Local in-time existence and regularity of solutions of the - PowerPoint PPT Presentation

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Local in-time existence and regularity of solutions of the Navier-Stokes equations via discretization Jo˜ ao Teixeira Departamento de Matem´ atica, Instituto Superior T´ ecnico, Lisboa, Portugal June 5, 2008 Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions 1 The Navier-Stokes equations in T n 2 Discretization of the modified Navier-Stokes problem 3 Estimating the iterates 4 Existence and regularity of solutions Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Here, let T n = R n / Z n is the n -dimensional torus and D = T n × [0 , T ), with T > 0. Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Here, let T n = R n / Z n is the n -dimensional torus and D = T n × [0 , T ), with T > 0. Let ν ∈ R + . Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Here, let T n = R n / Z n is the n -dimensional torus and D = T n × [0 , T ), with T > 0. Let ν ∈ R + . Let u 0 : R n / Z n → R n be C 2 with second order partial derivatives Lipshitz continuous. Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Here, let T n = R n / Z n is the n -dimensional torus and D = T n × [0 , T ), with T > 0. Let ν ∈ R + . Let u 0 : R n / Z n → R n be C 2 with second order partial derivatives Lipshitz continuous. Let f : R n / Z n → R be C 1 , with Lipshitz continuous partial derivatives. Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Here, let T n = R n / Z n is the n -dimensional torus and D = T n × [0 , T ), with T > 0. Let ν ∈ R + . Let u 0 : R n / Z n → R n be C 2 with second order partial derivatives Lipshitz continuous. Let f : R n / Z n → R be C 1 , with Lipshitz continuous partial derivatives. Without loss of generality, div f = 0. (Otherwise, replace f by f − ∇ p 0 , with ∆ p 0 = − div f ). Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions The Navier-Stokes problem The initial value problem for the Navier-Stokes equations is:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  div u = 0 in D (1) on T n × { 0 } u = u 0  Here, let T n = R n / Z n is the n -dimensional torus and D = T n × [0 , T ), with T > 0. Let ν ∈ R + . Let u 0 : R n / Z n → R n be C 2 with second order partial derivatives Lipshitz continuous. Let f : R n / Z n → R be C 1 , with Lipshitz continuous partial derivatives. Without loss of generality, div f = 0. (Otherwise, replace f by f − ∇ p 0 , with ∆ p 0 = − div f ). The unique (if it exists) strong solution of (1) is a pair of “sufficiently regular” functions u : D → R n , p : D → R satisfying (1) Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Modified Navier-Stokes equations Consider a cut-off function χ M ∈ C ∞ ([0 , ∞ )) such that: � 1 if r < M χ M ( r ) = 0 if r > 2 M Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Modified Navier-Stokes equations Consider a cut-off function χ M ∈ C ∞ ([0 , ∞ )) such that: � 1 if r < M χ M ( r ) = 0 if r > 2 M Let � u � 1 , 2 = max � u x i � L ∞ ( T n × [0 , T ]) + max i , j � u x i x j � L ∞ ( T n × [0 , T ]) . i Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Modified Navier-Stokes equations Consider a cut-off function χ M ∈ C ∞ ([0 , ∞ )) such that: � 1 if r < M χ M ( r ) = 0 if r > 2 M Let � u � 1 , 2 = max � u x i � L ∞ ( T n × [0 , T ]) + max i , j � u x i x j � L ∞ ( T n × [0 , T ]) . As i long as � u � 1 , 2 < M , the Navier-Stokes equations are equivalent to:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  ∆ p = − χ M ( � u � 1 , 2 ) tr ( D x u ) 2 in D (2) on T n × { 0 } u = u 0  Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Modified Navier-Stokes equations Consider a cut-off function χ M ∈ C ∞ ([0 , ∞ )) such that: � 1 if r < M χ M ( r ) = 0 if r > 2 M Let � u � 1 , 2 = max � u x i � L ∞ ( T n × [0 , T ]) + max i , j � u x i x j � L ∞ ( T n × [0 , T ]) . As i long as � u � 1 , 2 < M , the Navier-Stokes equations are equivalent to:  u t − ν ∆ u + ( u · ∇ ) u = −∇ p + f in D  ∆ p = − χ M ( � u � 1 , 2 ) tr ( D x u ) 2 in D (2) on T n × { 0 } u = u 0  n ∂ u i ∂ u j Note that div ( u · ∇ ) u = tr ( D x u ) 2 = � ∂ x j ∂ x i i , j =1 Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Discretized space and time • Discretization of T n (for any n ∈ N ): � n � T n = 0 , h , 2 h , . . . , ( M − 1) h , 1 M h ( Z mod M ) n ; = with M ∈ N 1 , and h = 1 M . Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Discretized space and time • Discretization of T n (for any n ∈ N ): � n � T n = 0 , h , 2 h , . . . , ( M − 1) h , 1 M h ( Z mod M ) n ; = with M ∈ N 1 , and h = 1 M . Given any x = ( m 1 , m 2 , . . . , m n ) h and y = ( l 1 , l 2 , . . . , l n ) h in T n M , let: � � x + y = ( m 1 + l 1 ) mod M , . . . , ( m n + l n ) mod M h Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity

The Navier-Stokes equations in T n Discretization of the modified Navier-Stokes problem Estimating the iterates Existence and regularity of solutions Discretized space and time • Discretization of T n (for any n ∈ N ): � n � T n = 0 , h , 2 h , . . . , ( M − 1) h , 1 M h ( Z mod M ) n ; = with M ∈ N 1 , and h = 1 M . Given any x = ( m 1 , m 2 , . . . , m n ) h and y = ( l 1 , l 2 , . . . , l n ) h in T n M , let: � � x + y = ( m 1 + l 1 ) mod M , . . . , ( m n + l n ) mod M h • Discretization of time: with T ∈ R + and K ∈ N 1 , let k = T K , and define: � � I T K = 0 , k , 2 k , . . . , ( K − 1) k = k ( N ∩ [0 , K )); Jo˜ ao Teixeira Local in-time Navier-Stokes eqn. existence and regularity