Discontinuous Solutions for the Degasperis-Procesi Equation. - PDF document

Discontinuous Solutions for the Degasperis-Procesi Equation. Giuseppe Maria Coclite Department of Mathematics University of Bari Via Orabona 4, 70125 Bari, Italy e-mail : coclitegm@dm.uniba.it joint work with Professor K. H. Karlsen (University of Oslo and C.M.A. - Oslo, Norway) 0

Consider the equation ∂ 3 u ∂ 2 u ∂x 2 + u∂ 3 u ∂u ∂t∂x 2 + 4 u∂u ∂x = 3 ∂u ∂t − ( DP ) ∂x 3 , ∂x where t ≥ 0 , x ∈ I u ( t, x ) ∈ I R, R. (DP) is termed Degasperis - Procesi Equation. Deduction. Degasperis and Procesi in 1999 studied the following family of third order dispersive nonlinear equations , indexed over six constants c 0 , γ, α, c 1 , c 2 , c 3 ∈ I R : � � � ∂u � 2 ∂x + γ ∂ 3 u ∂x 3 − α 2 ∂ 3 u + c 3 u∂ 2 u ∂u ∂u ∂t∂x 2 = ∂ c 1 u 2 + c 2 ∂t + c 0 . ∂x 2 ∂x ∂x Using the method of asymptotic integrability , they found that only three equations from this family were asymptotically integrable up to third order : • the KdV equation : α = c 2 = c 3 = 0; • the Camassa-Holm equation : c 1 = − 3 c 3 2 α 2 , c 2 = c 3 2 ; • one new equation : c 1 = − 2 c 3 α 2 , c 2 = c 3 (after a proper scaling) � ∂ 3 u � ∂x + ∂ 3 u ∂ 2 u 2 u∂ 3 u ∂u ∂t + ∂u ∂x +6 u∂u ∂t∂x 2 + 9 ∂u ∂x 2 + 3 ∂x 3 − α 2 = 0 . ∂x 3 2 ∂x 1

After • rescaling • shifting the dependent variable • applying a Galilean boost the three equations read • the KdV equation : ∂x + ∂ 3 u ∂u ∂t + u∂u ∂x 3 = 0 ( KdV ) • the Camassa-Holm equation : ∂ 3 u ∂x 2 + u∂ 3 u ∂ 2 u ∂u ∂t∂x 2 + 3 u∂u ∂x = 2 ∂u ∂t − ( CH ) ∂x 3 ∂x • the Degasperis - Procesi equation : ∂ 3 u ∂ 2 u ∂x 2 + u∂ 3 u ∂u ∂t∂x 2 + 4 u∂u ∂x = 3 ∂u ∂t − ( DP ) ∂x 3 ∂x Physics: Shallow Water Waves. ⇒ depth of the water Shallow Water Waves ⇐ < 1 < length of the wave • Unidirectional shallow water waves • u ≡ wave height (KdV) / velocity (CH - DP) above the bottom • Flat bottom 2

Essential Literature. Degasperis - Holm - Khon (2002): • exact integrability (by constructing a Lax pair) • bi-Hamiltonian structure • two infinite sequences of conserved quantities • “non-smooth” solutions: superpositions of multipeakons • some special explicit solution Lundmark - Szmigielski (2003): • n-peakon solutions (via inverse scattering) Mustafa (2005): • smooth solutions have infinite speed of propagation Yin (2003-2003-2004-2004): • Cauchy Problem: local and global well-posedness u 0 ∈ H r ( S 1 ) , r > 3 u 0 ∈ H s ( I R ) , s ≥ 3 or 2 � � u 0 − ∂ 2 u 0 sign constant ∂x 2 Remark. The signum of the vorticity u − ∂ 2 u ∂x 2 is conserved � � � � u 0 − ∂ 2 u 0 u ( t, · ) − ∂ 2 u ⇒ sign ∂x 2 ( t, · ) sign > 0 = > 0 ∂x 2 3

