generic singularities of solutions to some nonlinear … singularities of solutions to some...
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Generic Singularities of Solutions tosome Nonlinear Wave Equations
Alberto Bressan
Department of Mathematics, Penn State University
(Oberwolfach, June 2016)
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Singularity formation
For several nonlinear wave equations, solutions with smooth initial datadevelop singularities in finite time: |ux | → ∞
‖u(t, ·)‖C1(R) → ∞ or ‖u(t, ·)‖Hs(R) → ∞
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Generic singularities
Prove that, for “generic” smooth initial data, singularities arelocalized along finitely many points, or curves
Give a local asymptotic description of (structurally stable) singularities
generic ⇐⇒ valid on a countable intersection of open dense sets in Ck
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Three basic settings
hyperbolic systems of conservation laws: ut + f (u)x = 0
Burgers-Hilbert equation: ut + (u2/2)x = H[u]
variational wave equations: utt − c(u)(c(u)ux)x = 0
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Generic regularity for scalar conservation laws
ut + f (u)x = 0 x ∈ R, t ∈ [0,T ]
u(0, x) = u(x)
Theorem (D. Schaeffer, 1973)
Assume f smooth, f ′′ > 0. For a generic initial data u ∈ C3(R), thesolution remains smooth outside finitely many shock curves.
D. Schaeffer, A regularity theorem for conservation laws. Adv. Math. 11 (1973),368–386.
C. Dafermos and X. Geng, Generalized characteristics uniqueness and regularity ofsolutions in a hyperbolic system of conservation laws. Ann. Inst. H. Poincare 8 (1991),231–269.
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ut + f ′(u)ux = 0 u(0, x) = u(x)
equations of characteristics:
x = f ′(u)u = 0ux = − f ′′(u)u2x
Along the characteristic starting at y :∣∣ux(t, x(t))∣∣ → ∞ as t → T blowup(y) =
−1
f ′′(u(y)) · ux(y)
New shocks can only form at positive local minima of the map
y 7→ T blowup(y)
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Example: Burgers’ equation
ut +
(u2
2
)x
= 0, u(0, x) = u(x)
New shocks are formed along characteristics originating from negative localminima of ux
ux has N local minima =⇒ at most N shock curves can appear
t
_
x
u(x)
x
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Piecewise regularity for hyperbolic systems of conservation laws?
Question. For generic initial data u ∈ C3, is the solution smooth outsidefinitely many shock curves?
3 x 3t
x
t
x
2 x 2
possibly true for 2× 2 systems
false for n × n systems, with n ≥ 3
L. Caravenna and L. Spinolo, Schaeffer’s regularity theorem for scalar conservation lawsdoes not extend to systems, Indiana U. Math. J., to appear
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Generic regularity for 2× 2 conservation laws ?
Detailed description of singularity formation:
De-Xing Kong, Formation and propagation of singularities for 2× 2 quasilinearhyperbolic systems. Trans. Amer. Math. Soc. 354 (2002), 3155–3179.
Generic regularity?
system
xx
t scalar conservation lawt
2 x 2
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The Burgers-Hilbert equation
ut +
(u2
2
)x
= H[u] , u(0, ·) = u (BH)
For u ∈ L2(R), the Hilbert transform is
H[u](x).
=1
πP.V.
∫u(x − y)
ydy = lim
ε→0+
1
π
∫|y |>ε
u(x − y)
ydy
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References
J. Biello and J. K. Hunter, Nonlinear Hamiltonian waves with constant frequencyand surface waves on vorticity discontinuities. Comm. Pure Appl. Math. 63(2009), 303–336.
Derivation of the model, for nonlinear waves with constant frequency.
J. K. Hunter and M. Ifrim, Enhanced life span of smooth solutions of aBurgers-Hilbert equation. SIAM J. Math. Anal. 44 (2012), 2039–2052.
