈❡ s♦♠❡ ✈❡rs✐♦♥s ♦❢ ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② t②♣❡ ❢♦r ❝♦♥✈♦❧✉t✐♦♥s
♦❢ t✇♦ ❢✉♥❝t✐♦♥s✱ s♦♠❡ ✈❡rs✐♦♥s ♦❢ ❍♦❧❞❡r✬s ✐♥❡q✉❛❧✐t② t②♣❡ ❢♦r ♣♦✐♥t✲✇✐s❡ ♣r♦❞✉❝t ♦❢
t✇♦ ❢✉♥❝t✐♦♥s✱ ❛♥❞ s♦♠❡ ✈❡rs✐♦♥s ♦❢ ❙♦❜♦❧❡✈✬s ✐♥❡q✉❛❧✐t②✳ ❲❡ ❛♣♣❧② t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s
t♦ st✉❞② ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❲❡ ❝♦♥str✉❝t ♠✐❧❞

s♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐♥ t❤❡s❡ s♣❛❝❡s ❜② ❛♣♣❧②✐♥❣ t❤❡ P✐❝❛r❞
❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ❛♥❞ s❤♦✇ t❤❛t ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❛r❡ ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡s❡
s♣❛❝❡s ✐♥ t❤❡ s❡♥s❡ ♦❢ ❍❛❞❛r♠❛r❞✳ ❲❡ ♣r♦✈❡ t❤❡ ✉♥✐q✉❡ ❣❧♦❜❛❧ ❡①✐st❡♥❝❡ ♦❢ ♠✐❧❞
s♦❧✉t✐♦♥s ✇❤❡♥ t❤❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐s s♠❛❧❧ ❡♥♦✉❣❤ ❛♥❞ t❤❡ ❧♦❝❛❧ ❡①✐st❡♥❝❡ ♦❢ ♠✐❧❞
s♦❧✉t✐♦♥s ❢♦r ❛♥ ❛r❜✐tr❛r② ✐♥✐t✐❛❧ ✈❛❧✉❡✳ ❚❤❡ r❡s✉❧ts ❤❛✈❡ ❛ st❛♥❞❛r❞ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥
❡①✐st❡♥❝❡ t✐♠❡ ❛♥❞ ❞❛t❛ s✐③❡✿ ❧❛r❣❡ t✐♠❡ ✇✐t❤ s♠❛❧❧ ❞❛t❛ ♦r ❧❛r❣❡ ❞❛t❛ ✇✐t❤ s♠❛❧❧
t✐♠❡✳
■♥ ❈❤❛♣t❡r ✸✱ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❋♦✐❛s✲❚❡♠❛♠✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ❍❛✉s❞♦r✛
❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s✐♥❣✉❧❛r s❡t ✐♥ t✐♠❡ ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s
♦♥ t❤❡ ✸❉ t♦r✉s✳


❈♦♥t❡♥ts
■♥tr♦❞✉❝t✐♦♥

✹

✶ Pr❡❧✐♠✐♥❛r✐❡s

✽

✶✳✶

✶✳✷
✶✳✸

❙♦♠❡ r❡s✉❧ts ♦❢ r❡❛❧ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✶✳

▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❞❡❝♦♠♣♦s✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✷✳

❇❡s♦✈ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✶✳✸✳

❖t❤❡r ✉s❡❢✉❧ ❢✉♥❝t✐♦♥ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✶✳✹✳

▼♦rr❡②✲❈❛♠♣❛♥❛t♦ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✶✳✺✳

▲♦r❡♥t③ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

❊❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣r❡ss✉r❡ ❛♥❞ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s
❡q✉❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✹

❙❝❛❧✐♥❣ ✐♥✈❛r✐❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✺

❖✉t❧✐♥❡ ♦❢ t❤❡ ❞✐ss❡rt❛t✐♦♥

✶✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ▼✐❧❞ s♦❧✉t✐♦♥s ✐♥ s♦♠❡ ❙♦❜♦❧❡✈ s♣❛❝❡s ♦✈❡r ❛ s❤✐❢t✲✐♥✈❛r✐❛♥t ❇❛♥❛❝❤ s♣❛❝❡ ✶✾

✷✳✶

❙♦❜♦❧❡✈ s♣❛❝❡s ♦✈❡r ❛ s❤✐❢t✲✐♥✈❛r✐❛♥t ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✷

▼✐❧❞ s♦❧✉t✐♦♥s ✐♥ ❝r✐t✐❝❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✷✳✷✳✶✳

❙♦♠❡ ❛✉①✐❧✐❛r② r❡s✉❧ts

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✷✳✷✳✷✳

❖♥ t❤❡ ❝♦♥t✐♥✉✐t② ❛♥❞ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❜✐❧✐♥❡❛r ♦♣❡r❛t♦r ❇ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✷✳✸✳

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
d
d
−1
−1
q
❝r✐t✐❝❛❧ s♣❛❝❡s Hq
(Rd ) ❛♥❞ H˙ qq (Rd ) ❢♦r 3 ≤ d ≤ 4, 2 ≤ q ≤ d ✳ ✳

✷✼

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
d
˙ qq −1 (Rd ) ❢♦r d ≥ 3 ❛♥❞ 2 < q ≤ d ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❝r✐t✐❝❛❧ s♣❛❝❡s H

✸✶

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
d
˙ qq −1 (Rd ) ❢♦r d ≥ 3 ❛♥❞ 1 < q ≤ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❝r✐t✐❝❛❧ s♣❛❝❡s H

✸✸

❈♦♥❝❧✉s✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹

▼✐❧❞ s♦❧✉t✐♦♥s ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s ♦❢ ♥❡❣❛t✐✈❡ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✷✳✷✳✹✳

✷✳✷✳✺✳

✷✳✷✳✻✳
✷✳✸

✶


✷
✷✳✸✳✶✳

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
˙ ps (Rd ) ❢♦r d ≥ 2, p > d , and d − 1 ≤ s < d ✳ ✳ ✳ ✳
❙♦❜♦❧❡✈ s♣❛❝❡s H
2
p
2p

✸✺

❈♦♥❝❧✉s✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

▼✐❧❞ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❙♦❜♦❧❡✈✲❋♦✉r✐❡r✲▲♦r❡♥t③ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

✷✳✹✳✶✳

❙♦❜♦❧❡✈✲❋♦✉r✐❡r✲▲♦③❡♥t③ ❙♣❛❝❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹

✷✳✹✳✷✳

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
d
−1
˙ pp,r
❝r✐t✐❝❛❧ s♣❛❝❡s H
(Rd ) ✇✐t❤ 1 < p ≤ d ❛♥❞ 1 ≤ r < ∞ ✳ ✳ ✳ ✳ ✳ ✳
L

✹✽

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
d
−1

˙ pp,r
❝r✐t✐❝❛❧ s♣❛❝❡s H
(Rd ) ✇✐t❤ d ≤ p < ∞ ❛♥❞ 1 ≤ r < ∞ ✳ ✳ ✳ ✳ ✳ ✳
L

✺✺

❝r✐t✐❝❛❧ s♣❛❝❡s

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
d
H˙ Ld−1
1,r (R ) ✇✐t❤ 1 ≤ r < ∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✾

❈♦♥❝❧✉s✐♦♥s

✻✸

✷✳✸✳✷✳
✷✳✹

✷✳✹✳✸✳

✷✳✹✳✹✳

✷✳✹✳✺✳
✷✳✺

▼✐❧❞ s♦❧✉t✐♦♥s ✐♥ ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✹

✷✳✺✳✶✳

❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✹

✷✳✺✳✷✳

❆✉①✐❧✐❛r② s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✺

✷✳✺✳✸✳

❖♥ t❤❡ ❝♦♥t✐♥✉✐t② ❛♥❞ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❜✐❧✐♥❡❛r ♦♣❡r❛t♦r

✻✽

✷✳✺✳✹✳

✳ ✳ ✳ ✳ ✳ ✳

❙♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ t❤❡
❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✶

❈♦♥❝❧✉s✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✹

▼✐❧❞ s♦❧✉t✐♦♥s ✐♥ ♠✐①❡❞✲♥♦r♠ ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✹

✷✳✻✳✶✳

▼✐①❡❞✲♥♦r♠ ▲♦r❡♥t③ s♣❛❝❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✺

✷✳✻✳✷✳

▼✐①❡❞✲♥♦r♠ ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✼

✷✳✻✳✸✳

Lp Lq,r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✾

✷✳✻✳✹✳

❯♥✐q✉❡♥❡ss t❤❡♦r❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✸

✷✳✻✳✺✳

❈♦♥❝❧✉s✐♦♥s

✽✹

✷✳✺✳✺✳
✷✳✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

s♦❧✉t✐♦♥s ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s❡t ♦❢ s✐♥❣✉❧❛r✐t✐❡s ❢♦r ✇❡❛❦ s♦❧✉t✐♦♥s
✸✳✶

❋✉♥❝t✐♦♥❛❧ s❡tt✐♥❣ ♦❢ t❤❡ ❡q✉❛t✐♦♥s

✸✳✷

❲❡❛❦ s♦❧✉t✐♦♥s ✐♥

Lr H α

✸✳✸

❲❡❛❦ s♦❧✉t✐♦♥s ✐♥

Lr W 1,q

✽✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✸

●❡♥❡r❛❧ ❈♦♥❝❧✉s✐♦♥s

✾✽

▲✐st ♦❢ t❤❡ ❛✉t❤♦r✬s ♣✉❜❧✐❝❛t✐♦♥s r❡❧❛t❡❞ t♦ t❤❡ ❞✐ss❡rt❛t✐♦♥

✾✾

❇✐❜❧✐♦❣r❛♣❤②

✶✵✶


✸

❋✉♥❝t✐♦♥ ❙♣❛❝❡s
Rd
Hqs
H˙ qs
S
S
Bqs,p
B˙ qs,p
Fqs,p
F˙ qs,p
M p,q
M˙ p,q
Lp,q
H˙ Ls q,r
H˙ Ls p,r
H˙ s q,r
L

d✲❞✐♠❡♥s✐♦♥❛❧

❊✉❝❧✐❞❡❛♥ s♣❛❝❡

■♥❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s
❍♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s
❚❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥
❙❝❤✇❛rt③ ❝❧❛ss
■♥❤♦♠♦❣❡♥❡♦✉s ❇❡s♦✈ s♣❛❝❡s
❍♦♠♦❣❡♥❡♦✉s ❇❡s♦✈ s♣❛❝❡s
■♥❤♦♠♦❣❡♥❡♦✉s ❚r✐❡❜❡❧✲▲✐③♦r❦✐♥ s♣❛❝❡s
❍♦♠♦❣❡♥❡♦✉s ❚r✐❡❜❡❧✲▲✐③♦r❦✐♥ s♣❛❝❡s
■♥❤♦♠♦❣❡♥❡♦✉s ▼♦rr❡②✲❈❛♠♣❛♥❛t♦ s♣❛❝❡s
❍♦♠♦❣❡♥❡♦✉s ▼♦rr❡②✲❈❛♠♣❛♥❛t♦ s♣❛❝❡s
▲♦r❡♥t③ s♣❛❝❡s
❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s
❙♦❜♦❧❡✈✲❋♦✉r✐❡r✲▲♦r❡♥t③ s♣❛❝❡s
❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s

◆♦t❛t✐♦♥
◆❙❊

◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s

u X
[x]
{x}
Ld
µD
X∗

u∗ , u (X ∗ ,X)
∇v
∆v
❞✐✈(u)
P
Λ˙
et∆
ˆ
ˇ
⊗
Rj
[·, ·]·

◆♦r♠ ♦❢

u

X

✐♥ t❤❡ ♥♦r♠❡❞ s♣❛❝❡

■♥t❡❣❡r ♣❛rt ♦❢

x

❋r❛❝t✐♦♥ ♣❛rt ♦❢

x

▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ✐♥

Rd

❉✲❞✐♠❡♥s✐♦♥❛❧ ❍❛✉s❞♦r✛ ♠❡❛s✉r❡s ♦❢ ❛ s❡t ✐♥
❉✉❛❧ s♣❛❝❡ ♦❢ t❤❡ ♥♦r♠❡❞ s♣❛❝❡ X
∗
❉✉❛❧✐t② ♣r♦❞✉❝t u (u) ♦❢ u ∈ X ❛♥❞
●r❛❞✐❡♥t ♦❢ t❤❡ s❝❛❧❛r ❢✉♥❝t✐♦♥

R1

u∗ ∈ X ∗

v
v

▲❛♣❧❛❝✐❛♥ ♦❢ t❤❡ s❝❛❧❛r ❢✉♥❝t✐♦♥

❉✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥

u

▲❡r❛② ♣r♦❥❡❝t✐♦♥ ♦♣❡r❛t♦r
❈❛❧❞❡r♦♥ ❤♦♠♦❣❡♥❡♦✉s ♣s❡✉❞♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r
❍❡❛t ❦❡r♥❡❧
❋♦✉r✐❡r tr❛♥s❢♦r♠
■♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠
❚❡♥s♦r ♣r♦❞✉❝t
❘✐❡s③ tr❛♥s❢♦r♠s
❈♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ t✇♦ s♣❛❝❡s

