Generic Singularities of Solutions to some Nonlinear Wave Equations

Transcription

1 Generic Singularities of Solutions to some Nonlinear Wave Equations Alberto Bressan Deartment of Mathematics, Penn State University (Oberwolfach, June 2016) Alberto Bressan (Penn State) generic singularities 1 / 36

2 Singularity formation For several nonlinear wave equations, solutions with smooth initial data develo singularities in finite time: u u(t, ) C 1 (R) or u(t, ) H s (R) Alberto Bressan (Penn State) generic singularities 2 / 36

3 Generic singularities Prove that, for generic smooth initial data, singularities are localized along finitely many oints, or curves Give a local asymtotic descrition of (structurally stable) singularities generic valid on a countable intersection of oen dense sets in C k Alberto Bressan (Penn State) generic singularities 3 / 36

4 Three basic settings hyerbolic systems of conservation laws: u t + f (u) = 0 Burgers-Hilbert equation: u t + (u 2 /2) = H[u] variational wave equations: u tt c(u)(c(u)u ) = 0 Alberto Bressan (Penn State) generic singularities 4 / 36

5 Generic regularity for scalar conservation laws u t + f (u) = 0 R, t [0, T ] u(0, ) = ū() Theorem (D. Schaeffer, 1973) Assume f smooth, f > 0. For a generic initial data ū C 3 (R), the solution remains smooth outside finitely many shock curves. D. Schaeffer, A regularity theorem for conservation laws. Adv. Math. 11 (1973), C. Dafermos and X. Geng, Generalized characteristics uniqueness and regularity of solutions in a hyerbolic system of conservation laws. Ann. Inst. H. Poincaré 8 (1991), Alberto Bressan (Penn State) generic singularities 5 / 36

6 u t + f (u)u = 0 u(0, ) = ū() equations of characteristics: ẋ = f (u) u = 0 u = f (u)u 2 Along the characteristic starting at y: u (t, (t)) as t T blowu (y) = New shocks can only form at ositive local minima of the ma y T blowu (y) 1 f (ū(y)) ū (y) Alberto Bressan (Penn State) generic singularities 6 / 36

7 Eamle: Burgers equation ( ) u 2 u t + 2 = 0, u(0, ) = ū() New shocks are formed along characteristics originating from negative local minima of ū ū has N local minima = at most N shock curves can aear _ u() t Alberto Bressan (Penn State) generic singularities 7 / 36

8 Piecewise regularity for hyerbolic systems of conservation laws? Question. For generic initial data ū C 3, is the solution smooth outside finitely many shock curves? t 3 3 t 2 2 ossibly true for 2 2 systems false for n n systems, with n 3 L. Caravenna and L. Sinolo, Schaeffer s regularity theorem for scalar conservation laws does not etend to systems, Indiana U. Math. J., to aear Alberto Bressan (Penn State) generic singularities 8 / 36

9 Generic regularity for 2 2 conservation laws? Detailed descrition of singularity formation: De-Xing Kong, Formation and roagation of singularities for 2 2 quasilinear hyerbolic systems. Trans. Amer. Math. Soc. 354 (2002), Generic regularity? t scalar conservation law t 2 2 system Alberto Bressan (Penn State) generic singularities 9 / 36

10 The Burgers-Hilbert equation ( ) u 2 u t + 2 = H[u], u(0, ) = ū (BH) For u L 2 (R), the Hilbert transform is H[u](). = 1 u( y) π P.V. dy = y 1 u( y) lim dy ε 0+ π y >ε y Alberto Bressan (Penn State) generic singularities 10 / 36

11 References J. Biello and J. K. Hunter, Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities. Comm. Pure Al. Math. 63 (2009), Derivation of the model, for nonlinear waves with constant frequency. J. K. Hunter and M. Ifrim, Enhanced life san of smooth solutions of a Burgers-Hilbert equation. SIAM J. Math. Anal. 44 (2012), Local eistence and uniqueness of smooth solutions, estimates on the blow-u time Alberto Bressan (Penn State) generic singularities 11 / 36

