Electronic Journal of Qualitative Theory of Differential Equations 2003, No. 18, 1-7; http://www.math.u-szeged.hu/ejqtde/

ON THE UNIFORMLY CONTINUITY OF THE SOLUTION MAP FOR TWO DIMENSIONAL WAVE MAPS

Svetlin Georgiev Georgiev Penka Vasileva Georgieva

Abstract

University of Sofia, Faculty of Mathematics and Informatics, Department of Differential Equations, 1000 Sofia, Bulgaria e-mail: sgg2000bg@yahoo.com, p g@abv.bg

Abstract

Abstract. The aim of this paper is to analyze the properties of the solution map to the Cauchy problem for the wave map equation with a source term, when the target is the hyperboloid H^{2} that is embedded in R^{3}. The initial data are in Ḣ^{1} × L^{2}. We prove that the solution map is not uniformly continuous.

Abstract

Subject classification: Primary 35L10, Secondary 35L50.

In this paper we study the properties of the solution map (u_{∘}, u_{1}, g) ⟶ u(t, x) to the Cauchy problem

(1)  u_{tt} − △u − (|u_{t}|^{2} − |∇_{x}u|^{2})u = g(t, x),

(2)  u(0, x) = u_{∘}(x) ∈ Ḣ^{1}(R^{2}), u_{t}(0, x) = u_{1}(x) ∈ L^{2}(R^{2})

in the case when x ∈ R^{2} and the target is the hyperboloid H^{2} : u_{1}^{2} + u_{2}^{2} − u_{3}^{2} = −1, H^{2} ↪ R^{3}. Here

|u_{t}|^{2} = u_{1t}^{2} + u_{2t}^{2} − u_{3t}^{2},
|∇_{x}u|^{2} = |∇_{x_{1}}u|^{2} + |∇_{x_{2}}u|^{2},
|∇_{x_{i}}u|^{2} = u_{1x_{i}}^{2} + u_{2x_{i}}^{2} − u_{3x_{i}}^{2}, i = 1, 2.

More precisely, we prove that the solution map (u_{∘}, u_{1}, g) ⟶ u(t, x) to the Cauchy problem (1),

(2) is not uniformly continuous.

In [1] is proved that the solution map isn’t uniformly continuous in the case when g ≡ 0. When we say that the solution map (u_{∘}, u_{1}, g) ⟶ u(t, x) is uniformly continuous we under-

stand: for every positive constant ϵ there exist positive constants δ and R such that for any two solutions u, v : R × R^{2} ⟶ H^{2} of (1), (2), with right hands g = g_{1}, g = g_{2} of (1), so that (3) E(0, u − v) ≤ δ, ||g_{1}||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ R, ||g_{2}||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ R, ||g_{1} − g_{2}||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ R, the following inequality holds

(4)  E(t, u − v) ≤ ϵ for ∀t ∈ [0, 1],

where

E(t, u) := ||∂_{t}u(t, ·)||_{L^{2}(R^{2})}^{2} + ||∇_{x}u(t, ·)||_{L^{2}(R^{2})}^{2}.

Here we prove Theorem 1.There exist constant ϵ > 0 such that for every pair of positive constants δ and R there exists smooth solutions u, v: R × R^{2} ⟶ H^{2} of (1), (2), with right hands g = g_{1}, g = g_{2} of (1), so that

E(0, u − v) ≤ δ, ||g_{1}||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ R, ||g_{2}||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ R, ||g_{1} − g_{2}||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ δ,

and

E(1, u − v) ≥ ϵ.

Proof. We suppose that the solution map (u_{∘}, u_{1}, g) ⟶ u(t, x) to the Cauchy problem (1), (2) is uniformly continuous. Then for every ϵ > 0 there exist positive constants δ and R such that for any solution u of (1), (2) with right hand g of (1) for which

(5)  E(0, u) ≤ δ, ||g||_{L^{1}([0,1]L^{2}(R^{2}))} ≤ R

and the inequality

(6)  E(t, u) ≤ ϵ

holds for every t ∈ [0, 1] (in this case v = 0, which is solution of (1) with right hand g = 0). Let

(⋆)

u = (u_{1}, u_{2}, u_{3}),
u_{1} = sinh χ cos ϕ_{1},
u_{2} = sinh χ sin ϕ_{1},
u_{3} = cosh χ, χ ≥ 0, ϕ_{1} ∈ [0, 2π],

χ = Y^{2}, where Y is solution to the Cauchy problem

(7)  Y_{tt} − △Y = 0,

(8)  Y (0, x) = 0, Y_{t}(0, x) = q(x),

q(x) = ∫_{R^{2}} sin(xξ)ϕ(ξ)dξ,

ϕ(ξ) ≡ ϕ_{N}(ξ) = H(A_{N})\frac{1}{\sqrt{|ξ|}},

H(·) is the characteristic function of correspond set, xξ = x_{1}ξ_{1} + x_{2}ξ_{2},