Local existence and uniqueness of smooth solutions, estimates on the blow-uptime
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Entropy-weak solutions in L2(R)(A.B., K.Nguyen, SIAM J. Math. Anal., 2014)
Theorem (global existence in L2)
Given any initial data u ∈ L2(R), the Cauchy problem (BH) has an entropy weaksolution u = u(t, x) defined for all (t, x) ∈ [0,∞[×R.
For this solution, the map t 7→ ‖u(t, ·)‖L2 is non-increasing,while ‖u(t, ·)‖L∞ ≤ C (1 + t−1/3) for every t > 0.
Theorem (uniqueness for spatially periodic, BV solutions)
Let u, v be spatially periodic entropy weak solutions with the same initial data.
Assume that the total variation of u(t, ·) and v(t, ·) over [0, 2π] remainsuniformly bounded for t ∈ [0,T ].
Then u and v coincide for all t ∈ [0,T ].
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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 piecewise smooth?
x
t
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Piecewise regular solutions
0
u(t,x)
x0
u( ,x)τ
Burgers Burgers − Hilbert
For Burgers’ equation, at the time τ when a new shock is formed:
u(τ, x) = a− b(x − x0)1/3 + · · · for x ≈ x0
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0
u(t,x)
x0
u( ,x)τ
Burgers Burgers − Hilbert
For Burgers-Hilbert, near a shock located at x = 0:
u(t, x) =
u− + 2|x| ln |x|π + b− x +O(1) · |x |3/2 if x < 0
u+ + 2|x| ln |x|π + b+ x +O(1) · |x |3/2 if x > 0
A.B., Tianyou Zhang, Piecewise smooth solutions to the Burgers-Hilbert equation.Comm. Math. Sci., to appear. (local existence and uniqueness)
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The variational wave equation
utt − c(u)(c(u)ux
)x
= 0
u(0, x) = u0(x)ut(0, x) = u1(x)
(u0, u1) ∈ H1(R)× L2(R)
c : R 7→ R+ is a smooth, uniformly positive function
±c(u) = wave speeds
Ping Zhang and Yuxi Zheng, Proc. Royal Soc. Edinburgh (2002),Ping Zhang and Yuxi Zheng, Arch. Rat. Mech. Anal. (2003),Ping Zhang and Yuxi Zheng, Ann. Inst. H. Poincare, (2004).
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Auxiliary variables
R
.= ut + c(u)ux ,
S.
= ut − c(u)ux ,
ut =R + S
2, ux =
R − S
2c
Evolution equation for R, S : Rt − cRx = c ′
4c (R2 − S2)
St + cSx = c ′
4c (S2 − R2)
Possible blow-up: |R|, |S | → ∞ in finite timec ′ ≡ 0 =⇒ D’Alembert solution of wave equation
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Conserved quantities (for smooth solutions)
Balance laws for R2,S2: (R2)t − (cR2)x = c′
2c(R2S − RS2)
(S2)t + (cS2)x = − c′
2c(R2S − RS2)
R2 and S2 represent 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
∫ (u2t + c2u2
x
)dx = constant
Natural domain: (u, ut) ∈ H1(R)× L2(R)
=⇒ solutions remain Holder continuous
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Recent results (A.B., Geng Chen, Tao Huang, Fang Yu)
For an open, dense set of initial data
(u0, u1) ∈ D ⊂ U .=(C3(R) ∩ H1(R)
)×(C2(R) ∩ L2(R)
)the conservative solution u = u(t, x) is C2 outside a finite set of singularpoints and C2 singular curves.
A detailed asymptotic description of u can be given near each point ofsingularity.
p
p
p
q
q
p
t
x
2
1
3
2
1
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Basic tools from differential geometry:
Sard’s theorem, Thom’s transversality theorem
apply to Ck maps.
For solutions to nonlinear wave equations, such regularity is notavailable.