Conservation Law Approach. This third order equation ∂ 3 u ∂ 2 u ∂x 2 + u∂ 3 u ∂u ∂t∂x 2 + 4 u∂u ∂x = 3 ∂u ∂t − ∂x 3 , ( DP ) ∂x is formally equivalent to the elliptic - hyperbolic system  ∂u ∂t + u∂u ∂x + ∂P   ∂x = 0 ,  − ∂ 2 P  ∂x 2 + P = 3   2 u 2 , to the integro - differential system  ∂u ∂t + u∂u ∂x + ∂P   ∂x = 0 ,  � P ( t, x ) = 3   e −| x − y | u 2 ( t, y ) dy,  4 I R and, finally, to the integro - differential equation � ∂x + 3 ∂u ∂t + u∂u e −| x − y | sign( y − x ) u 2 ( t, y ) dy = 0 . 4 I R 4

Comparison with the Camssa - Holm Equation. Degasperis - Procesi Equation. ∂ 3 u ∂ 2 u ∂x 2 + u∂ 3 u ∂u ∂t∂x 2 + 4 u∂u ∂x = 3 ∂u ∂t − ( DP ) ∂x 3 ∂x �  ∂u ∂t + u∂u ∂x + ∂P   ∂x = 0  ( wDP )  − ∂ 2 P ∂x 2 + P = 3   2 u 2 � � Functional setting L ∞ R + ; L 2 loc ( I R ) : discontinuous solutions. I loc Camassa - Holm Equation. ∂ 3 u ∂ 2 u ∂x 2 + u∂ 3 u ∂u ∂t∂x 2 + 3 u∂u ∂x = 2 ∂u ∂t − ( CH ) ∂x 3 ∂x �  ∂u ∂t + u∂u ∂x + ∂P  ∂x = 0   ( wCH ) � ∂u � 2 − ∂ 2 P  ∂x 2 + P = 1   + u 2 2 ∂x � � Functional setting L ∞ R + ; H 1 I loc ( I R ) : continuous solutions. loc 5

Definitions of Solutions. Weak Solutions. We call u : I R + × I R → I R weak solution of the Cauchy problem  ∂ 3 u ∂x 2 + u∂ 3 u ∂ 2 u ∂u ∂t∂x 2 + 4 u∂u ∂x = 3 ∂u   ∂t − ∂x 3 , ∂x ( CP )   u (0 , x ) = u 0 ( x ) , if and only if i ) u ∈ L ∞ � � R + ; L 2 ( I I R ) ; ii ) u satisfies  ∂u ∂t + u∂u ∂x + ∂P   ∂x = 0 ,     − ∂ 2 P ∂x 2 + P = 3 ( wCP ) 2 u 2 ,       u (0 , x ) = u 0 ( x ) , in the sense of distributions. 6

entropy weak solutions. R + × I R → I We call u : I R entropy weak solution of the Cauchy problem  ∂ 3 u ∂x 2 + u∂ 3 u ∂ 2 u  ∂u ∂t∂x 2 + 4 u∂u ∂x = 3 ∂u  ∂t − ∂x 3 , ∂x ( CP )   u (0 , x ) = u 0 ( x ) , if and only if i ) u ∈ L ∞ � � ∩ L ∞ � � R + ; L 2 ( I [0 , T ]; BV ( I R ) I R ) , T > 0; ii ) u satisfies  ∂u ∂t + u∂u ∂x + ∂P   ∂x = 0 ,     − ∂ 2 P ∂x 2 + P = 3 ( wCP ) 2 u 2 ,       u (0 , x ) = u 0 ( x ) , in the sense of distributions; iii ) for any convex C 2 entropy η : I R → I R with corresponding R defined by q ′ ( u ) = η ′ ( u ) u there holds R → I entropy flux q : I ∂η ( u ) + ∂q ( u ) + η ′ ( u ) ∂P ∂x ≤ 0 , ∂t ∂x in the sense of distributions on I R + × I R. Remark. u entropy weak solution = ⇒ u weak solution . 7