■♥tr♦❞✉❝t✐♦♥
❋♦r♠✉❧❛t❡❞ ❛♥❞ ✐♥t❡♥s✐✈❡❧② st✉❞✐❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♥✐♥❡t❡❡♥t❤ ❝❡♥t✉r②✱ t❤❡
❝❧❛ss✐❝❛❧ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s r❡♣r❡s❡♥t t❤❡ ❢♦✉♥❞❛t✐♦♥ ♦❢
♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ ✇❛✈❡s✱ ❤❡❛t ❝♦♥❞✉❝t✐♦♥✱ ❤②❞r♦❞②♥❛♠✐❝s ❛♥❞ ♦t❤❡r ♣❤②s✐❝❛❧ ♣r♦❜❧❡♠s✳
❚❤❡✐r st✉❞② ♣r♦♠♣t❡❞ ❢✉rt❤❡r ✇♦r❦ ❜② ♠❛t❤❡♠❛t✐❝❛❧ r❡s❡❛r❝❤❡rs ❛♥❞✱ ✐♥ t✉r♥✱ ❜❡♥❡✜t❡❞
❢r♦♠ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♥❡✇ ♠❡t❤♦❞s ✐♥ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s✳ ■t ✐s ❛ ✈❛st s✉❜❥❡❝t✱ ✐♥t✐♠❛t❡❧②
❝♦♥♥❡❝t❡❞ t♦ ✈❛r✐♦✉s s❝✐❡♥❝❡s s✉❝❤ ❛s P❤②s✐❝s✱ ▼❡❝❤❛♥✐❝s✱ ❈❤❡♠✐str②✱ ❊♥❣✐♥❡❡r✐♥❣ ❙❝✐❡♥❝❡s✱
✇✐t❤ ❛ ❝♦♥s✐❞❡r❛❜❧❡ ♥✉♠❜❡r ♦❢ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥❞✉str✐❛❧ ♣r♦❜❧❡♠s✳ ❆❧t❤♦✉❣❤ t❤❡ t❤❡♦r② ♦❢
♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❤❛s ✉♥❞❡r❣♦♥❡ ❛ ❣r❡❛t ❞❡✈❡❧♦♣♠❡♥t ✐♥ t❤❡ t✇❡♥t✐❡t❤ ❝❡♥t✉r②✱
s♦♠❡ ❢✉♥❞❛♠❡♥t❛❧ q✉❡st✐♦♥s r❡♠❛✐♥ ✉♥r❡s♦❧✈❡❞✳ ❚❤❡② ❛r❡ ❡ss❡♥t✐❛❧❧② ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡
❣❧♦❜❛❧ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s✱ ❛s ✇❡❧❧ ❛s t❤❡✐r ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r✳
❋r♦♠ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ✈✐❡✇♣♦✐♥t✱ ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♥tr✐❣✉✐♥❣ ✉♥r❡s♦❧✈❡❞ q✉❡st✐♦♥s ❝♦♥❝❡r♥✐♥❣
t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❛♥❞ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t✉r❜✉❧❡♥❝❡ ♣❤❡♥♦♠❡♥❛ ✐s t❤❡ r❡❣✉❧❛r✐t②
❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥s t♦ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❣✐✈❡♥ ❛ s♠♦♦t❤
❞❛t✉♠ ❛t t✐♠❡ ③❡r♦✱ ✇✐❧❧ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝♦♥t✐♥✉❡ t♦ ❜❡ s♠♦♦t❤
❛♥❞ ✉♥✐q✉❡ ❢♦r ❛❧❧ t✐♠❡❄ ❚❤✐s q✉❡st✐♦♥ ✇❛s ♣♦s❡❞ ✐♥ ✶✾✸✹ ❜② ❏✳ ▲❡r❛② ❬✹✼✱ ✹✾❪ ❛♥❞ ✐s st✐❧❧
✇✐t❤♦✉t ❛♥s✇❡r✱ ♥❡✐t❤❡r ✐♥ t❤❡ ♣♦s✐t✐✈❡ ♥♦r ✐♥ t❤❡ ♥❡❣❛t✐✈❡✳
❚❤❡r❡ ✐s ♥♦ ✉♥✐q✉❡♥❡ss ♣r♦♦❢ ❡①❝❡♣t ❢♦r ♦✈❡r s♠❛❧❧ t✐♠❡ ✐♥t❡r✈❛❧s ❛♥❞ ✐t ❤❛s ❜❡❡♥ q✉❡st✐♦♥❡❞
✇❤❡t❤❡r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s r❡❛❧❧② ❞❡s❝r✐❜❡ ❣❡♥❡r❛❧ ✢♦✇s✳ ❇✉t t❤❡r❡ ✐s ♥♦ ♣r♦♦❢
❢♦r ♥♦♥✲✉♥✐q✉❡♥❡ss ❡✐t❤❡r✳
❯♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐s t❤❡ ❝♦r♥❡rst♦♥❡ ♦❢ ❝❧❛ss✐❝❛❧
❞❡t❡r♠✐♥✐s♠ ❬✶✽❪✳

■❢ ♠♦r❡ t❤❛♥ ♦♥❡ s♦❧✉t✐♦♥ ✇❡r❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ❞❛t❛✱

t❤❡ ❝♦♠♠✐tt❡❞ ❞❡t❡r♠✐♥✐st ✇✐❧❧ s❛② t❤❛t t❤❡ s♣❛❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✐s t♦♦ ❧❛r❣❡✱ ❜❡②♦♥❞ t❤❡
r❡❛❧ ♣❤②s✐❝❛❧ ♣♦ss✐❜✐❧✐t②✱ ❛♥❞ t❤❛t ✉♥✐q✉❡♥❡ss ❝❛♥ ❜❡ r❡st♦r❡❞ ✐❢ t❤❡ ✉♥♣❤②s✐❝❛❧ s♦❧✉t✐♦♥s
❛r❡ ❡①❝❧✉❞❡❞✳

■♥ t❤❡ ♥✐♥❡t❡❡♥t❤ ❝❡♥t✉r②✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦❜❧❡♠s ❛r✐s✐♥❣ ❢r♦♠ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ✇❡r❡
st✉❞✐❡❞ ✇✐t❤ t❤❡ ❛✐♠ ♦❢ ✜♥❞✐♥❣ ❡①❛❝t s♦❧✉t✐♦♥s t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉❛t✐♦♥s✳ ❚❤✐s ✐s ♦♥❧②
♣♦ss✐❜❧❡ ✐♥ ♣❛rt✐❝✉❧❛r ❝❛s❡s✳

❋♦r ✐♥st❛♥❝❡✱ ✈❡r② ❢❡✇ ❡①❛❝t s♦❧✉t✐♦♥s ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s

❡q✉❛t✐♦♥s ✇❡r❡ ❢♦✉♥❞ ❛♥❞✱ ❡①❝❡♣t ❢♦r s♦♠❡ ❡①❛❝t st❛t✐♦♥❛r② s♦❧✉t✐♦♥s✱ ❛❧♠♦st ❛❧❧ ♦❢ t❤❡♠
❞♦ ♥♦t ✐♥✈♦❧✈❡ t❤❡ s♣❡❝✐✜❝❛❧❧② ♥♦♥❧✐♥❡❛r ❛s♣❡❝ts ♦❢ t❤❡ ♣r♦❜❧❡♠✱ s✐♥❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣
♥♦♥❧✐♥❡❛r t❡r♠s ✐♥ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✈❛♥✐s❤✳ ■♥ t❤❡ t✇❡♥t✐❡t❤ ❝❡♥t✉r②✱ t❤❡ str❛t✲
❡❣② ❝❤❛♥❣❡❞✳ ■♥st❡❛❞ ♦❢ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ✐♥ ♣❛rt✐❝✉❧❛r ❝❛s❡s✱ t❤❡ ♣r♦❜❧❡♠s ✇❡r❡ st✉❞✐❡❞ ✐♥
❛❧❧ t❤❡✐r ❣❡♥❡r❛❧✐t②✳ ❚❤✐s ❧❡❞ t♦ t❤❡ ❝♦♥❝❡♣t ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s✳ ❚❤❡ ♣r✐❝❡ t♦ ♣❛② ✐s t❤❛t ♦♥❧②
t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ❡♥s✉r❡❞✳ ■♥ ❢❛❝t✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s
❛s t❤❡ ❧✐♠✐t ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s s♦❧✉t✐♦♥s ♦♣❡♥s t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t t❤❡r❡ ✐s ♠♦r❡ t❤❛♥ ♦♥❡
✇❡❛❦ s♦❧✉t✐♦♥s✳
❆ q✉❡st✐♦♥ ✐♥t✐♠❛t❡❧② r❡❧❛t❡❞ t♦ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦❜❧❡♠ ✐s t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥✳
❉♦ t❤❡ s♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❜❧♦✇✲✉♣ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❚❤❡ s♦❧✉t✐♦♥ ✐s
✐♥✐t✐❛❧❧② r❡❣✉❧❛r ❛♥❞ ✉♥✐q✉❡✱ ❜✉t ❛t t❤❡ ✐♥st❛♥t ❚ ✇❤❡♥ ✐t ❝❡❛s❡s t♦ ❜❡ ✉♥✐q✉❡ ✭✐❢ s✉❝❤ ❛♥
✐♥st❛♥t ❡①✐sts✮✱ t❤❡ r❡❣✉❧❛r✐t② ❝♦✉❧❞ ❛❧s♦ ❜❡ ❧♦st✳

✹


✺
❖♥❡ ♠❛② ✐♠❛❣✐♥❡ t❤❛t ❜❧♦✇✲✉♣ ♦❢ ✐♥✐t✐❛❧❧② r❡❣✉❧❛r s♦❧✉t✐♦♥s ♥❡✈❡r ❤❛♣♣❡♥s✱ ♦r t❤❛t ✐t
❜❡❝♦♠❡s ♠♦r❡ ❧✐❦❡❧② ❛s t❤❡ ✐♥✐t✐❛❧ ♥♦r♠ ✐♥❝r❡❛s❡s✱ ♦r t❤❛t t❤❡r❡ ✐s ❜❧♦✇✲✉♣✱ ❜✉t ♦♥❧② ♦♥ ❛
✈❡r② t❤✐♥ s❡t ♦❢ ♣r♦❜❛❜✐❧✐t② ③❡r♦✳ ❚❤❡ ❜❡st r❡s✉❧t ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ❝♦♥❝❡r♥✐♥❣ t❤❡ ♣♦ss✐❜❧❡
❧♦ss ♦❢ s♠♦♦t❤♥❡ss ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇❛s ♦❜t❛✐♥❡❞ ❜② ▲✳ ❈❛✛❛r❡❧❧✐✱ ❘✳ ❑♦❤♥
❛♥❞ ▲✳ ◆✐r❡♥❜❡r❣ ❬✾✱ ✹✺❪✱ ✇❤♦ ♣r♦✈❡❞ t❤❛t t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❍❛✉s❞♦r✛ ♠❡❛s✉r❡ ♦❢ t❤❡
s✐♥❣✉❧❛r s❡t ✐s ③❡r♦✳
❲❡ ❝❛♥ s❛② t❤❛t ✐❢ ✑s♦♠❡ q✉❛♥t✐t②✑ t✉r♥s ♦✉t t♦ ✑❜❡ s♠❛❧❧✑✱ t❤❡♥ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s

❡q✉❛t✐♦♥s ❛r❡ ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❍❛❞❛♠❛r❞ ✭❡①✐st❡♥❝❡✱ ✉♥✐q✉❡♥❡ss ❛♥❞ st❛❜✐❧✐t②
♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥s✮✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ✉♥✐q✉❡ ❣❧♦❜❛❧ s♦❧✉t✐♦♥ ❡①✐sts ✇❤❡♥ t❤❡
✐♥✐t✐❛❧ ✈❛❧✉❡ ❛♥❞ t❤❡ ❡①t❡r✐♦r ❢♦r❝❡ ❛r❡ s♠❛❧❧ ❡♥♦✉❣❤✱ ❛♥❞ t❤❡ s♦❧✉t✐♦♥ ✐s s♠♦♦t❤ ❞❡♣❡♥❞✐♥❣
♦♥ s♠♦♦t❤♥❡ss ♦❢ t❤❡s❡ ❞❛t❛✳ ❆♥♦t❤❡r q✉❛♥t✐t② t❤❛t ❝❛♥ ❜❡ s♠❛❧❧ ✐s t❤❡ ❞✐♠❡♥s✐♦♥✳ ■❢ ✇❡
t❤❡ s✐t✉❛t✐♦♥ ✐s ❡❛s✐❡r t❤❛♥ ✐♥ ❞✐♠❡♥s✐♦♥ n = 3 ❛♥❞ ❝♦♠♣❧❡t❡❧②
3
✉♥❞❡rst♦♦❞ ❬✹✶✱ ✻✸❪✳ ❋✐♥❛❧❧②✱ ✐❢ t❤❡ ❞♦♠❛✐♥ Ω ⊂ R ✐s s♠❛❧❧✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t Ω = ω × (0, )
❛r❡ ✐♥ ❞✐♠❡♥s✐♦♥

n = 2✱

✐s t❤✐♥ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥✱ s❛②✱ t❤❡♥ t❤❡ q✉❡st✐♦♥ ✐s ❛❧s♦ s❡tt❧❡❞ ❬✻✻❪✳
■♥ t❤✐s t❤❡s✐s✱ ✇❡ st✉❞② ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ♦❢ ✐♥❝♦♠♣r❡ss✐❜❧❡ ◆❛✈✐❡r✲
❙t♦❦❡s ❡q✉❛t✐♦♥s


 ∂t u = ∆u − ∇ · (u ⊗ u) − ∇p,
div(u) = 0,

u0 (0, x) = u0 ,

✭✵✳✶✮

t ∈ R+ , x ∈ Rd (d ≥ 2), u = (u1 , u2 , ..., ud ) ❞❡♥♦t❡ t❤❡ ✢♦✇ ✈❡❧♦❝✐t② ✈❡❝t♦r ❛♥❞
p(t, x) ❞❡s❝r✐❜❡s t❤❡ s❝❛❧❛r ♣r❡ss✉r❡✱ ∇ = (∂1 , ∂2 , ..., ∂d ) ✐s t❤❡ ❣r❛❞✐❡♥t ♦♣❡r❛t♦r✱ ∆ =
∂12 + ∂22 + ... + ∂d2 ✐s t❤❡ ▲❛♣❧❛❝✐❛♥✱ u0 (x) = (u01 , u02 , ..., u0d ) ✐s ❛ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❞❛t✉♠ ✇✐t❤
div(u0 ) = ∂1 u01 + ∂2 u02 + ... + ∂d u0d = 0✳ ❋♦r ❛ t❡♥s♦r F = (Fij ) ✇❡ ❞❡✜♥❡ t❤❡ ✈❡❝t♦r ∇ · F
d
❜② (∇ · F )i =
j=1 ∂j Fij ❛♥❞ ❢♦r t✇♦ ✈❡❝t♦rs u ❛♥❞ v ✱ ✇❡ ❞❡✜♥❡ t❤❡✐r t❡♥s♦r ♣r♦❞✉❝t

