Analysis on Keller-Segel Models in Chemotaxis Li CHEN Universit at - PowerPoint PPT Presentation

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Analysis on Keller-Segel Models in Chemotaxis Li CHEN Universit¨ at Mannheim 03.2019, Potsdam Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Keller-Segel model with linear diffusion 1 Keller-Segel equation with nonlinear diffusion in multi-D 2 Keller-Segel model with nonlinear nonlocal reaction 3 Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Keller-Segel model with linear diffusion Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction The parabolic-elliptic Keller-Segel model with linear diffusion is � � case ∇ K = C x ρ t = ∆ ρ − div ( ρ ∇ c ) , − ∆ c = ρ. . | x | n Typical quantities of the system � � Conservation of mass m 0 ( t ) = ρ ( x , t ) dx = ρ 0 ( x ) dx = m 0 � ( ρ ln ρ − ρ c Entropy dissipation relation for F ( ρ ) = 2 ) dx � d ρ |∇ ln ρ − ∇ c | 2 dx = 0 . ⇒ F ( ρ ) ≤ F ( ρ 0 ) dt F ( ρ ) + Key feature of the system: Global existence vs. finite time blow up. Since 1990’s, J¨ ager and Luckhaus, Biler, Herrero, Horstmann, Medina, Nagai, Stevens, Velazquez, Winkler...... Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction The critical mass 8 π in 2 -D � c = − 1 1 ρ t = ∆ ρ − div ( ρ ∇ c ) , log | x − y | ρ ( y ) dy 2 π � � � ρ ln ρ dx − 1 1 Entropy: F ( ρ ) = ρ ( x , t ) ρ ( y , t ) log | x − y | 2 dxdy . 8 π Logarithmic Hardy-Littlewood-Sobolev inequality � � � ρ log ρ dx − 1 1 ρ ( x ) ρ ( y ) log | x − y | 2 dxdy + C ( m 0 ) ≥ 0 , m 0 � where m 0 = ρ ( x ) dx , C ( m 0 ) := m 0 (1 + log π − log m 0 ). A direct application of this inequality gives � F ( ρ ( · , t )) ≥ (1 − m 0 ρ log ρ dx − m 0 8 π C ( m 0 ) , 8 π ) � � F ( ρ ( · , t )) ≥ ( 1 m 0 − 1 1 | x − y | 2 dxdy − C ( m 0 ) . 8 π ) ρ ( x , t ) ρ ( y , t ) log m 0 < 8 π : global existence , Blanchet, Dolbeault, Perthame, 2006. Another proof: Carrillo, Chen , Liu, and Wang, based on Delort’s theory of 2-D incompressible Euler equation, 2012. Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction blow up in finite time 2-D, m 0 > 8 π , Dolbeault, Perthame, 2004. � | x | 2 Idea: Second moment, m 2 ( t ) := 2 ρ dx , 2 ( t ) = 2 m 0 (1 − m 0 m ′ 8 π ) < 0 . n multi-D, m 2 < Cm n − 2 , Perthame, 2005. 0 n − 2 2 − 2 1 multi-D, m n (0) < ( n − 1)2 n | S n − 1 | m n , Chen , Siedentop, 2017. n 0 � | x | n ρ dx , Idea: n-th moment m n ( t ) := 0 − n 2 1 − n n − 2 2 m ′ | S n − 1 | m 2 n ( t ) ≤ 2 n ( n − 1) m 0 . n m n n Remark: The global existence only holds for “small” initial data. It is an interesting question whether the reverse inequality would imply global existence. If not, what additional condition can help? Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Main idea with the n-th moment The following Keller-Segel system is considered (with an additional point source) ∂ t ρ − ∇ · ( ∇ ρ − ρ ∇ c ) = 0 , x ∈ R n , t ≥ 0 , − ∆ c = ρ − Z δ, ρ ( x , 0) = ρ 0 ( x ) ≥ 0 . Use | x | n as test function � � nZ m ′ | x | n − 2 ρ ( x ) dx + n ( t ) = 2 n ( n − 1) ρ ( x ) dx | S n − 1 | � � n ( | x | n − 1 x | x | − | y | n − 1 y | y | ) x − y − ρ ( x ) ρ ( y ) dxdy 2 | S n − 1 | | x − y | n � �� � =: V We can show that V ≥ 2 2 − n , therefore n ( t ) ≤ 2 n ( n − 1) m n − 2 − n 2 1 − n nZ m ′ | S n − 1 | m 2 0 + | S n − 1 | m 0 . Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Further interpolation shows that 0 − n 2 1 − n n − 2 2 nZ | S n − 1 | m 2 m ′ n ( t ) ≤ 2 n ( n − 1) m n m n 0 + | S n − 1 | m 0 . n In particular, we have a shrinking n -th moment, if the initial moments fulfill 1 n − 2 2 − 2 Z 1 − 2 m n (0) < ( n − 1)2 n | S n − 1 | m n − 2( n − 1) | S n − 1 | m n . n 0 0 In case of no external point source ( Z = 0) and n = 2, we have 1 < m 0 8 π . Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Keller-Segel equation with nonlinear diffusion in multi-D Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction To balance the aggregation, porus media type nonlinear diffusion is considered since 2005, ∆ ρ m − div ( ρ ∇ c ) ρ t = m ≥ 1 � � m �� m − 1 ρ m − 1 − c , x ∈ R n , t ≥ 0 . = ρ ∇ div m ∗ = 2 − 2 n , Idea: scaling invariance of the total mass. (if ρ ( x , t ) is a solution, then λ n ρ ( λ x , t ) is also a solution). Many results for m = m ∗ since 2005: Bedrossian, Bertozzi, Blanchet, Carrillo, Cie´ slak, Horstmann, Ishida, Kowalczyk, Laurencot, Luckhaus, Rodr´ i guez, Sugiyama, Szyma´ nska ,Winkler, Yao, Yokota ... However , the results are not as “beautiful” as the 2-D case. 2 n m c = n +2 , Idea: stationary solutions and conformal invariant of the entropy, Chen , Liu, Wang, 2012. Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Stationary solutions in 2-D, ln ρ − c = 0, − ∆ c = e c in R 2 , has a family of solutions C λ, x 0 ( x ), and the stationary solution for ρ is � � 2 λ U λ, x 0 ( x ) = e C λ, x 0 ( x ) = 8 , with � U λ, x 0 � L 1 = 8 π. λ 2 + | x − x 0 | 2 m − 1 ρ m − 1 − c = 0, m Stationary solutions in Multi-D, − ∆ c = ( m − 1 1 1 m − 1 c m − 1 in R n . ) m n − 2 , then c ≡ 0. m ∗ belongs to this case. m − 1 < n +2 1 Gidas, Spruck (1981) 1 ≤ m − 1 = n +2 1 2 n n − 2 , or m = n +2 , then c = C λ, x 0 ( x ), and the stationary solution is � n +2 � λ 2 n +2 n +2 4 n with � U λ, x 0 � m U λ, x 0 ( x ) = 2 L m = K ( n ) . 2 λ 2 + | x − x 0 | 2 Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Results with exponent m c = 2 n / ( n + 2). Radially symmetric solutions, Chen , Liu, Wang (2012) t →∞ → 0 in L 1 loc ( R n ). ρ 0 ( | x | ) < U λ 0 ( | x | ), global existence , ρ ( | x | , t ) − ρ 0 ( | x | ) > U λ 0 ( | x | ), blow-up , ∃ t ∗ ≤ + ∞ , i.e., ∃ r ( t ) t → t ∗ − → 0 s.t. � ρ ( | x | , t ) dx ≥ C > 0 . B (0 , r ( t )) For general initial data, Chen , Liu, Wang (2012) � ρ 0 � L mc < C s < � U λ � L mc , global existence , for t ≫ 1, it holds β = 2 m 2 c − 3 m c + 2 1 � ρ ( · , t ) � L mc ≤ Ct − mc ( β − 1) , > 1 . m c ( m c − 1) m 2 (0) < ∞ , F ( ρ 0 ) < F ( U λ ) and � ρ 0 � L mc > � U λ � L mc , blow up in the sense that ∃ T ∗ < ∞ s.t. t → T ∗ � ρ ( · , t ) � L mc = + ∞ . lim Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction Results with exponent m c < m < m ∗ . Chen , Wang, (2014) 2 n For ρ 0 ∈ L 1 + ( R n ) ∩ L n +2 ( R n ), F ( ρ 0 ) < F ∗ , the following holds, n − 2 2 n ( m − 1) , global existence n +2 ( R n ) < ( s ∗ ) � ρ 0 � 2 n L n − 2 2 n ( m − 1) , m 2 (0) < ∞ , finite time blow up . n +2 ( R n ) > ( s ∗ ) � ρ 0 � 2 n L where F ∗ = K 1 ( n , m ) m α 1 ( m , n ) s ∗ = K 2 ( n , m ) m α 2 ( m , n ) > 0 , . 0 0 Remarks n − 2 2 n n +2 norm of the initial data can not be ( s ∗ ) 2 n ( m − 1) . If F ( ρ 0 ) < F ∗ , L Thus the classification of the initial data is complete. F ( ρ 0 ) < F ∗ gives a relation between the mass and the free energy, 2 − 2 � 2 n 2 α ( n ) n ( m − 1) n − m � m ( n +2) − 2 n 2 n − 2 − mn . F ( ρ 0 ) M 2 n − 2 − mn < 0 ( m − 1)(1 − 2 C ( n ) n ) This tells that in this exponent region, total mass is not the appropriate quantity to classify global existence and blow up. Li CHEN Keller-Segel Models in Chemotaxis