12 Entroy-weak solutions in L 2 (R) (A.B., K.Nguyen, SIAM J. Math. Anal., 2014) Theorem (global eistence in L 2 ) Given any initial data ū L 2 (R), the Cauchy roblem (BH) has an entroy weak solution u = u(t, ) defined for all (t, ) [0, [ R. For this solution, the ma t u(t, ) L 2 is non-increasing, while u(t, ) L C(1 + t 1/3 ) for every t > 0. Theorem (uniqueness for satially eriodic, BV solutions) Let u, v be satially eriodic entroy weak solutions with the same initial data. Assume that the total variation of u(t, ) and v(t, ) over [0, 2π] remains uniformly bounded for t [0, T ]. Then u and v coincide for all t [0, T ]. Alberto Bressan (Penn State) generic singularities 12 / 36

13 Generic singularities for the Burgers-Hilbert equation Describe the local behavior of a solution near a shock Describe how a shock is formed Describe the interaction of two shocks Is a generic solution iecewise smooth? t Alberto Bressan (Penn State) generic singularities 13 / 36

14 Piecewise regular solutions Burgers Burgers Hilbert u( τ,) u(t,) 0 0 For Burgers equation, at the time τ when a new shock is formed: u(τ, ) = a b( 0 ) 1/3 + for 0 Alberto Bressan (Penn State) generic singularities 14 / 36

15 Burgers Burgers Hilbert u( τ,) u(t,) 0 0 For Burgers-Hilbert, near a shock located at = 0: u(t, ) = u + u ln π + b + O(1) 3/2 if < 0 2 ln π + b + + O(1) 3/2 if > 0 A.B., Tianyou Zhang, Piecewise smooth solutions to the Burgers-Hilbert equation. Comm. Math. Sci., to aear. (local eistence and uniqueness) Alberto Bressan (Penn State) generic singularities 15 / 36

16 The variational wave equation u tt c(u) ( c(u)u ) = 0 { u(0, ) = u0 () u t (0, ) = u 1 () (u 0, u 1 ) H 1 (R) L 2 (R) c : R R + is a smooth, uniformly ositive function ±c(u) = wave seeds Ping Zhang and Yui Zheng, Proc. Royal Soc. Edinburgh (2002), Ping Zhang and Yui Zheng, Arch. Rat. Mech. Anal. (2003), Ping Zhang and Yui Zheng, Ann. Inst. H. Poincaré, (2004). Alberto Bressan (Penn State) generic singularities 16 / 36

17 Auiliary variables {. R = ut + c(u)u,. S = u t c(u)u, u t = R + S 2, u = R S 2c Evolution equation for R, S: R t cr = c 4c (R2 S 2 ) S t + cs = c 4c (S 2 R 2 ) Possible blow-u: R, S in finite time c 0 = D Alembert solution of wave equation Alberto Bressan (Penn State) generic singularities 17 / 36

18 Conserved quantities (for smooth solutions) Balance laws for R 2, S 2 : (R 2 ) t (cr 2 ) = c 2c (R2 S RS 2 ) (S 2 ) t + (cs 2 ) = c 2c (R2 S RS 2 ) R 2 and S 2 reresent the energy of backward and forward moving waves. Energy is transferred from forward to backward waves, and vice-versa Total energy: E(t) = 1 2 (u 2 t + c 2 u 2 ) d = constant Natural domain: (u, u t) H 1 (R) L 2 (R) = solutions remain Hölder continuous Alberto Bressan (Penn State) generic singularities 18 / 36

19 Recent results (A.B., Geng Chen, Tao Huang, Fang Yu) For an oen, dense set of initial data (u 0, u 1 ) D U =. ( ) C 3 (R) H 1 (R) ( ) C 2 (R) L 2 (R) the conservative solution u = u(t, ) is C 2 outside a finite set of singular oints and C 2 singular curves. A detailed asymtotic descrition of u can be given near each oint of singularity. t 3 q 2 2 q 1 Alberto Bressan (Penn State) generic singularities 19 / 36 1