(u ⊗ v)ij = ui vj ✳ ■t ✐s t♦ s❡❡ t❤❛t ✭✵✳✶✮ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ❢♦r♠✿

✇❤❡r❡

∂t u = ∆u − P∇ · (u ⊗ u),
u0 (0, x) = u0 ,
✇❤❡r❡ t❤❡ ♦♣❡r❛t♦r

✭✵✳✷✮

P ✐s t❤❡ ❍❡❧♠❤♦❧t③✲▲❡r❛② ♣r♦❥❡❝t✐♦♥ ♦♥t♦ t❤❡ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ✜❡❧❞s✳ ▲❡t
∂j
2
Rj ❛r❡ ❞❡✜♥❡❞ ❜② Rj = √−∆
✱ ✐✳ ❡✳✱ ❢♦r f ∈ L ❜②

✉s r❡❝❛❧❧ t❤❛t t❤❡ ❘✐❡s③ tr❛♥s❢♦r♠s

∧
iξj
f (ξ)✳ ❚❤❡♥ P ✐s ❞❡✜♥❡❞ ♦♥ (L2 )d ❛s P = Id + R ⊗ R ✇❤❡r❡
|ξ|
t❤❡ ❘✐❡s③ tr❛♥s❢♦r♠❛t✐♦♥s✿ (Pf )j = fj +
1≤k≤d Rj Rk fk ✱

(Rj f )∧ =

R

✐s t❤❡ ✈❡❝t♦r ♦❢

■t ✐s ❦♥♦✇♥ t❤❛t ✭✵✳✷✮ ✐s ❡ss❡♥t✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✿

t

u = et∆ u0 −

e(t−τ )∆ P∇ · (u ⊗ u)dτ,

✭✵✳✸✮

0
✇❤❡r❡ t❤❡ ❤❡❛t ❦❡r♥❡❧

et∆

✐s ❞❡✜♥❡❞ ❛s

et∆ u(x) = ((4πt)−d/2 e−|·|

2 /4t

∗ u)(x).

◆♦t❡ t❤❛t ✭✵✳✶✮ ✐s s❝❛❧✐♥❣ ✐♥✈❛r✐❛♥t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ✐❢ u s♦❧✈❡s ✭✵✳✶✮✱ s♦ ❞♦❡s uλ (t, x) =
λu(λ2 t, λx) ❛♥❞ pλ (t, x) = λ2 p(λ2 t, λx) ✇✐t❤ ✐♥✐t✐❛❧ ❞❛t❛ λu0 (λx)✳ ❆ ❢✉♥❝t✐♦♥ s♣❛❝❡ X
d
❞❡✜♥❡❞ ✐♥ R ✐s s❛✐❞ t♦ ❜❡ ❛ ❝r✐t✐❝❛❧ s♣❛❝❡ ❢♦r ✭✵✳✶✮ ✐❢ ✐ts ♥♦r♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥
♦❢ t❤❡ s❝❛❧✐♥❣

f (x) → λf (λx)

❢♦r ❛♥②

λ > 0✱

✐✳❡✳✱

f (·) = λf (λx)

✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t

t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❛❝❡s ❛r❡ ❝r✐t✐❝❛❧ s♣❛❝❡s ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✿
d

−1,∞
d
−1,∞
H˙ 2 −1 (Rd ) → Ld (Rd ) → B˙ qq
(Rd )(q<∞) → BM O−1 (Rd ) → B˙ ∞
(Rd ).

✭✵✳✹✮


✻
■t ✐s r❡♠❛r❦❛❜❧❡ ❢❡❛t✉r❡ t❤❛t t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❛r❡ ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ s❡♥s❡ ♦❢
❍❛❞❛r♠❛r❞ ✭❡①✐st❡♥❝❡✱ ✉♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥t✐♥✉♦✉s ❞❡♣❡♥❞❡♥❝❡ ♦♥ ❞❛t❛✮ ✇❤❡♥ t❤❡ ✐♥✐t✐❛❧
−1,∞
˙∞

✮ ❧✐st❡❞
❞❛t❛ ❛r❡ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ❛♥❞ ❜❡❧♦♥❣ t♦ t❤❡ ❝r✐t✐❝❛❧ ❢✉♥❝t✐♦♥ s♣❛❝❡s ✭❡①❝❡♣t B
d
−1,∞
d
˙ 2 −1 (Rd )✱ Ld (Rd )✱ ❛♥❞ B˙ qq
✐♥ ✭✵✳✹✮ ✭s❡❡ ❬✶✶❪ ❢♦r H
(Rd )✱ s❡❡ ❬✹✵❪ ❢♦r BM O−1 (Rd )✱ ❛♥❞
−1,∞
˙∞
(Rd )✮✳ ❱❡r② r❡❝❡♥t❧②✱ ✐❧❧✲♣♦s❡❞♥❡ss ♦❢ ◆❛✈✐❡r✲
t❤❡ r❡❝❡♥t ✐❧❧✲♣♦s❡❞♥❡ss r❡s✉❧t ✐♥ ❬✸❪ ❢♦r B
−1
˙ ∞,q ✇❛s ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✻✽❪ ❛♥❞ ✜♥✐t❡ t✐♠❡
❙t♦❦❡s ❡q✉❛t✐♦♥s ✐♥ ❝r✐t✐❝❛❧ ❇❡s♦✈ s♣❛❝❡s B
❜❧♦✇✉♣ ❢♦r ❛♥ ❛✈❡r❛❣❡❞ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥ ✇❛s ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✻✺❪✳
■♥ t❤❡ ✶✾✻✵s✱ ♠✐❧❞ s♦❧✉t✐♦♥s ✇❡r❡ ✜rst ❝♦♥str✉❝t❡❞ ❜② ❑❛t♦ ❛♥❞ ❋✉❥✐t❛ ❬✸✹✱ ✸✺❪ t❤❛t ❛r❡
d
s
d
❝♦♥t✐♥✉♦✉s ✐♥ t✐♠❡ ❛♥❞ t❛❦❡ ✈❛❧✉❡s ✐♥ t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡s H (R ), (s ≥
− 1)✱ s❛② u ∈
2
s
d
s
d
C([0, T ]; H (R ))✳ ■♥ ✶✾✾✷✱ ❛ ♠♦❞❡r♥ tr❡❛t♠❡♥t ❢♦r ♠✐❧❞ s♦❧✉t✐♦♥s ✐♥ H (R ), (s ≥ d2 − 1)
✇❛s ❣✐✈❡♥ ❜② ❈❤❡♠✐♥ ❬✶✻❪✳ ■♥ ✶✾✾✺✱ ✉s✐♥❣ t❤❡ s✐♠♣❧✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❜✐❧✐♥❡❛r ♦♣❡r❛t♦r✱
˙ s (Rd ), (s ≥ d − 1)✱ s❡❡ ❬✶✶❪✳ ❘❡s✉❧ts

❈❛♥♥♦♥❡ ♣r♦✈❡❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✐❧❞ s♦❧✉t✐♦♥s ✐♥ H
2
q
d
♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✐❧❞ s♦❧✉t✐♦♥s ✇✐t❤ ✈❛❧✉❡ ✐♥ L (R ), (q > d) ✇❡r❡ ❡st❛❜❧✐s❤❡❞ ✐♥ t❤❡
♣❛♣❡rs ♦❢ ❋❛❜❡s✱ ❏♦♥❡s ❛♥❞ ❘✐✈✐èr❡ ❬✶✾❪ ❛♥❞ ♦❢ ●✐❣❛ ❬✷✼❪✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛
∞
✐♥ t❤❡ s♣❛❝❡ L ✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐❧❞ s♦❧✉t✐♦♥ ✇❛s ♦❜t❛✐♥❡❞ ❜② ❈❛♥♥♦♥❡ ❛♥❞ ▼❡②❡r
✐♥ ❬✶✶✱ ✶✹❪✳

▼♦r❡♦✈❡r✱ ✐♥ ❬✶✶✱ ✶✹❪✱ t❤❡② ❛❧s♦ ♦❜t❛✐♥❡❞ t❤❡♦r❡♠s ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✐❧❞
q
d
s♦❧✉t✐♦♥s ✇✐t❤ ✈❛❧✉❡ ✐♥ t❤❡ ▼♦rr❡②✲❈❛♠♣❛♥❛t♦ s♣❛❝❡ M2 (R ), (q > d) ❛♥❞ t❤❡ ❙♦❜♦❧❡✈
1
s
1
s
d
s♣❛❝❡ Hq (R ), (q < d,
− d < d )✱ ❛♥❞ ✐♥ ❣❡♥❡r❛❧ ✐♥ t❤❡ s♦✲❝❛❧❧❡❞ ✇❡❧❧✲s✉✐t❡❞ s♣❛❝❡ W
q
❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ❚❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐♥ t❤❡ ▼♦rr❡②✲❈❛♠♣❛♥❛t♦
s♣❛❝❡s ✇❡r❡ ❛❧s♦ tr❡❛t❡❞ ❜② ❑❛t♦ ❬✸✽❪ ❛♥❞ ❚❛②❧♦r ❬✻✷❪✳ ■♥ ✶✾✽✶✱ ❲❡✐ss❧❡r ❬✻✼❪ ❣❛✈❡ t❤❡ ✜rst
3
3
❡①✐st❡♥❝❡ r❡s✉❧t ❢♦r ♠✐❧❞ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❤❛❧❢ s♣❛❝❡ L (R+ )✳ ❚❤❡♥ ●✐❣❛ ❛♥❞ ▼✐②❛❦❛✇❛
3
3
❬✷✽❪ ❣❡♥❡r❛❧✐③❡❞ t❤❡ r❡s✉❧t t♦ L (Ω)✱ ✇❤❡r❡ Ω ✐s ❛♥ ♦♣❡♥ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ R ✳ ❋✐♥❛❧❧②✱

✐♥ ✶✾✽✹✱ ❑❛t♦ ❬✸✼❪ ♦❜t❛✐♥❡❞✱ ❜② ♠❡❛♥s ♦❢ ❛ ♣✉r❡❧② ❛♥❛❧②t✐❝❛❧ t♦♦❧ ✭✐♥✈♦❧✈✐♥❣ ♦♥❧② ❍¨
♦❧❞❡r
❛♥❞ ❨♦✉♥❣ ✐♥❡q✉❛❧✐t✐❡s ❛♥❞ ✇✐t❤♦✉t ✉s✐♥❣ ❛♥② ❡st✐♠❛t❡ ♦❢ ❢r❛❝t✐♦♥❛❧ ♣♦✇❡rs ♦❢ t❤❡ ❙t♦❦❡s
3
3
♦♣❡r❛t♦r✮✱ ❛♥ ❡①✐st❡♥❝❡ t❤❡♦r❡♠ ✐♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡ L (R )✳ ■♥ ❬✶✶✱ ✶✷✱ ✶✸❪✱ ❈❛♥♥♦♥❡ s❤♦✇❡❞
❤♦✇ t♦ s✐♠♣❧✐❢② ❑❛t♦✬s ♣r♦♦❢✳ ❚❤❡ ✐❞❡❛ ✐s t♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❜✐❧✐♥❡❛r
t∆
♦♣❡r❛t♦r ✐♥ ✐ts s❝❛❧❛r ❢♦r♠✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❞✐✈❡r❣❡♥❝❡ ∇ ❛♥❞ ❤❡❛t e
♦♣❡r❛t♦rs ❝❛♥ ❜❡
tr❡❛t❡❞ ❛s ❛ s✐♥❣❧❡ ❝♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r✳ ■♥ ✶✾✾✹✱ ❑❛t♦ ❛♥❞ P♦♥❝❡ ❬✸✾❪ s❤♦✇❡❞ t❤❛t t❤❡
◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❛r❡ ✇❡❧❧✲♣♦s❡❞ ✇❤❡♥ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ❜❡❧♦♥❣ t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s
d
˙ qq −1 (Rd ), (d ≤ q < ∞)✳
❙♦❜♦❧❡✈ s♣❛❝❡s H
■♥ t❤✐s t❤❡s✐s✱ ✇❡ ✉s❡ t❤❡ ♣r♦❣r❡ss ❛❝❤✐❡✈❡❞ ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ❢♦r t❤❡
❧❛st ✜❢t❡❡♥ ②❡❛rs t♦ st✉❞② t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳ ◆❛♠❡❧②✱ ✇❡ ✉s❡ t❤❡ t♦♦❧s ♦❢ ❋♦✉r✐❡r
❆♥❛❧②s✐s ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐♥ ♦r❞❡r t♦ st✉❞② t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳
❈❤❛♣t❡r ✶ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ r❡❝❛❧❧✐♥❣ ♦❢ s♦♠❡ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧ts ♦❢ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s✳
■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ❛♣♣❧② t❤❡s❡ t♦♦❧s t♦ t❤❡ st✉❞② ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t❤❡ ◆❛✈✐❡r✲
❙t♦❦❡s ❡q✉❛t✐♦♥s✳
❙❡❝t✐♦♥ ✷✳✶ ♣r❡s❡♥ts t❤❡ ❣❡♥❡r❛❧ s❤✐❢t✲✐♥✈❛r✐❛♥t s♣❛❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ❛♥❞ s♦♠❡ ❙♦❜♦❧❡✈
s♣❛❝❡s ♦✈❡r ❛ s❤✐❢t✲✐♥✈❛r✐❛♥t ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥s✳
❋r♦♠ ❙❡❝t✐♦♥ ✷✳✷ t♦ ❙❡❝t✐♦♥ ✷✳✻✱ ✇❡ ❝♦♥str✉❝t ♠✐❧❞ s♦❧✉t✐♦♥s t♦ ✭✵✳✸✮✱ ❛ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s
t (t−τ )∆
t∆
t♦ ✐t❡r❛t❡ t❤❡ tr❛♥s❢♦r♠ u → e u0 −
e
P∇ · (u ⊗ u)dτ ❛♥❞ t♦ ✜♥❞ ❛ ✜①❡❞ ♣♦✐♥t u ❢♦r
0