Key idea: By a change of dependent and independent coordinates,one obtains an equivalent system whose solutions remain globallysmooth
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Coordinate change: independent variables
Equations for characteristics
x+ = c(u) , x− = −c(u)
s 7→ x+(s, t, x) x 7→ x−(s, t, x)
As coordinates (X ,Y ) of a point (t, x) we use the quantities
X.
= x−(0, t, x) , Y.
= − x+(0, t, x)
x (s,x,t)+s
X = const.Y = const.
x (s,x,t)
t
x
−
(x,t)
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Coordinate change: dependent variables
w.
= 2 arctanR , z.
= 2 arctanS
w , z ∈ R/(2πZZ )
R , S → ±∞ ⇐⇒ w , z → π
p.
=1 + R2
Xx, q
.=
1 + S2
−Yx
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A semilinar system in characteristic variables
A.B., Yuxi Zheng, Conservative solutions to a nonlinear variational wave equation,Comm. Math. Phys. 266 (2006), 471–497.
wY = c′(u)
8c2(u)(cos z − cos w) q
zX = c′(u)8c2(u)
(cos w − cos z) p
pY = c′(u)
8c2(u)(sin z − sin w) pq
qX = c′(u)8c2(u)
(sin w − sin z) pq
uX =sin w
4c(u)p uY =
sin z
4c(u)q
xX = (1+cosw) p
4
xY = − (1+cos z) q4
tX = (1+cosw) p
4c(u)
tY = (1+cos z) q4c(u)
Λ : (X ,Y ) 7→ (x , t)
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Boundary data - compatible solutions
0
Y
0 X
γ
Along the curve γ0.
= X + Y = 0 corresponding to t = 0, the boundary data(w , z , p, q, u) ∈ L∞ are defined by
w = 2 arctanR(x , 0)z = 2 arctanS(x , 0)
p = 1 + R2(x , 0)q = 1 + S2(x , 0)
x = X = − Y , u = u0(x)
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Global conservative solutions
Theorem (A.B. - Yuxi Zheng, 2006)
Given smooth initial data (u, ut)∣∣∣t=0
= (u0, u1), the semilinear system has a
unique smooth solution (x , t, u,w , z , p, q)(X ,Y ) defined for all (X ,Y ) ∈ R2.The function u = u(x , t) whose graph is
graph(u) =
(x(X ,Y ) , t(X ,Y ) , u(X ,Y )) ; (X ,Y ) ∈ R2
is the unique conservative solution to the wave equation
utt − c(u)(c(u)ux)x = 0
Singularities can only arise because the map Λ : (X ,Y ) 7→ (x , t) is not smoothlyinvertible
DΛ =
(xX xYtX tY
)=
(1+cosw) p4 − (1+cos z) q
4
(1+cosw) p4c(u)
(1+cos z) q4c(u)
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Structure of the singular set
The set of points (x , t) where u is not smooth is contained in the image of thelevel sets
Sw .= (X ,Y ) ; w(X ,Y ) = π , Sz .
= (X ,Y ) ; z(X ,Y ) = π
P
w
w
Y
0
γ0
X
2 P
Q1
Q2
P1
P3
w = π
< π
> π z = π
> πz
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Generic regularity
utt − c(u)(c(u)ux)x = 0 (∗)
(A) The function c is smooth and uniformly positive. Moreover,
c ′(u) = 0 =⇒ c ′′(u) 6= 0
Theorem (A.B., Geng Chen, Ann. Inst. H.Poincare, 2016)
Let (A) hold. Then there exists an open dense set of initial data
D ⊂(C3(R) ∩ H1(R)
)×(C2(R) ∩ L2(R)
)such that the solution u = u(t, x) is piecewise smooth in the x-t plane.