20 Basic tools from differential geometry: Sard s theorem, Thom s transversality theorem aly to C k mas. For solutions to nonlinear wave equations, such regularity is not available. Key idea: By a change of deendent and indeendent coordinates, one obtains an equivalent system whose solutions remain globally smooth Alberto Bressan (Penn State) generic singularities 20 / 36

21 Coordinate change: indeendent variables Equations for characteristics ẋ + = c(u), ẋ = c(u) s + (s, t, ) (s, t, ) As coordinates (X, Y ) of a oint (t, ) we use the quantities X =. (0, t, ), Y =. + (0, t, ) t X = const. (,t) Y = const. s + (s,,t) (s,,t) Alberto Bressan (Penn State) generic singularities 21 / 36

22 Coordinate change: deendent variables w. = 2 arctan R, z. = 2 arctan S w, z R/(2π Z) R, S ± w, z π. = 1 + R2 X, q. = 1 + S 2 Y Alberto Bressan (Penn State) generic singularities 22 / 36

23 A semilinar system in characteristic variables A.B., Yui Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys. 266 (2006), w Y = c (u) (cos z cos w) q 8c 2 (u) Y = c (u) (sin z sin w) q 8c 2 (u) z X = c (u) (cos w cos z) 8c 2 (u) q X = c (u) (sin w sin z) q 8c 2 (u) u X = sin w 4c(u) u Y = sin z 4c(u) q X = Y = (1+cos w) 4 (1+cos z) q 4 t X = t Y = (1+cos w) 4c(u) (1+cos z) q 4c(u) Λ : (X, Y ) (, t) Alberto Bressan (Penn State) generic singularities 23 / 36

24 Boundary data - comatible solutions Y 0 X γ 0. Along the curve γ 0 = {X + Y = 0} corresonding to {t = 0}, the boundary data ( w, z,, q, ū) L are defined by { w = 2 arctan R(, 0) { = 1 + R 2 (, 0) z = 2 arctan S(, 0) q = 1 + S 2 (, 0) = X = Y, ū = u 0 () Alberto Bressan (Penn State) generic singularities 24 / 36

25 Global conservative solutions Theorem (A.B. - Yui Zheng, 2006) Given smooth initial data (u, u t ) = (u 0, u 1 ), the semilinear system has a t=0 unique smooth solution (, t, u, w, z,, q)(x, Y ) defined for all (X, Y ) R 2. The function u = u(, t) whose grah is grah(u) = {((X, Y ), t(x, Y ), u(x, Y )) ; (X, Y ) R 2} is the unique conservative solution to the wave equation u tt c(u)(c(u)u ) = 0 Singularities can only arise because the ma Λ : (X, Y ) (, t) is not smoothly invertible ( ) (1+cos w) (1+cos z) q X DΛ = Y 4 4 = t X t Y (1+cos w) 4c(u) (1+cos z) q 4c(u) Alberto Bressan (Penn State) generic singularities 25 / 36

26 Structure of the singular set The set of oints (, t) where u is not smooth is contained in the image of the level sets S w. = {(X, Y ) ; w(x, Y ) = π}, S z. = {(X, Y ) ; z(x, Y ) = π} w = π Y w > π z = π P 3 Q 2 P P 2 z > π γ 0 w < π Q 1 0 P 1 X Alberto Bressan (Penn State) generic singularities 26 / 36

27 Generic regularity u tt c(u)(c(u)u ) = 0 ( ) (A) The function c is smooth and uniformly ositive. Moreover, c (u) = 0 = c (u) 0 Theorem (A.B., Geng Chen, Ann. Inst. H.Poincaré, 2016) Let (A) hold. Then there eists an oen dense set of initial data ( ) ( ) D C 3 (R) H 1 (R) C 2 (R) L 2 (R) such that the solution u = u(t, ) is iecewise smooth in the -t lane. Alberto Bressan (Penn State) generic singularities 27 / 36

28 Classification of generic singularities w = π Y w > π z = π P 3 Q 2 P P 2 z > π γ 0 w < π Q 1 0 P 1 X t 3 q 2 2 q 1 Alberto Bressan (Penn State) generic singularities 28 / 36 1