t❤✐s tr❛♥s❢♦r♠✳ ❚❤✐s ✐s t❤❡ s♦✲❝❛❧❧❡❞ P✐❝❛r❞ ❝♦♥tr❛❝t✐♦♥ ♠❡t❤♦❞ ❛❧r❡❛❞② ✐♥ ✉s❡ ❜② ❖s❡❡♥
❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✷✵t❤ ❝❡♥t✉r② t♦ ❡st❛❜❧✐s❤ t❤❡ ❧♦❝❛❧ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥
t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❢♦r ❛ r❡❣✉❧❛r ✐♥✐t✐❛❧ ✈❛❧✉❡✱ s❡❡ ❖s❡❡♥ ❬✺✹❪✳ ❲❡ r❡✇r✐t❡ t❤❡
❡q✉❛t✐♦♥ ✭✵✳✸✮ ❛s ❢♦❧❧♦✇s

u = U0 − B(u, u),

✭✵✳✺✮


✼
✇❤❡r❡

t

e(t−s)∆ P∇ · (u ⊗ v)ds

B(u, v)(t) =

✭✵✳✻✮

0
❛♥❞

U0 = et∆ u0 .
u ❢♦r t❤❡ ❡q✉❛t✐♦♥
d
✭✵✳✺✮✱ ✇❡ ♥❡❡❞ t♦ tr② t♦ ✜♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ET ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ (0, T ) × R s♦ t❤❛t
t❤❡ ❜✐❧✐♥❡❛r ♦♣❡r❛t♦r B ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❜② ✭✵✳✻✮ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ET ×ET → ET ✳ ❙❡❝t✐♦♥ ✷✳✷
t♦ ❙❡❝t✐♦♥ ✷✳✻ ❛r❡ ❞❡✈♦t❡❞ t♦ ❝♦♥str✉❝t ❡①❛♠♣❧❡s ♦❢ s✉❝❤ s♣❛❝❡s ET ✳ ❚❤❡ ♦❜t❛✐♥❡❞ r❡s✉❧ts

❇② ❚❤❡♦r❡♠ ✶✳✺✳✶ ✭s❡❡ ❙❡❝t✐♦♥ ✶✳✺ ♦❢ ❈❤❛♣t❡r ✶✮✱ t♦ ✜♥❞ ❛ ✜①❡❞ ♣♦✐♥t

❤❛✈❡ ❛ st❛♥❞❛r❞ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❡①✐st❡♥❝❡ t✐♠❡ ❛♥❞ ❞❛t❛ s✐③❡✿ ❧❛r❣❡ t✐♠❡ ✇✐t❤ s♠❛❧❧ ❞❛t❛
♦r ❧❛r❣❡ ❞❛t❛ ✇✐t❤ s♠❛❧❧ t✐♠❡✳
■♥ ❙❡❝t✐♦♥ ✷✳✷✱ ✇❡ st✉❞② ❧♦❝❛❧ ❛♥❞ ❣❧♦❜❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s
d
˙ qq −1 (Rd ) ❢♦r d ≥ 2, 1 < q ≤ d✳ ❚❤❡
✇✐t❤ ✐♥✐t✐❛❧ ❞❛t❛ ✐♥ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s H
♦❜t❛✐♥❡❞ r❡s✉❧t ✐♠♣r♦✈❡s t❤❡ ❦♥♦✇♥ ♦♥❡s ❢♦r

q = 2 ❛♥❞ q = d✳

❚❤❡s❡ ❝❛s❡s ✇❡r❡ ❝♦♥s✐❞❡r❡❞

❜② ♠❛♥② ❛✉t❤♦rs✱ s❡❡ ❬✶✶✱ ✶✸✱ ✶✻✱ ✶✼✱ ✸✹✱ ✸✺✱ ✸✼✱ ✹✻✱ ✺✼❪✳
■♥ ❙❡❝t✐♦♥ ✷✳✸✱ ✇❡ st✉❞② ❧♦❝❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤
˙ s (Rd ) ❢♦r d ≥ 2, p > d , and d − 1 ≤
❛r❜✐tr❛r② ✐♥✐t✐❛❧ ❞❛t❛ ✐♥ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s H
p
2
p
d
s < 2p ✳ ❚❤❡ ♦❜t❛✐♥❡❞ r❡s✉❧t ✐♠♣r♦✈❡s t❤❡ ❦♥♦✇♥ ♦♥❡s ❢♦r p > d ❛♥❞ s = 0 ✭s❡❡ ❬✶✶✱ ✶✹❪✮✳
d
■♥ t❤❡ ❝❛s❡ ♦❢ ❝r✐t✐❝❛❧ ✐♥❞❡①❡s s =
− 1✱ ✇❡ ♣r♦✈❡ ❣❧♦❜❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r ◆❛✈✐❡r✲❙t♦❦❡s
p
❡q✉❛t✐♦♥s ✇❤❡♥ t❤❡ ♥♦r♠ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐s s♠❛❧❧ ❡♥♦✉❣❤✳ ❚❤✐s r❡s✉❧t ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥
d
♦❢ t❤❡ ♦♥❡s ✐♥ ❬✶✸❪ ❛♥❞ ❬✹✻❪ ✐♥ ✇❤✐❝❤ (p = d, s = 0) ❛♥❞ (p > d, s =

− 1)✱ r❡s♣❡❝t✐✈❡❧②✳
p
■♥ ❙❡❝t✐♦♥ ✷✳✹✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥❞ st✉❞② ❙♦❜♦❧❡✈✲❋♦✉r✐❡r✲▲♦r❡♥t③ s♣❛❝❡s

H˙ Ls p,r (Rd )✳

❲❡

t❤❡♥ st✉❞② ❧♦❝❛❧ ❛♥❞ ❣❧♦❜❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧ ❞❛t❛
d
−1
˙ pp,r
(Rd ) ✇✐t❤ d ≥ 2, 1 ≤ p < ∞✱ ❛♥❞ 1 ≤ r < ∞✳
✐♥ ❝r✐t✐❝❛❧ s♣❛❝❡s H
L
■♥ ❙❡❝t✐♦♥ ✷✳✺✱ ✇❡ st✉❞② ❧♦❝❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡
˙ s q,r (Rd ) := (−∆)−s/2 Lq,r
❛r❜✐tr❛r② ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s H
L
d
d
❢♦r d ≥ 2, q > 1, s ≥ 0✱ 1 ≤ r ≤ ∞✱ ❛♥❞
−
1
≤
s
<
✳ ❚❤✐s r❡s✉❧t ✐♠♣r♦✈❡s t❤❡ ❦♥♦✇♥
q
q

d
d
r❡s✉❧ts ❢♦r q > d, r = q, s = 0 ✭s❡❡ ❬✶✶✱ ✶✹❪✮ ❛♥❞ ❢♦r q = r = 2, − 1 < s <
✭s❡❡ ❬✶✶✱ ✶✻❪✮✳
2
2
d
■♥ t❤❡ ❝❛s❡ ♦❢ ❝r✐t✐❝❛❧ ✐♥❞❡①❡s ✭s =
− 1✮✱ ✇❡ ♣r♦✈❡ ❣❧♦❜❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ◆❛✈✐❡r✲
q
❙t♦❦❡s ❡q✉❛t✐♦♥s ✇❤❡♥ t❤❡ ♥♦r♠ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐s s♠❛❧❧ ❡♥♦✉❣❤✳ ❚❤❡ r❡s✉❧t ✐s ❛
❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ r❡s✉❧t ✐♥ ❬✶✷❪ ❢♦r

q = r = d, s = 0✳

0 ≤ m < ∞ ❛♥❞ ✐♥❞❡① ✈❡❝t♦rs q = (q1 , q2 , ..., qd ), r = (r1 , r2 , ..., rd )✱
✇❤❡r❡ 1 < qi < ∞, 1 ≤ ri ≤ ∞, ❛♥❞ 1 ≤ i ≤ d✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥❞ st✉❞② ♠✐①❡❞✲
˙ m , ✳ ❚❤❡♥ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢
♥♦r♠ ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s H
L
p
˙ m , ) ✇❤❡r❡
s♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐♥ t❤❡ s♣❛❝❡s Q := QT = L ([0, T ]; H
L
d d
p > 2, T > 0✱ ❛♥❞ ✐♥✐t✐❛❧ ❞❛t❛ ✐s t❛❦❡♥ ✐♥ t❤❡ ❝❧❛ss I = {u0 ∈ (S (R )) , div(u0 ) = 0 :
e·∆ u0 Q < ∞}✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡♥ m = 0, q1 = q2 = ... = qd = r1 = r2 = ... = rd ✱ ♦✉r
■♥ ❙❡❝t✐♦♥ ✷✳✻✱ ❢♦r

q r

q r

r❡s✉❧ts r❡❝♦✈❡r t❤♦s❡ ♦❢ ❋❛❜❡r✱ ❏♦♥❡s ❛♥❞ ❘✐✈✐❡r❡ ❬✶✾❪✳
■♥ ❈❤❛♣t❡r ✸✱ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❋♦✐❛s✲❚❡♠❛♠✱ ✇❡ s❤♦✇ t❤❡ ✈❛♥✐s❤✐♥❣ ♦❢ ❍❛✉s❞♦r✛
♠❡❛s✉r❡ ♦❢ t❤❡ s✐♥❣✉❧❛r s❡t ✐♥ t✐♠❡ ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐♥ ✸❉
t♦r✉s✳


❈❤❛♣t❡r ✶

Pr❡❧✐♠✐♥❛r✐❡s
❚❤r♦✉❣❤♦✉t t❤❡ t❤❡s✐s✱ ❛ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥
✈✐❛t❡❞ ✐t ❛s

Rd ✱

s❛②

E(Rd )✱

✇✐❧❧ ❜❡ ❛❜❜r❡✲

E✳

❲❡ ❞♦ ♥♦t ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❛♥❞ s❝❛❧❛r✲✈❛❧✉❡❞ s♣❛❝❡s
M
♦❢ ❢✉♥❝t✐♦♥s✳ ❋♦r ❛♥② ❝♦❧❧❡❝t✐♦♥ ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s (Xm )m=1 ❛♥❞ X = X1 ∩ ... ∩ Xm ✱ ✇❡
1
m=M

g 2Xm 2 ✳ ❙✐♠✐❧❛r❧②✱ ❢♦r ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ f = (f1 , ..., fM )✱ ✇❡
s❡t g X =
m=1
1
m=M
fm 2X 2 ✳ ❲❡ s♦♠❡t✐♠❡s ✉s❡ t❤❡ ♥♦t❛t✐♦♥ A B ❛s ❛♥ ❡q✉✐✈❛❧❡♥t
❞❡✜♥❡ f X =
m=1
t♦ A ≤ CB ✇✐t❤ ❛ ✉♥✐❢♦r♠ ❝♦♥st❛♥t C ✳ ❚❤❡ ♥♦t❛t✐♦♥ A
B ♠❡❛♥s t❤❛t A
B ❛♥❞

B

A✳

✶✳✶ ❙♦♠❡ r❡s✉❧ts ♦❢ r❡❛❧ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s
❚❤✐s s❡❝t✐♦♥ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ r❡❝❛❧❧✐♥❣ ♦❢ s♦♠❡ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧ts ♦❢ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s✳

✶✳✶✳✶✳ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❞❡❝♦♠♣♦s✐t✐♦♥
❲❡ t❛❦❡ ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥
❢♦r♠

ϕˆ

ϕ

✐♥ t❤❡ ❙❝❤✇❛rt③ ❝❧❛ss

S(Rd )

❛♥❞ ✇❤♦s❡ ❋♦✉r✐❡r tr❛♥s✲

✐s s✉❝❤ t❤❛t

3
3
0 ≤ ϕ(ξ)
ˆ
≤ 1, ϕ(ξ)
ˆ
= 1 if |ξ| ≤ , ϕ(ξ)
ˆ
= 0 if |ξ| ≥ ,
4
2
❛♥❞ ❧❡t

ψ(x) = 2d ϕ(2x) − ϕ(x),
ϕj (x) = 2dj ϕ(2j x), j ∈ Z,
ψj (x) = 2dj ψ(2j x), j ∈ Z.
Sj ❛♥❞ ∆j ✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs ✇✐t❤ ϕj
{Sj , ∆j }j∈Z ✐s t❤❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❞❡❝♦♠♣♦s✐t✐♦♥✱ s♦ t❤❛t

❲❡ ❞❡♥♦t❡ ❜②
t❤❡ s❡t

I = S0 +

∆j .

❛♥❞

ψj

✳ ❋✐♥❛❧❧②✱

✭✶✳✶✮

j≥0
❚♦ ❜❡ ♠♦r❡ ♣r❡❝✐s❡✱ ✇❡ s❤♦✉❧❞ s❛② ✬❛ ❞❡❝♦♠♣♦s✐t✐♦♥✬✱ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t ♣♦ss✐❜❧❡
✭❡q✉✐✈❛❧❡♥t✮ ❝❤♦✐❝❡s ❢♦r t❤❡ ❢✉♥❝t✐♦♥

ϕ✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥ ❛r❜✐tr❛r② t❡♠♣❡r❡❞

✽


✾
❞✐str✐❜✉t✐♦♥

f✱

t❤❡ ❧❛st ✐❞❡♥t✐t② ❣✐✈❡s

f = S0 f +

∆j f.