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Classification of generic singularities
P
w
w
Y
0
γ0
X
2 P
Q1
Q2
P1
P3
w = π
< π
> π z = π
> πz
p
p
p
q
q
p
t
x
2
1
3
2
1
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Three types of singular points (X ,Y )
Type 1: w = π, wX 6= 0
(points along a singular curve)
Type 2: w = π, wX = 0 =⇒ wY 6= 0, wXX 6= 0
(points were two singular curves of the same family originate or terminate)
Type 3: w = π, z = π =⇒ wX 6= 0, zY 6= 0
(points where two curves of opposite families cross)
Note: the implication “=⇒” is true for a generic solution
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Thom’s transversality theorem =⇒
Fix a bounded domain Ω in the X -Y plane. Then there is an open dense set of“compatible” solutions (u, x , t,w , z , p, q) to the semilinear system such that thefollowing values are NEVER attained on Ω:
(w ,wX ,wXX ) = (π, 0, 0),
(z , zY , zYY ) = (π, 0, 0),(1)
(w , z ,wX ) = (π, π, 0),
(w , z , zY ) = (π, π, 0),(2)
(w ,wX , c
′(u)) = (π, 0, 0),
(z , zY , c′(u)) = (π, 0, 0).
(3)
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_
X
f(X)
yf
For a fixed y = (y1, y2, y3), a generic smooth map f : R2 7→ R3
does NOT attain the value y .
BUT: a generic solution of a system containing the equation
wY =c ′(u)
8c2(u)(cos z − cosw) q
can still attain the value (w , z ,wY ) = (0, 0, 0).
Results on a “generic solution” to a system of PDEs require more detailedanalysis.
J. Damon, Generic properties of solutions to partial differential equations.Arch. Rational Mech. Anal. 140 (1997) 353–403.
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Asymptotic description of singularities
P
w
w
Y
0
γ0
X
2 P
Q1
Q2
P1
P3
w = π
< π
> π z = π
> πz
p
p
p
q
q
p
t
x
2
1
3
2
1
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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 types ofsingular points (x0, t0).
At points of Type 1 (along a singular curve γ) one has
u(x , t) = u0 − a ·[c(u0)(t − t0) + (x − x0)
]2/3+O(1) ·
(|t − t0|+ |x − x0|
)At points of Type 2 (where two new singular curves γ−, γ+ originate) one has
u(x , t) = u0 + a ·[c(u0)(t − t0) + (x − x0)
]3/5+O(1) ·
(|t − t0|+ |x − x0|
)4/5At points of Type 3 (where two singular curves γ, γ cross), one has
u(x , t) = u0 + a1 ·[c(u0)(t − t0) + (x − x0)
]2/3+a2 ·
[c(u0)(t − t0)− (x − x0)
]2/3+O(1) ·
(|t − t0|+ |x − x0|
).
Alberto Bressan (Penn State) generic singularities 33 / 36
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At a time t0 when a new singularity forms:
u(x , t0) ≈ u0 − a · (x − x0)3/5
After the singularity has formed:
u(x , t0) ≈ u0 + a · (x − x0)2/3
0
u(x,t)
0u
x x0
0u(x,t )
0u
x x
p
p
p
q
q
p
t
x
2
1
3
2
1
Alberto Bressan (Penn State) generic singularities 34 / 36
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Singular curves and characteristics
x
Y
X
w= π
γP
p
x
0t
t
0
Characteristics curves satisfy x(t) = ± c(u(t, x(t))
Singular curves are envelopes of characteristics
The distance between a singular curve γ(·) and the characteristic x(·) passingthrough the same point (x0, t0) is
x(t)− γ(t) ≈ κ · (t − t0)3
Alberto Bressan (Penn State) generic singularities 35 / 36
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New singular curves
t
P
Y
XX0
0Y
0
P1 P2
w= π
t = t
t = τ > t
0
0
t0
p0
p1
p2
_γ
γ +
xx0
At the point (x0, t0) where two new singular curves γ−, γ+ are formed, theirdistance is
γ+(t)− γ−(t) = κ · (t − t0)5/2 +O(1) · (t − t0)3
Alberto Bressan (Penn State) generic singularities 36 / 36