29 Three tyes of singular oints (X, Y ) Tye 1: w = π, w X 0 (oints along a singular curve) Tye 2: w = π, w X = 0 = w Y 0, w XX 0 (oints were two singular curves of the same family originate or terminate) Tye 3: w = π, z = π = w X 0, z Y 0 (oints where two curves of oosite families cross) Note: the imlication = is true for a generic solution Alberto Bressan (Penn State) generic singularities 29 / 36

30 Thom s transversality theorem = Fi a bounded domain Ω in the X -Y lane. Then there is an oen dense set of comatible solutions (u,, t, w, z,, q) to the semilinear system such that the following values are NEVER attained on Ω: { (w, wx, w XX ) = (π, 0, 0), (z, z Y, z YY ) = (π, 0, 0), { (w, z, wx ) = (π, π, 0), (w, z, z Y ) = (π, π, 0), { (w, wx, c (u)) = (π, 0, 0), (z, z Y, c (u)) = (π, 0, 0). (1) (2) (3) Alberto Bressan (Penn State) generic singularities 30 / 36

31 f y X f(x) For a fied ȳ = (ȳ 1, ȳ 2, ȳ 3 ), a generic smooth ma f : R 2 R 3 does NOT attain the value ȳ. BUT: a generic solution of a system containing the equation w Y = c (u) 8c 2 (cos z cos w) q (u) can still attain the value (w, z, w Y ) = (0, 0, 0). Results on a generic solution to a system of PDEs require more detailed analysis. J. Damon, Generic roerties of solutions to artial differential equations. Arch. Rational Mech. Anal. 140 (1997) Alberto Bressan (Penn State) generic singularities 31 / 36

32 Asymtotic descrition of singularities w = π Y w > π z = π P 3 Q 2 P P 2 z > π γ 0 w < π Q 1 0 P 1 X t 3 q 2 2 q 1 1 Alberto Bressan (Penn State) generic singularities 32 / 36

33 Theorem (A.B., T.Huang, F. Yu, Bull. Inst. Math. Acad. Sinica, 2015) Let (A) hold. Then a generic solution to the wave equation has only three tyes of singular oints ( 0, t 0). At oints of Tye 1 (along a singular curve γ) one has [ ] 2/3 ( ) u(, t) = u 0 a c(u 0)(t t 0) + ( 0) + O(1) t t At oints of Tye 2 (where two new singular curves γ, γ + originate) one has [ ] 3/5+ ( ) 4/5 u(, t) = u 0 + a c(u 0)(t t 0) + ( 0) O(1) t t At oints of Tye 3 (where two singular curves γ, γ cross), one has [ ] 2/3 u(, t) = u 0 + a 1 c(u 0)(t t 0) + ( 0) [ ] 2/3 ( ) +a 2 c(u 0)(t t 0) ( 0) + O(1) t t Alberto Bressan (Penn State) generic singularities 33 / 36

34 At a time t 0 when a new singularity forms: u(, t 0 ) u 0 a ( 0 ) 3/5 After the singularity has formed: u(, t 0 ) u 0 + a ( 0 ) 2/3 u(,t ) 0 u(,t) u 0 u t 3 q 2 2 q 1 1 Alberto Bressan (Penn State) generic singularities 34 / 36

35 Singular curves and characteristics Y t t 0 P w= π X 0 γ Characteristics curves satisfy ẋ(t) = ± c(u(t, (t)) Singular curves are enveloes of characteristics The distance between a singular curve γ( ) and the characteristic ( ) assing through the same oint ( 0, t 0 ) is (t) γ(t) κ (t t 0 ) 3 Alberto Bressan (Penn State) generic singularities 35 / 36

36 New singular curves Y w= π Y 0 P 1 P 2 P 0 t = τ > t t = t 0 0 t 1 γ 2 _ γ + t 0 0 X 0 X 0 At the oint ( 0, t 0 ) where two new singular curves γ, γ + are formed, their distance is γ + (t) γ (t) = κ (t t 0 ) 5/2 + O(1) (t t 0 ) 3 Alberto Bressan (Penn State) generic singularities 36 / 36