✭✶✳✷✮

j≥0
❚❤❡ ✐♥t❡r❡st ✐♥ ❞❡❝♦♠♣♦s✐♥❣ ❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ✐♥t♦ ❛ s✉♠ ♦❢ ❞②❛❞✐❝ ❜❧♦❝❦s

∆j f ✱

✇❤♦s❡ s✉♣♣♦rt ✐♥ ❋♦✉r✐❡r s♣❛❝❡ ✐s ❧♦❝❛❧✐③❡❞ ✐♥ ❛ ❝♦r♦♥❛✱ ❝♦♠❡s ❢r♦♠ t❤❡ ♥✐❝❡ ❜❡❤❛✈✐♦r ♦❢
t❤❡s❡ ❜❧♦❝❦s ✇✐t❤ r❡s♣❡❝t t♦ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t✐♦♥s✳ ❚❤✐s ❢❛❝t ✐s ✐❧❧✉str❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣
d
❝❡❧❡❜r❛t❡❞ ❇❡r♥st❡✐♥s ❧❡♠♠❛ ✐♥ R ✱ ✇❤♦s❡ ♣r♦♦❢ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✹❪✳

▲❡♠♠❛ ✶✳✶✳✶✳

▲❡t

1≤p≤q≤∞

❛♥❞

k ∈ N✱

sup ∂ α f

t❤❡♥ ♦♥❡ ❤❛s

Rk f

p

p

|α|=k

❛♥❞

f

1

✇❤❡♥❡✈❡r ❢ ✐s ❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ✐♥
t❤❡ ❝♦r♦♥❛

|ξ|

1

Rd( p − q ) f

q

S

p

✇❤♦s❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠

fˆ(ξ)

✐s s✉♣♣♦rt❡❞ ✐♥

R✳

■♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇❤♦s❡ s✉♣♣♦rt ✐s ❛ ❜❛❧❧ ✭❛s✱ ❢♦r ✐♥st❛♥❝❡✱ ❢♦r

Sj f ✮

t❤❡ ❧❡♠♠❛

r❡❛❞s ❛s ❢♦❧❧♦✇s

▲❡♠♠❛ ✶✳✶✳✷✳

▲❡t

1≤p≤q≤∞

❛♥❞

k ∈ N✱

sup ∂ α f

t❤❡♥ ♦♥❡ ❤❛s

Rk f

p

p

|α|=k

❛♥❞

f

1

✇❤❡♥❡✈❡r ❢ ✐s ❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ✐♥
t❤❡ ❜❛❧❧

|ξ|

1

Rd( p − q ) f

q

S

p

✇❤♦s❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠

fˆ(ξ)

✐s s✉♣♣♦rt❡❞ ✐♥

R✳

▲❡t ✉s ❣♦ ❜❛❝❦ t♦ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✉♥✐t② ❡q✉❛t✐♦♥s ✭✶✳✶✮ ❛♥❞ ✭✶✳✷✮✳ ■t ✇❛s
p
✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❡❛r❧② ✶✾✸✵s ❜② ▲✐tt❧❡✇♦♦❞ ❛♥❞ P❛❧❡② t♦ ❡st✐♠❛t❡ t❤❡ L ✲♥♦r♠ ♦❢ tr✐❣♦♥♦✲
♠❡tr✐❝ ❋♦✉r✐❡r s❡r✐❡s ✇❤❡♥

1 < p < ∞✳

■❢ ✇❡ ♦♠✐t t❤❡ tr✐✈✐❛❧ ❝❛s❡ p = 2✱ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦
Lp ❜② s✐♠♣❧② ✉s✐♥❣ ✐ts ❋♦✉r✐❡r

❡♥s✉r❡ t❤❡ ❜❡❧♦♥❣✐♥❣ ♦❢ ❛ ❣❡♥❡r✐❝ ❋♦✉r✐❡r s❡r✐❡s t♦ t❤❡ s♣❛❝❡

❝♦❡✣❝✐❡♥ts✱ ❜✉t t❤✐s ❜❡❝♦♠❡s tr✉❡ ✐❢ ✇❡ ❝♦♥s✐❞❡r ✐♥st❡❛❞ ✐ts ❞②❛❞✐❝ ❜❧♦❝❦s✳ ■♥ t❤❡ ❝❛s❡ ♦❢
❛ ❢✉♥❝t✐♦♥

f

✭♥♦t ♥❡❝❡ss❛r✐❧② ♣❡r✐♦❞✐❝✮✱ t❤✐s ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥❝❡

if 1 < p < ∞ then f

S0 f

p

|∆j f (·)|2

p+
j≥0

■t ✐s ❡✈❡♥ ❡❛s✐❡r t♦ ♣r♦✈❡ t❤❛t t❤❡ ❝❧❛ss✐❝❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡s

1
2

.

✭✶✳✸✮

p

H s = H2s , s ∈ R

❝❛♥ ❜❡

❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ♥♦r♠

f

S0 f

Hs

22js ∆j f

2+

2
2

1
2

.

✭✶✳✹✮

j≥0
s

, s ∈ R, 1 < p < ∞ ❝♦rr❡s♣♦♥❞✐♥❣
p
s
t♦ t❤❡ ❙♦❜♦❧❡✈✲❇❡ss❡❧ s♣❛❝❡s Hp ✭t❤❛t✱ ✇❤❡♥ s ✐s ❛♥ ✐♥t❡❣❡r✱ r❡❞✉❝❡ t♦ t❤❡ ✇❡❧❧✲❦♥♦✇♥
s
α
❙♦❜♦❧❡✈ s♣❛❝❡s Wp ✇❤♦s❡ ♥♦r♠s ❛r❡ ❣✐✈❡♥ ❜② f Wps =
|α|≤s ∂ f p ✇❡ ✇✐❧❧ s❡❡ ✐♥ t❤❡

❆s ❢❛r ❛s t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ♥♦r♠

f

Hps

:= (I − ∆) 2 f

♥❡①t s❡❝t✐♦♥ ❤♦✇ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✹✮ ❤❛s t♦ ❜❡ ♠♦❞✐✜❡❞✳


✶✵
❇❡❢♦r❡ ❞❡✜♥✐♥❣ t❤❡ ❇❡s♦✈ s♣❛❝❡s t❤❛t ✇✐❧❧ ♣❧❛② ❛ ❦❡② r♦❧❡ ✐♥ ♦✉r st✉❞② ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s
❡q✉❛t✐♦♥s✱ ❧❡t ✉s r❡❝❛❧❧ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✉♥✐t②✱ ❛♥❛❧♦❣♦✉s t♦ t❤❡
❡q✉❛t✐♦♥ ✭✶✳✶✮✱ ❜✉t ❝♦♥t❛✐♥✐♥❣ ❛❧s♦ ❛❧❧ t❤❡ ❧♦✇ ❢r❡q✉❡♥❝✐❡s

I=

(j < 0)✱

s❛②

∆j .
j∈Z

■❢ ✇❡ ❛♣♣❧② t❤✐s ✐❞❡♥t✐t② t♦ ❛♥ ❛r❜✐tr❛r② t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥

f✱

✇❡ ♠❛② ❜❡ t❡♠♣t❡❞ t♦

✇r✐t❡

f=

∆j f,

✭✶✳✺✮

j∈Z
❜✉t✱ ❛t ✈❛r✐❛♥❝❡ ✇✐t❤ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✷✮✱ t❤✐s ✐❞❡♥t✐t② ❤❛s ♥♦ ♠❡❛♥✐♥❣ ✐♥
r❡❛s♦♥s✳

S

❢♦r s❡✈❡r❛❧

S
g ∈ S ✇❤♦s❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ❡q✉❛❧ t♦ ✶
q✉❛♥t✐t② < ∆j f, g > ✐s✱ ❢♦r ❛❧❧ j
0✱ ❛ ♣♦s✐t✐✈❡

❋✐rst ♦❢ ❛❧❧✱ t❤❡ s✉♠ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✺✮ ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♥✈❡r❣❡ ✐♥

❛s ✇❡ ❝❛♥ s❡❡ ✐❢ ✇❡ ❝♦♥s✐❞❡r ❛ t❡st ❢✉♥❝t✐♦♥
♥❡❛r t❤❡ ♦r✐❣✐♥✱ ❜❡❝❛✉s❡ ✐♥ t❤✐s ❝❛s❡ t❤❡
❝♦♥st❛♥t ♥♦t ❞❡♣❡♥❞✐♥❣ ♦♥

j✳

❆♥❞✱ ❡✈❡♥ ✇❤❡♥ t❤❡ s✉♠ ✐s ❝♦♥✈❡r❣❡♥t✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❤❛s

t♦ ❜❡ ✉♥❞❡rst♦♦❞ ♠♦❞✉❧♦ ♣♦❧②♥♦♠✐❛❧s✱ ❜❡❝❛✉s❡✱ ❢♦r t❤❡s❡ ♣❛rt✐❝✉❧❛r ❢✉♥❝t✐♦♥s

∆j P = 0

❢♦r ❛❧❧

❢♦r♠❛❧ s❡r✐❡s

j ∈ Z✳

P✱

✇❡ ❤❛✈❡

❆ ✇❛② t♦ r❡st♦r❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s t♦ ✑s✉✣❝✐❡♥t❧②✑ ❞❡r✐✈❡ t❤❡

j∈Z ❛s ✐t st❛t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ✭s❡❡ ❬✹✱ ✺✱ ✺✺❪ ❢♦r ❛ s✐♠♣❧❡ ♣r♦♦❢ ✮✳

▲❡♠♠❛ ✶✳✶✳✸✳ ❋♦r ❛♥② t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r m s✉❝❤ t❤❛t ❢♦r ❛♥②
α, |α| ≥ m

t❤❡ s❡r✐❡s

j<0

∂ α (∆j f )

❝♦♥✈❡r❣❡s ✐♥

S

✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦r♦❧❧❛r②✱ ✇❤♦s❡ ♣r♦♦❢ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❧❡♠♠❛✱ ❣✐✈❡s t❤❡ ❝♦rr❡❝t
♠❡❛♥✐♥❣ t♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ✭✶✳✺✮✱ t❤❛t ✐s ♠♦❞✉❧♦ ♣♦❧②♥♦♠✐❛❧s✳

❈♦r♦❧❧❛r② ✶✳✶✳✹✳ ❋♦r ❛♥② ✐♥t❡❣❡r ◆✱ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ PN
t❤❡ q✉❛♥t✐t②

j≥−N ∆j f − PN

■♥ s✉❝❤ ❛ ✇❛②✱ t❤❡ s❡r✐❡s

S

❝♦♥✈❡r❣❡s ✐♥

∆j f

✇❤❡♥

♦❢ ❞❡❣r❡❡


s✉❝❤ t❤❛t

N → ∞✳

✐s ❛❧✇❛②s ✇❡❧❧✲❞❡✜♥❡❞❀ ❢✉rt❤❡r♠♦r❡✱ ✐t ✐s ♥♦t ❞✐✣❝✉❧t t♦

f−

j∈Z ∆j f ❤❛s ✐ts s♣❡❝tr✉♠ r❡❞✉❝❡❞ t♦ ③❡r♦❀ ✐♥ ♦t❤❡r ✇♦r❞s✱
✐t ✐s ❛ ♣♦❧②♥♦♠✐❛❧✳ ■♥ t❤✐s ✇❛②✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ✭✶✳✺✮✱ t❤❛t ❢❛✐❧s t♦ ❜❡ ✈❛❧✐❞ ✐♥ S ✱ ✐s
♣r♦✈❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡

❡♥s✉r❡❞ ✐♥ t❤❡ q✉♦t✐❡♥t s♣❛❝❡

S /P

✳

✶✳✶✳✷✳ ❇❡s♦✈ s♣❛❝❡s
❚❤❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s ✈❡r② ✉s❡❢✉❧ ❜❡❝❛✉s❡ ✇❡ ❝❛♥ ❞❡✜♥❡ ✭✐♥❞❡♣❡♥✲
❞❡♥t❧② ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❢✉♥❝t✐♦♥

ϕ✮

t❤❡ ❢♦❧❧♦✇✐♥❣ ✭✐♥❤♦♠♦❣❡♥❡♦✉s✮ ❇❡s♦✈ s♣❛❝❡s

❬✷✸✱ ✺✻❪

❉❡✜♥✐t✐♦♥ ✶✳✶✳✶✳

▲❡t

0 < p, q ≤ ∞ ❛♥❞ s ∈ R✳ ❚❤❡♥
Bqs,p ✐❢ ❛♥❞ ♦♥❧② ✐❢

❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ❢ ❜❡❧♦♥❣s t♦

t❤❡ ✭✐♥❤♦♠♦❣❡♥❡♦✉s✮ ❇❡s♦✈ s♣❛❝❡

S0 f

q

sj

+

2

∆j f

p
q

1
p

< ∞.

j≥0
❋♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣❧❡t❡♥❡ss✱ ✇❡ ❛❧s♦ ❞❡✜♥❡ t❤❡ ✭✐♥❤♦♠♦❣❡♥❡♦✉s✮ ❚r✐❡❜❡❧✲ ▲✐③♦r❦✐♥
s♣❛❝❡s✱ ❡✈❡♥ ✐❢ ✇❡ ✇✐❧❧ ♥♦t ♠❛❦❡ ❛ ❣r❡❛t ✉s❡ ♦❢ t❤❡♠ ✐♥ t❤❡ st✉❞② ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s
❡q✉❛t✐♦♥s✳


✶✶

❉❡✜♥✐t✐♦♥ ✶✳✶✳✷✳

▲❡t

0 < p ≤ ∞, 0 < q < ∞✱

❜❡❧♦♥❣s t♦ t❤❡ ✭✐♥❤♦♠♦❣❡♥❡♦✉s✮ ❚r✐❡❜❡❧✲▲✐③♦r❦✐♥

S0 f

q

s ∈ R✳
s,p
s♣❛❝❡ Fq

❛♥❞

sj

2 |∆j f |

+

1
p

p

❚❤❡♥ ❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ❢
✐❢ ❛♥❞ ♦♥❧② ✐❢

< ∞.
q

j≥0

p, q ≥ 1 ❛♥❞ ❛ q✉❛s✐✲♥♦r♠ ✐♥
L∞
q = ∞ ✐♥ t❤❡ s❡❝♦♥❞ ❞❡✜♥✐t✐♦♥

■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤❡ ❛❜♦✈❡ q✉❛♥t✐t✐❡s ❞❡✜♥❡ ❛ ♥♦r♠ ✐❢
❣❡♥❡r❛❧✱ ✇✐t❤ t❤❡ ✉s✉❛❧ ❝♦♥✈❡♥t✐♦♥ t❤❛t

p=∞

✐♥ ❜♦t❤ ❝❛s❡s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✉s✉❛❧

♥♦r♠✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❤❛✈❡ ♥♦t ✐♥❝❧✉❞❡❞ t❤❡ ❝❛s❡
∞
❜❡❝❛✉s❡ t❤❡ L
♥♦r♠ ❤❛s t♦ ❜❡ r❡♣❧❛❝❡❞ ❤❡r❡ ❜② ❛ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❈❛r❧❡s♦♥ ♠❡❛s✉r❡
✭s❡❡ ❬✷✸❪✮✳

