Yoshie Sugiyama (Tsuda university)

Transcription

1 Measure valued soluions of he D Keller Segel sysem oin work: Sephan Luckhaus (Leipzig Univ.) Yoshie Sugiyama (Tsuda universiy) Juan J.L.Velázquez (ICMAT in Madrid) The 3rd OCAMI-TIMS Join Inernaional Workshop March 3,

2 Keller-Segel sysem [97] diffusion aggregaion (KS) ì ï í ï ï î u =Du -div uñv, ( ) -D v = u - ò udx W W u( x, ) =, v( x, ) = n n u( x,) = u ( x),,, xîw, > xîw, xî W, xîw > > u ( x, ) v( x, ) densiy of amoebae concenraion of he chemical subsances WÌR :bounded W:smooh

3 I. Known resul local classical soluion u Î L ( W), W Ì R : bounded $ T >, $ ( u, v) : classical sol of (KS) on [, T ) Remark scale invarian norm: u L u L $ ( u, v) : classical sol of (KS) on [, ) small daa global exisence: large daa finie ime blow-up: limsup T * u( ) L u L = * for some T <

4 Scaling invarian ransform where { u, v}: sol of (KS) { ul, vl}: sol of (KS) for " l > ul x l u lx l vl ( x, ) : = v( lx, l ) (, ) : = (, ), " N ³ u u N N N L ( R N ) l = for " l >, L ( R ) u u N = l = for " l >, L ( R ) L ( R ) 3

5 Mass conservaion law { u, v}: sol of (KS) on W [, T ) u u( ) L W L W = " < T ( ) ( )

6 (locaion of blow-up poins) sharp e - regulariy hm Nagai-Senba-Suzuki, Luckhaus-S.-Velazquez Mass cons. law u( ) L ( W) L ( W) Scaling invaran norm = u sup u( ) = sup u ( ) L ( R ) l L ( R ) < < < < " < new resul sup < < T ò B (( x,, r )) u ( x, ) ) dx dx < 8p indep. of x sup sup u( x, ) < u( xc, ) < ( x( x,, ) Î ) B (( xx,, r ) ) (, T- ) c r ) ( (, ) C indep. of x 5

7 x i N Î R : blow-up poin of u a he ime ò B(, r ) x i u( x, ) dx > 8p for " r > x i blow-up se consiss of finiely many poins u = u( ) a he ime L ( W) L ( W) Quesion (i) Locaion of blow-up poins (ii) Number of blow-up poins x i sandard ε-reg hm sharp ε-reg hm (iii) δ-funcional singulariy w ([, *]; ( )) C T L W 6

8 u C T L Î W Quesion w ([, *]; ( )) (i) Locaion of blow-up poins (ii) Number of blow-up poins (iii) δ-funcional singulariy * $ u Î L ( W) s.. ò W u( x, ) ( x) dx u * ( x) ( x) dx ò W as T* for " Î L ( W) 7

9 = T * : blow-up ime Nagai Senba Suzuki Herrero Velazquez u > 8 L p» a ( T* ) d x T k ( ) *» a ( T* ) d x T ( ) * : he number of he blow-up poins ai( T* ) ³ 8 p, i =,,, Quesion (i) Locaion of blow-up poins (ii) Number of blow-up poins (iii) δ-funcional singulariy k k u 8p» a ( T ) d k L ( W) k * x ( T ) sandard ε-reg hm sharp ε-reg hm C T L W w ([, *]; ( )) * 8

10 The exisence of minimal immersions of -spheres J.Sacks, and K.Uhlenbeck Ann. of Mah, Vol. 7, No., 98, pp Parial regulariy of suiable weak soluions of he Navier-Sokes equaions L.A.Caffarelli, R.Kohn, and L.Nirenberg Comm. Pure Appl. Mah., Vol. 35, 98, pp

11 II. Purpose Consrucion of Thm for large iniial daa u s.. Thm 3 "measure valued sol" of (KS) on [, ) wih a simple expression of soluion u L Remark such as ha a T * () srong soluion is unique () measure valued soluion is unique? Thm, Thm 4 (non-uniquenss of measure valued sol)

12 Thm, Thm 3 (consrucion of measure valued sol) Thm, Thm 4 (non-uniquenss of measure valued sol) x ( ) x ( ) ( i) ( ii) m k u e T * = + " ³ T * m, dm = dmd» a ( ) d x å a ( ) d x ( )( ) (, ) u dx a.e. Î [, ) x ( ) ÎS (, ; L ( )) u Î L W x ( ) ( iii) S º S {( x, ); = } consiss of a mos finiely many poins