Theorem. (G.M.C. - K. H. Karlsen (JFA - 2006)) • (Existence) If u 0 ∈ L 1 ( I R ) ∩ BV ( I R ) , then there exists an entropy weak solution to the Cauchy prob- lem (CP). • (Stability and uniqueness) Fix any T > 0 , and let u, v be two entropy weak solutions to (CP) with initial data u 0 , v 0 ∈ L 1 ( I R ) ∩ BV ( I R ) , respectively. Then for any t ∈ (0 , T ) R ) ≤ e M T t � u 0 − v 0 � L 1 ( I � u ( t, · ) − v ( t, · ) � L 1 ( I R ) , � � M T := 3 � u � L ∞ ((0 ,T ) × I R ) + � v � L ∞ ((0 ,T ) × I < ∞ . R ) 2 In particular, there exists at most one entropy weak solution to (CP). • (Time L 1 -continuity) For any T > 0 : � u ( t 2 , · ) − u ( t 1 , · ) � L 1 ( I R ) ≤ C T | t 2 − t 1 | , ∀ t 1 , t 2 ∈ [0 , T ] , � � 2 R ) + 12 T � u 0 � 2 + 12 � u 0 � 2 � u 0 � L 1 ( I C T := R ) . L 2 ( I R ) L 2 ( I • (Oleinik type estimate) For a.e. ( t, x ) ∈ (0 , T ] × I R , ∂u ∂x ( t, x ) ≤ 1 t + K T , � � 2 � 1 / 2 � R ) + 3 6 � u 0 � 2 TV ( u 0 ) + 24 T � u 0 � 2 K T := . L 2 ( I L 2 ( I R ) 2 8

Vanishing Viscosity Approximation. We approximate (wCP) with the following elliptic - parabolic system  ∂x = ε∂ 2 u ε ∂u ε ∂u ε ∂x + ∂P ε   ∂t + u ε ∂x 2 ,     − ∂ 2 P ε ∂x 2 + P ε = 3 2 u 2 ε ,       u ε (0 , x ) = u ε, 0 ( x ) , that is equivalent to the fourth order problem  ∂t − ∂ 3 u ε ∂u ε ∂u ε   ∂t∂x 2 + 3 u ε   ∂x   ∂ 2 u ε ∂x 2 + u∂ 3 u ε ∂x 3 + ε∂ 2 u ε ∂x 2 − ε∂ 4 u ε = 3 ∂u ε ∂x 4 ,    ∂x    u ε (0 , x ) = u ε, 0 ( x ) . • Existence and Uniqueness of smooth solutions if u ε, 0 ∈ H 2 ( I R ). • Lipschitz continuity with respect to the viscosity coefficient ε and the initial condition u ε, 0 . • The Lipschitz constant depends on ε . (see G.M.C. - H. Holden - K. H. Karlsen (DCDS - 2005)) 9

L 2 - estimate. � ε ( t, y ) dy = ε∂ 2 u ε ∂x + 3 ∂u ε ∂u ε e −| x − y | sign( y − x ) u 2 ∂t + u ε ∂x 2 4 I R Viscous Burgers equation with a nonlocal source. Estimate on the nonlocal term: L 2 bound on u ε . • multiplying by u ε and integrating on I R is not working. • Hamiltonian structure: conserved quantities for (DP). Let v = v ( t, x ) be defined by the equation − ∂ 2 v ∂x 2 + 4 v = u � � � u − ∂ 2 u is a conserved quantity for (DP) v dx ∂x 2 I R Observe that � � � � � � − 1 � � u − ∂ 2 u 4 − ∂ 2 1 − ∂ 2 v dx = u u dx = ∂x 2 ∂x 2 ∂x 2 I R I R � � � ∂v � 2 � ∂ 2 v � 2 � 4 v 2 + 5 = + dx ∂x 2 ∂x I R ≃ � v � 2 R ) ≃ � u � 2 H 2 ( I L 2 ( I R ) 10