❆s ✇❡ ❤❛✈❡ ❛❧r❡❛❞② r❡♠❛r❦❡❞ ❜❡❢♦r❡ ❢♦r s♦♠❡ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡s ♦❢

s, p, q,

s❡❡

t❤❡ ❡q✉❛t✐♦♥s✳ ✭✶✳✸✮ ❛♥❞ ✭✶✳✹✮✱ t❤❡ ❇❡s♦✈ ❛♥❞ ❚r✐❡❜❡❧✲▲✐③♦r❦✐♥ s♣❛❝❡s ❣❡♥❡r❛❧✐③❡ t❤❡ ✉s✉❛❧
▲❡❜❡s❣✉❡ ♦♥❡s✱ ❢♦r ✐♥st❛♥❝❡

Lq = Fq0,2 , 1 < q < ∞,

❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ t❤❡ ❙♦❜♦❧❡✈✲❇❡ss❡❧ s♣❛❝❡s

Hqs = Fqs,2 , s ∈ R, 1 < q < ∞.
▲❡t

1
s < d/q ✳

❛♥❞

❲❡ ❞❡✜♥❡ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡

H˙ qs

❛s t❤❡ ❝❧♦s✉r❡

♦❢ t❤❡ s♣❛❝❡

S0 = f ∈ S : 0 ∈
/ Suppfˆ
✐♥ t❤❡ ♥♦r♠

f

H˙ qs

= Λ˙ s f

q

√
˙ = −∆ ❞❡♥♦t❡s t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❈❛❧❞❡ró♥ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r✳
✇❤❡r❡✱ ❛s ✉s✉❛❧✱ Λ
˙ qs ✐s ❛ s♣❛❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥s
❋✐♥❛❧❧②✱ ✇❤❡♥ d/q + m ≤ s < d/q + m + 1 ❛♥❞ m ✐s ❛♥ ✐♥t❡❣❡r✱ H
♠♦❞✉❧♦ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ≤ m✳
❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ ❞❡✜♥❡ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ✈❡rs✐♦♥ ♦❢ t❤❡ ❇❡s♦✈ ❛♥❞ ❚r✐❡❜❡❧✲▲✐③♦r❦✐♥

m ∈ Z✱ ✇❡ ❞❡♥♦t❡ ❜② Pm t❤❡ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ≤ m ✇✐t❤
t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t Pm = ∅ ✐❢ m < 0✳ ■❢ p = 1 ❛♥❞ s − d/q ∈ Z✱ ✇❡ ♣✉t m = s − d/q − 1❀
✐❢ ♥♦t✱ ✇❡ ♣✉t m = [s − d/q]✱ ✇✐t❤ t❤❡ ❜r❛❝❦❡ts ❞❡♥♦t✐♥❣ t❤❡ ✐♥t❡❣❡r ♣❛rt ❢✉♥❝t✐♦♥✳

s♣❛❝❡s ❬✹✱ ✺✱ ✷✸✱ ✺✻❪✳ ■❢

❉❡✜♥✐t✐♦♥ ✶✳✶✳✸✳

▲❡t

0 < p, q ≤ ∞ ❛♥❞ s ∈ R✳ ❚❤❡♥
B˙ qs,p ✐❢ ❛♥❞ ♦♥❧② ✐❢

❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ❢ ❜❡❧♦♥❣s t♦

t❤❡ ✭❤♦♠♦❣❡♥❡♦✉s✮ ❇❡s♦✈ s♣❛❝❡

2sj ∆j f

1
p

p

< ∞ and f =

q

j∈Z

❉❡✜♥✐t✐♦♥ ✶✳✶✳✹✳

▲❡t

j∈Z

0 < p ≤ ∞, 0 < q < ∞✱

❛♥❞

❜❡❧♦♥❣s t♦ t❤❡ ✭❤♦♠♦❣❡♥❡♦✉s✮ ❚r✐❡❜❡❧✲▲✐③♦r❦✐♥ s♣❛❝❡
sj

2 |∆j f |
j∈Z

∆j f in S /Pm .

p

1

p

s ∈ R✳ ❚❤❡♥ ❛ t❡♠♣❡r❡❞
F˙ s,p ✐❢ ❛♥❞ ♦♥❧② ✐❢
q

< ∞ and f =
q

∆j f in S /Pm ,
j∈Z

✇✐t❤ ❛♥ ❛♥❛❧♦❣♦✉s ♠♦❞✐✜❝❛t✐♦♥ ❛s ✐♥ t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s ❝❛s❡ ✇❤❡♥
❆s ❡①♣❡❝t❡❞✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐✜❝❛t✐♦♥s✿

Lq = F˙ q0,2 , 1 < q < ∞,
❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②

H˙ qs = F˙ qs,2 , s ∈ R, 1 < q < ∞,

q = ∞✳

❞✐str✐❜✉t✐♦♥ ❢


✶✷
0,2
BM O = F˙ ∞
,
❛♥❞

−1,2
BM O−1 = F˙ ∞
.
▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❜❡❞❞✐♥❣s ✭s❡❡ ❬✶✱ ✷✱ ✽❪✮✳

▲❡♠♠❛ ✶✳✶✳✺✳ ❋♦r 1 ≤ p, q, r ≤ ∞ ❛♥❞ s ∈ R✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❜❡❞❞✐♥❣ ♠❛♣♣✐♥❣s✳
✭❛✮ ■❢

1
t❤❡♥

B˙ qs,q → H˙ qs → B˙ qs,2 , Bqs,q → Hqs → Bqs,2 .
✭❜✮ ■❢

2≤q<∞

t❤❡♥

B˙ qs,2 → H˙ qs → B˙ qs,q , Bqs,2 → Hqs → Bqs,q .
✭❝✮ ■❢

1 ≤ p1 < p2 ≤ ∞

t❤❡♥

B˙ qs,p1 → B˙ qs,p2 , Bqs,p1 → Bqs,p2 , F˙ qs,p1 → F˙ qs,p2 , Fqs,p1 → Fqs,p2 .
✭❞✮ ■❢

s1 > s2 , 1 ≤ q1 , q2 ≤ ∞✱

❛♥❞

d
q1

s1 −

= s2 −

d
t❤❡♥
q2

B˙ qs11 ,p → B˙ qs22 ,p , Bqs11 ,p → Bqs22 ,p , F˙ qs11 ,p → F˙ qs22 ,r , Fqs11 ,p → Fqs22 ,r .
✭❡✮ ■❢

p≤q

t❤❡♥

Bqs,p → Fqs,p , B˙ qs,p → F˙ qs,p .
✭❢ ✮ ■❢

q≤p

t❤❡♥

Fqs,p → Bqs,p , F˙ qs,p → B˙ qs,p .

✭❣✮

Fqs,q = Bqs,q , F˙ qs,q = B˙ qs,q .
✭❤✮ ■❢

1Hqs = Fqs,2 , H˙ qs = F˙ qs,2 .

❲❡ r❡❝❛❧❧ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts

▲❡♠♠❛ ✶✳✶✳✻✳

▲❡t

1 ≤ p, q ≤ ∞
∞

❛♥❞

s

(t− 2 et∆ f
0

Lq

)p

s < 0✳
1

p

dt
t

❚❤❡♥ t❤❡ t✇♦ q✉❛♥t✐t✐❡s

and f

B˙ qs,p

are equivalent.

Pr♦♦❢✳ ❙❡❡ Pr♦♣♦s✐t✐♦♥ ✸ ✐♥ ✭❬✶✺❪✱ ♣✳ ✶✽✷✮✱ ♦r s❡❡ ❚❤❡♦r❡♠ ✺✳✹ ✐♥ ✭❬✹✻❪✱ ♣✳ ✹✺✮✳

▲❡♠♠❛ ✶✳✶✳✼✳

▲❡t

1 ≤ p, q ≤ ∞
∞

s

t− 2 et∆ f

❛♥❞

s < 0✳

p dt

0

1
p

t

Lq

❚❤❡♥ t❤❡ t✇♦ q✉❛♥t✐t✐❡s

and f

F˙qs,p

are equivalent.

Pr♦♦❢✳ ❙❡❡ Pr♦♣♦s✐t✐♦♥ ✹ ✐♥ ✭❬✶✺❪✱ ♣✳ ✶✽✸✮✳
❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▲❡♠♠❛ ✶✳✶✳✻✳

▲❡♠♠❛ ✶✳✶✳✽✳

▲❡t

1 ≤ p, q ≤ ∞, α ≥ 0✱

∞

s
α
(t− 2 et∆ t 2 Λ˙ α f

0

Pr♦♦❢✳ ◆♦t❡ t❤❛t
♣r♦✈❡ t❤❡ ❧❡♠♠❛✳

Λ˙ s0

Lq

)p

❛♥❞

dt
t

✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠

1
p

s < α✳

❚❤❡♥ t❤❡ t✇♦ q✉❛♥t✐t✐❡s

and f

B˙ qs,p

♦♥t♦

B˙ qs,p

are equivalent.

B˙ qs−s0 ,p ✱

s❡❡ ❬✽❪✱ t❤❡♥ ✇❡ ❝❛♥ ❡❛s✐❧②


✶✸

✶✳✶✳✸✳ ❖t❤❡r ✉s❡❢✉❧ ❢✉♥❝t✐♦♥ s♣❛❝❡s
■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r❡s❡♥t ♦t❤❡r ❢✉♥❝t✐♦♥❛❧ s♣❛❝❡s✱ t❤❛t ✇✐❧❧ ❜❡ ✉s❡❢✉❧ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣
❝❤❛♣t❡rs✳

✶✳✶✳✹✳ ▼♦rr❡②✲❈❛♠♣❛♥❛t♦ s♣❛❝❡s
1 ≤ p ≤ q ≤ ∞✱ t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s ▼♦rr❡②✲❈❛♠♣❛♥❛t♦
p
s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s f ✇❤✐❝❤ ❛r❡ ❧♦❝❛❧❧② ✐♥ L ❛♥❞ s✉❝❤ t❤❛t

❋♦r
t❤❡

sup sup R

d/q−d/p

p

|f (y)| dy

x∈Rd 0
1
p

s♣❛❝❡

M p,q

✐s ❞❡✜♥❡❞ ❛s

< ∞,

|y−x|
p,q
✇❤❡r❡ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤✐s ✐♥❡q✉❛❧✐t② ✐s t❤❡ ♥♦r♠ ♦❢ f ✐♥ M
✳ ❚❤❡ ❤♦♠♦❣❡♥❡♦✉s
p,q
˙
▼♦rr❡②✲❈❛♠♣❛♥❛t♦ s♣❛❝❡ M
✐s ❞❡✜♥❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✱ ❜② t❛❦✐♥❣ t❤❡ s✉♣r❡♠✉♠ ♦✈❡r
❛❧❧

R ∈ (0, ∞)

✐♥st❡❛❞

R ∈ (0, 1]✳

✶✳✶✳✺✳ ▲♦r❡♥t③ s♣❛❝❡s
p,q
t❤❡ ▲♦r❡♥t③ s♣❛❝❡ L
✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳
p,q
❆ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ f ∈ L
✐❢ ❛♥❞ ♦♥❧② ✐❢
❋♦r

1 ≤ p, q ≤ ∞✱

f
♦❢ ❝♦✉rs❡✱ ✐❢

q=∞

=

Lp,q

q
p

∞

1

(t p f ∗ (t))q
0

dt
t

1
q

< ∞,

t❤✐s ♠❡❛♥s

f

1

Lp,∞

= sup t p f ∗ (t) < ∞,
t>0

✇❤❡r❡

f∗

✐s t❤❡ ❞❡❝r❡❛s✐♥❣ r❡❛rr❛♥❣❡♠❡♥t ♦❢

f✿

f ∗ (t) = inf s ≥ 0 : {x : |f (x)| > s} ≤ t , t ≥ 0.
❲❡ r❡❝❛❧❧ t❤❡ s♦♠❡ r❡s✉❧ts ✐♥ ❬✹✻❪✳

❚❤❡♦r❡♠ ✶✳✶✳✾✳
▲❡t

1 < p < ∞

✭P♦✐♥t✇✐s❡ ♣r♦❞✉❝t ✐♥ t❤❡ ▲♦r❡♥t③ s♣❛❝❡s✮✳

❛♥❞

1 ≤ q ≤ ∞, 1/p + 1/p = 1

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❛ ❜♦✉♥❞❡❞ ❜✐❧✐♥❡❛r ♦♣❡r❛t♦r✿
p,q
✭❛✮ ❢r♦♠ L
× L∞ t♦ Lp,q ✱
p,q
✭❜✮ ❢r♦♠ L
× Lp ,q t♦ L1 ✱
p,q
✭❝✮ ❢r♦♠ L
× Lp1 ,q1 t♦ Lp2 ,q2 ✱ ❢♦r 1 < p, p1 , p2