13 III. Our resuls Regularizaion: I u =Du -div f ( u) Ñv, (KS) e ì ï í ï ï î u ( ) e -D v = fe ( u) - ò fe ( u) dx, W W ( ) v( x ) u x, =,, = n ( ) = u ( x) u x,, f ( u) = ò min{, ( / -s) } ds e e + u n ìu -div ( ) Ñ = í îu ( fe u v) fe ( u) xîw, > /e u >> xîw, u << xî W, xîw. > > u

14 Theorem [Luckhaus-S-Velázquez.] e e f u e e ( u ) u $ : sol. of (KS) e on [, ), $ Radon measure m, m - Î M + ( W [, )) and $ { e k } k = k s.. u e k m, f ( u e e ) m - in he weak-* opology k and m - m - - Moreover, dm = dm d, dm = dm d Moreover, he singular se S of m, m - Ì W [ T*, ) for some < T* < and wih where m = m + u, m - = m - + u, m, m - Î M + ( W [, )) u ÎC (( W [, ))\ S) W = ò m m wih m ( ) u ( x) dx, - for all W saisfying < < (KS) in W [, T ) * 3

15 Moreover, for a.e. Î [, ) S º S {( x, ); = } and () () m m - ( ) ÎS consiss of a mos finiely many poins = å a ( ) d x ( ) + u(, ) dx a.e. Î [, ) x ( ) ÎS ( ) - = å b ( ) d x ( ) + u(, ) dx a.e. Î [, ) x ( ) - - dm = dm d, dm = dm d - Here a ( ) ³ b ( ) ³ for all < < In paricular, a ( )= b - ( )= for all < < T * u Î L (, ; L ( W)) wih In addiion, ò W u( x, ) dx u ( x) dx ò W for all < < 4

16 Theorem [Luckhaus-S-Velázquez.] Suppose ha $ A Ì [ T, ) of posiive measure - s.. b ( ) > 8p for a.e. Î A * wih some in () - a ( ) > b ( ) for a.e. Î A in () and () k u e k m, f ( u e e ) m - -, a ( ) ³ b ( ) ³ k for all < < () () m m - = å a ( ) d x ( ) + u(, ) dx a.e. Î [, ) x ( ) ÎS ( ) ÎS ( ) - = å b ( ) d x ( ) + u(, ) dx a.e. Î [, ) x ( )

17 Regularizaion: II (KS) e ì ï í ï ï î u =D u+ eu - uñv 7/6 ( ) div, -D v = u - ò udx, W W ( ) v( x ) ( ) = u ( x) u x,, ( ) u x, =,, = n n xîw, > xîw, xî W, xîw. > > 6

18 Theorem 3 [Luckhaus-S-Válazquez.] $ u e : sol. of (KS) on [, ), k s.. u e m, Moreover, Moreover, he singular se S of m, m + and wih where e $ Radon measure m, m + Î M + ( W [, )) u e + + dm = d m d, dm = dm d m = m + u, m + = m + + u, $ { } e k k and = and [ T*, ) for some T* Ì W < < m, m + Î M + ( W [, )) u ÎC ( W [, ) \ S) k saisfying in he weak-* opology wih m ( ) u ( x) dx, for all W = ò W + ek 7 / 6 e k ( u ) m + m m + e e e u u +e( u ) m m + 7/6 < < (KS) in W [, T ) * 7

19 Moreover, for a.e. Î [, ) S º S {( x, ); = } and (3) (4) consiss of a mos finiely many poins m = å a ( ) d x ( ) + u (, ) dx a.e. Î [, ) m + ( ) x ( ) ÎS = å b + x + ( ) x ( ) ÎS + + d m = d m d, dm = dm d ( ) d ( ) u (, ) dx a.e. Î [, ) + Here a ( ) b ( ) In paricular, a ( )= b + ( )= u Î L (, ; L ( W)) In addiion, wih for all < < for all < < T * ò W u ( x, ) dx u ( x) dx ò W for all < < 8

20 Theorem 4 [Luckhaus-S-Velázquez.] Suppose ha $ A Ì [ T, ) of posiive measure s.. a ( ) > 8p for a.e. Î A * wih some in (4) + b ( ) > a ( ) for a.e. Î A in (3) and (4) m = å a ( ) d x ( ) + u (, ) dx, ( ) cf x ( ) ÎS Theorem [Luckhaus-S-Velázquez.] Suppose ha $ A Ì [ T, ) of posiive measure - s.. b ( ) > 8p for a.e. Î A * m + = b + ( ) d ( ) + u (, ) dx x ( ) ÎS å x ( ) wih some in () - a ( ) > b ( ) for a.e. Î A in () and () 9

21 Remark T * " ³ T * = non-uniquenss of measure valued sol " < < Regularizaion: I Regularizaion: II Regularizaion: I Regularizaion: II ( m ) = D m + [ m - ] Q + ( m ) = D m + [ m ] Q òò Q[ m ] y dxd» òò 4 p x- y > m ( x) m ( y) y [( x - y) ( Ñ ( x, ) - Ñy ( y, ))] d x - y + òò ( n Ñ y ( x, ) n ) dmˆ ( x, n ) d 4p W S y C Î ( W (, ))

