Generic Singularities of Solutions to some Nonlinear … Singularities of Solutions to some Nonlinear Wave Equations Alberto Bressan Department of Mathematics, Penn State University

Generic Singularities of Solutions tosome Nonlinear Wave Equations

Alberto Bressan

Department of Mathematics, Penn State University

(Oberwolfach, June 2016)

Alberto Bressan (Penn State) generic singularities 1 / 36

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) → ∞


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


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


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.


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)


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


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


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


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


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


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 ].


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


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


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)


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).


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


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


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


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


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)


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


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)


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)


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)


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


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.


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


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


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)


_

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.


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


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|

).


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


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


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


Documents

Generic Singularities of Solutions to some Nonlinear … Singularities of Solutions to some Nonlinear Wave Equations Alberto Bressan Department of Mathematics, Penn State University