❛♥❞

1/q + 1/q = 1✳

❚❤❡♥ ♣♦✐♥t✇✐s❡

< ∞, 1 ≤ q, q1 , q2 ≤ ∞,

1/p2 = 1/p + 1/p1 , 1/q2 = 1/q + 1/q1 ✳

❚❤❡♦r❡♠ ✶✳✶✳✶✵✳
▲❡t

1
❛♥❞

✭❈♦♥✈♦❧✉t✐♦♥ ♦❢ t❤❡ ▲♦r❡♥t③ s♣❛❝❡s✮✳

1 ≤ q ≤ ∞, 1/p + 1/p = 1

❜♦✉♥❞❡❞ ❜✐❧✐♥❡❛r ♦♣❡r❛t♦r✿
p,q
✭❛✮ ❢r♦♠ L
× L1 t♦ Lp,q ✱
p,q
✭❜✮ ❢r♦♠ L
× Lp ,q t♦ L∞ ✱
p,q
p ,q
p ,q
✭❝✮ ❢r♦♠ L ×L 1 1 t♦ L 2 2 ✱ ❢♦r

1/q2 = 1/q + 1/q1 ✳

❛♥❞

1/q + 1/q = 1✳

❚❤❡♥ ❝♦♥✈♦❧✉t✐♦♥ ✐s ❛

1 < p, p1 , p2 < ∞, 1 ≤ q, q1 , q2 ≤ ∞, 1/p2 +1 = 1/p+1/p1 ,


✶✹

✶✳✷ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s
❲❡ ❝♦♥s✐❞❡r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✭◆❙❊✮ ✐♥

d ❞✐♠❡♥s✐♦♥s ✐♥ t❤❡ s♣❡❝✐❛❧ s❡tt✐♥❣ ♦❢

❛ ✈✐s❝♦✉s✱ ❤♦♠♦❣❡♥❡♦✉s✱ ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞ t❤❛t ✜❧❧s t❤❡ ❡♥t✐r❡ s♣❛❝❡ ❛♥❞ ✐s ♥♦t s✉❜♠✐tt❡❞
t♦ ❡①t❡r♥❛❧ ❢♦r❝❡s✳ ❚❤✉s✱ t❤❡ ❡q✉❛t✐♦♥s ✇❡ ❝♦♥s✐❞❡r ❛r❡ t❤❡ s②st❡♠✿

∂t u = ∆u − ∇ · (u ⊗ u) − ∇p,
div(u) = 0,

✭✶✳✻✮

✇❤✐❝❤ ✐s ❛ ❝♦♥❞❡♥s❡❞ ❢♦r♠ ♦❢

d

l=1

For 1 ≤ k ≤ d, ∂t uk = ∆uk −
d
l=1 ∂l ul = 0.
❚❤❡ ✉♥❦♥♦✇♥ q✉❛♥t✐t✐❡s ❛r❡ t❤❡ ✈❡❧♦❝✐t②

x

❛♥❞ t❤❡ ♣r❡ss✉r❡

p✳

u(t, x)

∂l (ul uk ) − ∂k p,

✭✶✳✼✮

♦❢ t❤❡ ✢✉✐❞ ❡❧❡♠❡♥t ❛t t✐♠❡

t

❛♥❞ ♣♦s✐t✐♦♥

❚❛❦✐♥❣ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ ✭✶✳✻✮✱ ✇❡ ♦❜t❛✐♥✿

d

d

∆p = −∇ ⊗ ∇ · (u ⊗ u) = −

∂k ∂l (uk ul ).

✭✶✳✽✮

k=1 l=1
❚❤✉s✱ ✇❡ ❢♦r♠❛❧❧② ❣❡t t❤❡ ❡q✉❛t✐♦♥s

∂t u = ∆u − P∇ · (u ⊗ u),
div(u) = 0,
✇❤❡r❡

P

✭✶✳✾✮

✐s t❤❡ ❍❡❧♠❤♦❧t③✲▲❡r❛② ♣r♦❥❡❝t✐♦♥ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ❛s

Pf := f − ∇

1
∇⊗∇
(∇ · f ) = (I −
)f.
∆
∆

✭✶✳✶✵✮

❲❡ s❤❛❧❧ st✉❞② t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✾✮ ✭❧♦♦❦✐♥❣ ❢♦r ❛ s♦❧✉t✐♦♥ ♦♥
Rd ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ u0 ✮✱ ❛♥❞ tr❛♥s❢♦r♠ ✭✶✳✾✮ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥

u = et∆ u0 −
div(u0 ) = 0.

t (t−s)∆
e
P∇
0

(0, T )×

· (u ⊗ u)ds,

✭✶✳✶✶✮

❚❤❡ ♠❛✐♥ ♣r♦♣❡rt② ✇❡ ✉s❡ t❤r♦✉❣❤♦✉t t❤✐s t❤❡s✐s ✐s t❤❛t t❤❡ ♦♣❡r❛t♦r

et∆ P∇

✐s ❛ ♠❛tr✐①

♦❢ ❝♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs ✇✐t❤ ❜♦✉♥❞❡❞ ✐♥t❡❣r❛❜❧❡ ❦❡r♥❡❧s✳

▲❡♠♠❛ ✶✳✷✳✶✳

✭❚❤❡ ❖s❡❡♥ ❦❡r♥❡❧✮✳

1 ≤ j, k ≤ d ❛♥❞ t > 0✱ t❤❡ ♦♣❡r❛t♦r Oj,k,t = ∆1 ∂j ∂k et∆
Oj,k,t f = Kj,k,t ∗ f ✱ ✇❤❡r❡ t❤❡ ❦❡r♥❡❧ Kj,k,t s❛t✐s✜❡s Kj,k,t (x) =
❢✉♥❝t✐♦♥ Kj,k s✉❝❤ t❤❛t

❋♦r

✐s ❛ ❝♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r
1
K ( √xt ) ❢♦r ❛ s♠♦♦t❤
td/2 j,k

(1 + |x|)d+|α| ∂ α Kj,k ∈ L∞ (Rd ), f or all α ∈ Nd ,
(1 + |x|)d+m Λ˙ m Kj,k ∈ L∞ (Rd ), f or all m ≥ 0,
✇❤❡r❡
d

x2i

|x| =

1/2

, x = (x1 , x2 , ..., xd ) and Dxα denotes ∂x|α| = ∂ |α| /∂xα11 ∂xα22 ...∂xαdd .

i=1

Pr♦♦❢✳ ❙❡❡ Pr♦♣♦s✐t✐♦♥ ✶✶✳✶ ✐♥ ❬✹✻❪✱ ♣✳ ✶✵✼✳


✶✺

✶✳✸ ❊❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣r❡ss✉r❡ ❛♥❞ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡
◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s
❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✼✮ ✉♥❞❡r s♣❛t✐❛❧ tr❛♥s❧❛t✐♦♥s ❛♥❞
d
❞✐❧❛t✐♦♥s✱ ❛s ✇❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ♦♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡ R ✳ ❲❡ ❜❡❣✐♥ ❜② ❞❡✜♥✐♥❣ ✇❤❛t
✇❡ ❝❛❧❧ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✳

✭❲❡❛❦ s♦❧✉t✐♦♥s✮✳

❆ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ♦♥
u(t, x) ✐♥ (D ((0, T ) × Rd ))d s♦ t❤❛t ✿
d
✭❛✮ ✉ ✐s ❧♦❝❛❧❧② sq✉❛r❡ ✐♥t❡❣r❛❜❧❡ ♦♥ (0, T ) × R ✱
✭❜✮
✭❝✮

(0, T ) × Rd

✐s ❛ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ✜❡❧❞

div(u) = 0✱
∃p ∈ D ((0, T ) × Rd ), ∂t u = ∆u − ∇ · (u ⊗ u) − ∇p.
◆♦t✐❝❡ t❤❛t t❤✐s ✐s ♥♦t t❤❡ ✉s✉❛❧ ❞❡✜♥✐t✐♦♥ ❢♦r ✇❡❛❦ s♦❧✉t✐♦♥s ✭❛s ❣✐✈❡♥ ✐♥ t❤❡ ❜♦♦❦ ♦❢

❚❡♠❛♠ ❬✻✸❪ ❛♥❞ ✐♥ ❈❤❛♣t❡r ✸✮✳ ◆❡①t✱ ✇❡ r❡❝❛❧❧ s♦♠❡ r❡s✉❧ts ✐♥ ❬✹✻❪✳

❚❤❡♦r❡♠ ✶✳✸✳✶✳

✭❊❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣r❡ss✉r❡✮✳
d
✐s ✉♥✐❢♦r♠❧② ❧♦❝❛❧❧② sq✉❛r❡ ✲ ✐♥t❡❣r❛❜❧❡ ♦♥ (0, T ) × R ✭✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣
t1
2
1/2
2
❞❡✜♥✐t✐♦♥✿ Ut0 ,t1 (x) = ( t |u(t, x)| dt)
❜❡❧♦♥❣s t♦ t❤❡ ▼♦rr❡② s♣❛❝❡ Luloc ❢♦r ❛❧❧ 0 <
0
t0 < t1 < T ✱ ✇❤❡r❡ u Lp = supx0 ∈Rd ( |x−x |<1 |f |p dx)1/p )✱ t❤❡♥ P∇ · (u ⊗ u) ✐s ✇❡❧❧
✭❛✮ ■❢

u

uloc

0

(D ((0, T ) × Rd ))d ✱ ❛♥❞ t❤❡r❡ ❡①✐sts
P∇ · (u ⊗ u) = ∇ · (u ⊗ u) + ∇p✳ ❚❤✉s✱ ✐❢ u ✐s ❛
❞❡✜♥❡❞ ✐♥

❛ ❞✐str✐❜✉t✐♦♥

p ∈ D ((0, T ) × Rd )

s♦ t❤❛t

s♦❧✉t✐♦♥ ❢♦r ✭✶✳✾✮✱ t❤❡♥ ✐t ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥

❢♦r ✭✶✳✻✮✳
✭❜✮ ❈♦♥✈❡rs❡❧②✱ ✐❢

u

t❤❡ s❡♥s❡ t❤❛t ❢♦r ❛❧❧

✐s ❛ ✉♥✐❢♦r♠ ✇❡❛❦ s♦❧✉t✐♦♥ ❢♦r ✭✶✳✻✮✱ ❛♥❞ ✐❢

t0 < t1 ∈ (0, T )

1
R→∞x ∈Rd Rd
0
u

✈❛♥✐s❤❡s ❛t ✐♥✜♥✐t② ✐♥

✇❡ ❤❛✈❡
t1

|u|2 dxdt = 0,

lim sup

t❤❡♥

u

t0

|x−x0 |
✐s ❛ s♦❧✉t✐♦♥ ❢♦r ✭✶✳✾✮✳

❚❤❡♦r❡♠ ✶✳✸✳✷✳

✭❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ t❤❡♦r❡♠✮✳
2
2
d d
▲❡t u ∈ ∩t1 ✭❛✮ ✉ ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s

∂t u = ∆u − P∇ · (u ⊗ u),
div(u) = 0.
✭❜✮

u

✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s
d

d

∃u0 ∈ (S (R ))

u = et∆ u0 −

div(u0 ) = 0.

t (t−s)∆
e
P∇
0

· (u ⊗ u)ds,

✭✶✳✶✷✮

✶✳✹ ❙❝❛❧✐♥❣ ✐♥✈❛r✐❛♥❝❡
❚❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r ❛ ♣❛rt✐❝✉❧❛r ❝❤❛♥❣❡ ♦❢ t✐♠❡ ❛♥❞ s♣❛❝❡
d
s❝❛❧✐♥❣✳ ▼♦r❡ ❡①❛❝t❧②✱ ❛ss✉♠❡ t❤❛t✱ ✐♥ R × (0, ∞)✱ u(t, x) ❛♥❞ p(t, x) s♦❧✈❡ t❤❡ s②st❡♠

∂t u = ∆u − ∇ · (u ⊗ u) − ∇p,
div(u) = 0,


✶✻
t❤❡♥ t❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r t❤❡ r❡s❝❛❧❡❞ ❢✉♥❝t✐♦♥s

uλ (t, x) = λu(λ2 t, λx), pλ (t, x) = λ2 p(λ2 t, x).
❚❤❡ ❛❜♦✈❡ s❝❛❧✐♥❣ ✐♥✈❛r✐❛♥❝❡ ❧❡❛❞s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✶✳✹✳✶✳

❆ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥s ❳ ✐s

❝❛❧❧❡❞ ❛ ❝r✐t✐❝❛❧ s♣❛❝❡ ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✐❢ ✐ts ♥♦r♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡
❛❝t✐♦♥ ♦❢ t❤❡ s❝❛❧✐♥❣

f (x) → λf (λx)

❢♦r ❛♥②

λ > 0✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ r❡q✉✐r❡ t❤❛t

X →S
❛♥❞ t❤❛t ❢♦r ❛♥②

f ∈X
f (·)

λf (λ · −x0 ) , ∀λ > 0, ∀x0 ∈ Rd .

❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ❢❛♠✐❧②

Lp = Lp (Rd )

t❤❡ ❝r✐t✐❝❛❧ ✐♥✈❛r✐❛♥t s♣❛❝❡
˙ s = H˙ s (Rd ) t❤❡ ❝r✐t✐❝❛❧
❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✈❛❧✉❡ p = d✱ ❛♥❞ ✐♥ t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡ ❢❛♠✐❧② H
d
− 1✳
✐♥✈❛r✐❛♥t s♣❛❝❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✈❛❧✉❡ s =
2

Pr♦♣♦s✐t✐♦♥ ✶✳✹✳✶✳
s♣❛❝❡

■❢ ❳ ✐s ❛ ❝r✐t✐❝❛❧ s♣❛❝❡✱ t❤❡♥ ❳ ✐s ❝♦♥t✐♥✉♦✉s❧② ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ❇❡s♦✈

−1,∞
B˙ ∞
✳

✶✳✺ ❖✉t❧✐♥❡ ♦❢ t❤❡ ❞✐ss❡rt❛t✐♦♥
❚❤❡ ✐❞❡❛ ✐s t♦ ❝♦♥str✉❝t t❤❡ s♦❧✉t✐♦♥

u

❢♦r ◆❙❊ ❛s ❛ s♦❧✉t✐♦♥ ❢♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥
d
✭✶✳✶✶✮✳ ▲❡t ❛ ❇❛♥❛❝❤ s♣❛❝❡ ET ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ (0, T ) × R ❛♥❞ s✉❝❤ t❤❛t ET ⊆
d
d
∩t1 d
f ∈ ET ✐✛ f ∈ S (Rd ) ❛♥❞ (et∆ f )0≤t≤T ∈ ET ✳ ■❢ u ✐s ❛ s♦❧✉t✐♦♥ ♦❢ ✭✶✳✶✶✮✱ u ∈ ET ✱

u0 ∈ ET

t❤❡♥ ❛♣♣❧②✐♥❣ ❚❤❡♦r❡♠s ✶✳✸✳✶ ❛♥❞ ✶✳✸✳✷✱ ✇❡ ✐♠♣❧② t❤❛t

u

✐s ❛❧s♦ ❛ ✇❡❛❦ s♦❧✉t✐♦♥

♦❢ ✭✶✳✻✮ ❛♥❞ ✭✶✳✾✮✱ ✇❡ r❡✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✶✶✮ ❛s ❢♦❧❧♦✇s

u = U0 − B(u, u),
✇❤❡r❡

✭✶✳✶✸✮

t

e(t−s)∆ P∇ · (u ⊗ v)ds,

B(u, v)(t) =

✭✶✳✶✹✮

0
❛♥❞

U0 = et∆ u0 .
❚❤❡♥ ✇❡ ✇✐❧❧ ✜♥❞ ❛ ✜①❡❞ ♣♦✐♥t

u

❢♦r t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✶✸✮✳

❚❤✐s ✐s t❤❡ s♦✲❝❛❧❧❡❞ P✐❝❛r❞

❝♦♥tr❛❝t✐♦♥ ♠❡t❤♦❞✳

❚❤❡♦r❡♠ ✶✳✺✳✶✳

▲❡t ❳ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ ❧❡t

❜✐❧✐♥❡❛r ❢♦r♠ s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts

η

B : X×X → X

❜❡ ❛ ❝♦♥t✐♥✉♦✉s

s♦ t❤❛t

B(x, y) ≤ η x y ,
❢♦r ❛♥② ① ❛♥❞ ②

x = y − B(x, x)

∈ X✳

y ∈ X s✉❝❤ t❤❛t y < 1/(4η)✱
x ∈ X s❛t✐s❢②✐♥❣ x ≤ R✱ ✇✐t❤

❚❤❡♥ ❢♦r ❛♥② ✜①❡❞

❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥

R=

1−

1 − 4η y
.
2η

t❤❡ ❡q✉❛t✐♦♥


✶✼
Pr♦♦❢✳ ❙❡❡ ❚❤❡♦r❡♠ ✷✷✳✹ ✭❬✹✻❪✱ ♣✳ ✷✷✼✮✳

ET s♦ t❤❛t t❤❡ ❜✐❧✐♥❡❛r
B ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❜② ✭✶✳✶✹✮ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ET × ET → ET ✳
✷ ✐s ❞❡✈♦t❡❞ t♦ ❝♦♥str✉❝t ❡①❛♠♣❧❡s ♦❢ s✉❝❤ s♣❛❝❡s ET ✳ ❚❤❡ s♦❧✉t✐♦♥s t❤❛t ✇❡

❇② t❤❡ ❛❜♦✈❡ ❚❤❡♦r❡♠✱ ✇❡ ♥❡❡❞ t♦ tr② t♦ ✜♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡
♦♣❡r❛t♦r
❈❤❛♣t❡r

♦❜t❛✐♥ t❤r♦✉❣❤ t❤❡ P✐❝❛r❞ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ❛r❡ ❝❛❧❧❡❞ ♠✐❧❞ s♦❧✉t✐♦♥s✳ ❲❡ ❝❛❧❧ ❛ s♣❛❝❡

ET

✐❢ ✇❡ ♠❛② ❛♣♣❧② t❤❡ P✐❝❛r❞ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ❛s ❛♥ ❛❞♠✐ss✐❜❧❡ ♣❛t❤ s♣❛❝❡ ❢♦r t❤❡

◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ s♣❛❝❡

ET

❛s ❛♥ ❛❞❛♣t❡❞ ✈❛❧✉❡ s♣❛❝❡✳

▲❡t ✉s r❡✈✐❡✇ s♦♠❡ r❡s✉❧ts✳ ❲❡ ✇✐❧❧ ✐♥❞✐❝❛t❡ ✇❤❛t ❛r❡ t❤❡ ❛❞♠✐ss✐❜❧❡ ♣❛t❤ s♣❛❝❡
t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡

ET

❛♥❞

ET ✳

p
❈❧❛ss✐❝❛❧ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡s ❛r❡ ♣r♦✈✐❞❡❞ ❜② t❤❡ L t❤❡♦r② ♦❢ ❑❛t♦ ❬✸✼❪✿
p
✲ ❋♦r d < p < ∞, C([0; T ]; L ) ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡

•

✲ ❋♦r

p = d✱

Lp (Rd )✳

t❤❡ s♣❛❝❡

{f ∈ C([0; T ]; Ld ) : sup

√

t f

L∞ (dx)

√
< ∞ and lim t f
t→0

0✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡

L∞ (dx)

= 0}

Ld (Rd )✳

• Pr♦❞✐ ❬✺✷❪ ❣❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡s✱ ♣❧✉s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❛ss♦❝✐❛t❡❞
❛❞❛♣t❡❞ s♣❛❝❡
d
−1,q
d 2
ET = Lq ([0, T ], Lp ), ET = B˙ pp
with + = 1 and d < p < ∞.
p q

•

❈❛♥♥♦♥❡ ❬✶✷❪ st✉❞✐❡❞ t❤❡ s♣❛❝❡

α

α

{f ∈ C([0; T ]; Ld ) : sup t 2 f

Lq (dx)

< ∞ and lim t 2 f
t→0

0
Lq (dx)

= 0},

d
with q > d and α = 1 − ,
q
✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡

•

✭✶✳✶✺✮

✭✶✳✶✻✮

Ld (Rd )✳

■♥ ❬✷✺✱ ✷✻❪✱ ●❛❧❧❛❣❤❡r ❛♥❞ P❧❛♥❝❤♦♥ st✉❞✐❡❞ ❛ ❇❡s♦✈ s♣❛❝❡s s❝❛❧❡
2

ET = Lq B˙ pq

+ pd −1,q

d

, ET = B˙ pp

−1,q

with

d 2
+ > 1.
p q

■♥ ❈❤❛♣t❡r ✷ ♦❢ t❤✐s t❤❡s✐s ✇❡ st✉❞② s♦♠❡ ♦t❤❡r ❛❞♠✐ss✐❜❧❡ s♣❛❝❡s ✇✐t❤ ♦t❤❡r ❛ss♦❝✐❛t❡❞
❛❞❛♣t❡❞ s♣❛❝❡s✳

■♥ ❙❡❝t✐♦♥ ✷✳✷ ♦❢ ❈❤❛♣t❡r ✷✿
✲ ❋♦r

2 < q ≤ d ❛♥❞ p ❜❡ s✉❝❤ t❤❛t q < p < min

(d−2)q
,
d−q

d+2

✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡

s♣❛❝❡
2+d−p
p

Lp [0, T ]; H˙ p

∩ L∞ [0, T ]; H˙ qd/q−1

✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡
✲ ❋♦r

1
d/q−1
H˙ q
(Rd )✳

✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡

2q

L

[0, T ]; H˙

d+2−2q
q
dq
d+1−q

∩ L∞ [0, T ]; H˙ qd/q−1

✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡

d/q−1
H˙ q
(Rd )✳


✶✽

■♥ ❙❡❝t✐♦♥ ✷✳✸ ♦❢ ❈❤❛♣t❡r ✷✿

d
p > d2 , dp −1 ≤ s < 2p
, 1q = p1 − ds , and r > max{p, q}✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡
r
Kq,T
∩ L∞ ([0, T ]; H˙ ps ) ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡ H˙ ps (Rd )✱ ✇❤❡r❡
α
r
s♣❛❝❡ Kq,T ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s u(t, x) s✉❝❤ t❤❛t sup t 2 u(t, x)
< ∞ ❛♥❞
Lr
✲ ❋♦r

lim t

α
2

t→0

0
u(t, x)

Lr

=0

✇✐t❤

α=

d( 1q

−

1
)✳
r

■♥ ❙❡❝t✐♦♥ ✷✳✹ ♦❢ ❈❤❛♣t❡r ✷✿
t❤❡ ❙♦❜♦❧❡✈✲❋♦✉r✐❡r✲▲♦r❡♥t③ s♣❛❝❡s

✲ ❋♦r

1 < p ≤ d, 1 ≤ r < ∞✱

❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡

❛♥❞

Kp˜d
d]
[p

❋♦r s ∈ R
˙
HLs p,r (Rd )✳

p˜ ❜❡
,1,T

❛♥❞

s✉❝❤ t❤❛t

1
2p

1 ≤ p, r ≤ ∞
+

[ pd ]−1

2d

d
−1
p
Lp,r

∩L∞ [0, T ]; H˙

<

1
p˜

✇❡ ✐♥tr♦❞✉❝❡ ❛♥❞ st✉❞②

[ pd ] 1
,
d 2

< min

+

[ pd ]−1
2d

✱ ✇❡

✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦✲

d

−1
p˜
H˙ Lpp,r (Rd )✱ ✇❤❡r❡ t❤❡ s♣❛❝❡ Kp,r,T
✐s ♠❛❞❡ ✉♣ ❜② t❤❡ ❢✉♥❝t✐♦♥s u(t, x)
α
2
u(t, x) dp −1 < ∞ ❛♥❞ lim t u(t, x) pd −1 = 0 ✇✐t❤ α = d p1 − p1˜ ✳

❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡
s✉❝❤ t❤❛t

α

sup t 2
0
✲ ❋♦r

H˙

p ≥ d, r ≥ 1✱

❛♥❞

q > p✱

H˙

t→0

˜
Lp,r

˜
Lp,r

d

−1
q
Kd,1,T
∩ L∞ ([0, T ]; H˙ Lpp,r )

✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡
d

✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡

−1
H˙ Lpp,r (Rd )✳

∞
˙ d−1
✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡ Ks,r,T ∩ L ([0, T ]; H
L1,r )
d−1
d

˙
✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡ H 1,r (R )✱ ✇❤❡r❡ t❤❡ s♣❛❝❡ Ks,r,T ✐s
L α
♠❛❞❡ ✉♣ ❜② t❤❡ ❢✉♥❝t✐♦♥s u(t, x) s✉❝❤ t❤❛t
sup t 2 u(t, x) H˙ s
< ∞ ❛♥❞
L1,r
0✲ ❋♦r

d−1
α

lim t 2 u(t, x)
t→0

H˙ s 1,r

❛♥❞

r ≥ 1✱

=0

✇✐t❤

α = s + 1 − d.

L

■♥ ❙❡❝t✐♦♥ ✷✳✺ ♦❢ ❈❤❛♣t❡r ✷✿

q > 1, 1 ≤ r ≤ ∞ ❛♥❞ 0 ≤ s < dq ✱ ✇❡ ✐♥tr♦❞✉❝❡
˙ s q,r (Rd )✱ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ❝❧❛ss✐❝❛❧
❛♥❞ st✉❞② t❤❡ ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s H
L
˙ qs (Rd )✳ ❋♦r s ≥ 0, q > 1, r ≥ 1, s < 1 ≤ s+1 ✱ ❛♥❞ q˜ ❜❡ s✉❝❤ t❤❛t
❙♦❜♦❧❡✈ s♣❛❝❡s H
d
q
d
1 1
s
1
1
s 1
+
< < min + , ✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡ Ks,˜q ∩L∞ ([0, T ]; H˙ s q,r )
2

q

d

q˜

2

❋♦r

L

q,1,T

2d q

˙ s q,r (Rd )✱ ✇❤❡r❡ s♣❛❝❡ Ks,˜q ✐s ♠❛❞❡
✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ ❛❞❛♣t❡❞ s♣❛❝❡ H
L
q,r,T
α
α
✉♣ ❜② t❤❡ ❢✉♥❝t✐♦♥s u(t, x) s✉❝❤ t❤❛t sup t 2 u(t, x) ˙ s
<
∞
❛♥❞ limt 2 u(t, x) ˙ s
=0
H
H
✇✐t❤

α=

−

t→0

Lq˜,r

0
d( 1q

Lq˜,r

1
)✳
q˜

■♥ ❙❡❝t✐♦♥ ✷✳✻ ♦❢ ❈❤❛♣t❡r ✷✿ ❋♦r 0 ≤ m < ∞ ❛♥❞ ✐♥❞❡① ✈❡❝t♦rs q = (q1 , q2 , ..., qd )
❛♥❞ r = (r1 , r2 , ..., rd )✱ ✇❤❡r❡ 1 < qi < ∞, 1 ≤ ri ≤ ∞ for i = 1, 2, .., d✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥❞
˙ m ✳ ❋♦r q > ✶, r ≥ ✶, 2 < p < ∞✱ ❛♥❞ m ≥ 0
st✉❞② ♠✐①❡❞✲♥♦r♠ ❙♦❜♦❧❡✈✲▲♦r❡♥t③ s♣❛❝❡s H
L
q,r

❜❡ s✉❝❤ t❤❛t

m<

1
2

d
1
2
i=1 qi , p

−m+

d
1
i=1 qi

qi

≤ 1, 2 <
1−

< ∞, i = 1, 2, .., d,

m
1
d
i=1 qi

p
˙ mq,r ) ✇❤✐❝❤ ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞
✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡ L ([0, T ]; H
L
2
m− p ,p
❛❞❛♣t❡❞ s♣❛❝❡ BLq,r
✭❛ ❇❡s♦✈ s♣❛❝❡✮✳

■♥ ❈❤❛♣t❡r ✸✿

❯s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❋♦✐❛s✲❚❡♠❛♠✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ❍❛✉s❞♦r✛

❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s✐♥❣✉❧❛r s❡t ✐♥ t✐♠❡ ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s t♦ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✳