22 Theorem 5 [Luckhaus-S-Velázquez.] coninuiy for he singular se ( x, ) S, we have dm Î ò B ( R x )\{ x} m for some m and R > fixed. $ c >, $ L > depending only on u and W such ha S Ç B ( x ) Ì B ( x ) R L - L ( W) for Î[ - cr, ]

23 Remark 3 Key idea: e -regulariy hm ( x, ) Î S x x = L - S Ç B ( x ) Ì B ( x ) R L - for Î[ - cr, ]

24 IV. Skech of Proof Theorem 5 x ( ) Theorem x ( ) x u e k Theorem 3 T T * " ³ T * m, dm = dmd sup dm ò ò dm 8p < < T B( x, r ) B( x, r )» a ( ) d x x < ( ) ( i) ( ii) m å a ( ) d x ( )( ) (, ) u dx a.e. Î [, ) x ( ) ÎS (, ; L ( )) = + u Î L W ( iii) S º S {( x, ); = } consiss of a mos finiely many poins 3

25 x i N Î R : blow-up poin of u a he ime ò B( x, r ) i dm > 8p for " r > x i blow-up se consiss of finiely many poins u ( ) ( = dm W ) W L a he ime Quesion (i) Locaion of blow-up poins (ii) Number of blow-up poins (iii) dela measure x i sharp ε-reg hm sharp ε-reg hm 4

26 Theorem 5 x ( ) Theorem x ( ) x u e k Theorem 3 T * " ³ T * m, dm = dmd sup dm ò ò dm 8p < < T B( x, r ) B( x, r )» a ( ) d x x < ( ) ( i) ( ii) m å a ( ) d x ( )( ) (, ) u dx a.e. Î [, ) x ( ) ÎS (, ; L ( )) = + u Î L W ( iii) S º S {( x, ); = } consiss of a mos finiely many poins 5

27 Remark 4 Poupaud (Meh. Appl. An. ). Dolbeaul-Schmeiser (6). complicaed assumpion on measure Luckhaus-S-Velazquez m = å a ( ) d ( ) + u(, ) dx, x x ( ) ÎS ( ) by e -regulariy hm < < 6

28 Poupaud (Meh. Appl. An. ). Dolbeaul-Schmeiser (6). N M ( R ) is he space of Radon measures, N M( R ) is he space of bounded Radon measures l is he Lebesgue measure Definiion I Ì R : inerval N N DM + ( I; R ) : = {( m, n ); " Î I, m( ) Î M + ( R ), N N N n Î M ( I R ), m( ) is a ighly coninuous wih respec o, n is a non negaive, symmeric, marix valued measure, r( n (, x)) å ( m( )({ a}))( d ( x - a) l( ) } aîs a ( m ( )) 7

29 Remark 4 Poupaud (Meh. Appl. An. ). Dolbeaul-Schmeiser (6). complicaed assumpion on measure Luckhaus-S-Velazquez m = å a ( ) d ( ) + u(, ) dx, x x ( ) ÎS ( ) by e -regulariy hm < < 8

30 Previous resuls: (85-95). Seady saes (Schaap). Finie ime blow-up. (Jäger-Luckhaus). Dirac mass formaion. Asympoics (Herrero-Velazquez). Criical masses. (Biler, Nagai,...). Enropies. (Gaewsky-Zacharias). Mulispecies models (Wolansky). New resuls:... 9

31 Conrolling he moion of he mass: A crucial symmerizaion argumen. Senba-Suzuki (Adv. Diff. Equ. ). Poupaud (Meh. Appl. An. ). Dolbeaul-Schmeiser (6). 3

32 Thank you for your kind aenion 3

33 ò ò u y dxd = J» ò ò W W D u y dxd + uñ v Ñ ò ò ò ò u( x, ) N ( x, y)( u( y, ) u ) dy y ( x, ) dx = ò ò Ñ - W Ñ W W W W [( x - y) ( Ñy ( x, ) - Ñy ( y, ))] u( x, ) u( y, ) dxdy x - y x - y h( ) d = ò ò 4 m ( x) m ( y) dxdy òò p x- y > + Ñ 4p W S é x - y ù + ò ò -h( ) dxdy W W ê ë d ú û J dxd y [( x - y) ( Ñy ( x, ) - Ñy ( y, ))] d x - y 3 y C Î ( W (, )) x d ˆ x d ( n y (, ) n ) m (, n ) òò

34 Skech of proof of Th ò d ˆ m ( x, n ) = å g ( ) d x ( ) S x ( ) ÎS 8 pb ( ) ( b - ( )) < assump of Th Fac Fac g 8 pa ( ) ò d S m ˆ ( x, n ) ³ ( m - ) sing 33