A_{N} = \{ξ ∈ R^{2}, ξ_{1} = rcosϕ, ξ_{2} = rsinϕ, N_{∘} ≤ |ξ| ≤ N, ϕ ∈ (\frac{π}{6} , \frac{π}{4} )\},

N > N_{∘} > 0 are fixed such that N_{∘} is close enough to N, sin(ξη) ≥ a_{1}, cos(|η|) ≥ a_{2}, sin(|ξ|) ≥ a_{4} for ξ ∈ A_{N}, η ∈ A_{N}, where 0 < a_{1} < 1, 0 < a_{2} < 1, 0 < a_{4} < 1 (for instance N_{∘} = 1 − p, N = 1, p is close enough to zero), g = (g_{1}, g_{2}, g_{3}),

g_{1} = coshχcosϕ_{1}(χ_{tt} − Δχ) + \frac{1}{r^{2}} sinhχcosϕ_{1} − \frac{2x_{2}}{r^{2}} coshχsinϕ_{1}χ_{x_{1}}+

+\frac{2x_{1}}{r^{2}} coshχχ_{x_{2}}sinϕ_{1} + \frac{sinh^{3}χcosϕ_{1}}{r^{2}} ,

g_{2} = coshχsinϕ_{1}(χ_{tt} − Δχ) + \frac{1}{r^{2}} sinhχsinϕ_{1} + \frac{2x_{2}}{r^{2}} coshχcosϕ_{1}χ_{x_{1}}−

\frac{2x_{1}}{r^{2}} coshχχ_{x_{2}}cosϕ_{1} + \frac{sinh^{3}χsinϕ_{1}}{r^{2}} ,

g_{3} = sinhχf,

f = 2Y_{t}^{2} − 2Y_{r}^{2} + \frac{sinh(2Y^{2})}{2r^{2}},

x_{1} = rcosϕ_{1}, x_{2} = rsinϕ_{1}, r > 0. Then the function u which is defined with (⋆) is a solution to (1). We can to write the solution of the problem (7), (8) in the form

(9)  Y (t, x) = ∫_{R^{2}} sin(t|ξ|)sin(xξ)\frac{ϕ_{N}(ξ)}{|ξ|}dξ.

For the function Y, which is defined with (9), we have the following estimates

(10)  ||Y ||_{L^{2}(R^{2})} ≤ ||sin(t|x|)\frac{ϕ(x)}{|x|}||_{L^{2}(R^{2})} =

= (∫_{A_{N}} (sin(t|x|)\frac{ϕ_{N}(x)}{|x|})^{2}dx)^{\frac{1}{2}} ≤ \sqrt{\frac{π}{12}}(∫_{N_{∘}}^{N} \frac{1}{ρ^{2}} dρ)^{\frac{1}{2}} = \sqrt{\frac{π(N − N_{∘})}{12NN_{∘}}},

(11)  |Y (t, x)| ≤ ∫_{R^{2}} |sin(t|ξ|)sin(xξ)\frac{ϕ_{N}(ξ)}{|ξ|}|dξ ≤ ∫_{A_{N}} \frac{1}{|ξ|^{\frac{3}{2}}} dξ =

= ∫ \frac{π}{4} ; \frac{π}{6} ∫_{N_{∘}}^{N} \frac{ρ}{ρ^{\frac{3}{2}}} dρdϕ = \frac{π}{6} (\sqrt{N} − \sqrt{N_{∘}}),

(12)  |Y | ≤ |x|\frac{π}{18}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} ).

(13)  ||Y_{t}||_{L^{2}(R^{2})} ≤ ||cos(t|x|)ϕ(x)||_{L^{2}(R^{2})} ≤ (∫ \frac{π}{4} ; \frac{π}{6} ∫_{N_{∘}}^{N} \frac{1}{ρ}ρdρdϕ)^{\frac{1}{2}} = \sqrt{\frac{π}{12}(N − N_{∘})}.

(14)  |Y_{t}(t, x)| = |∫_{R^{2}} cos(t|ξ|)sin(xξ)ϕ_{N}(ξ)dξ| ≤ ∫_{R^{2}} ϕ_{N}(ξ)dξ = \frac{π}{18}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} ),

Similarly, we have

(15)  |Y_{x_{i}}| ≤ \frac{π}{18}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} ),

(16)  ||Y_{x_{i}}||_{L^{2}(R^{2})} ≤ \sqrt{\frac{π}{12}(N − N_{∘})}.

On the other hand

(17)  ||f||_{L^{2}(R^{2})} ≤ ||2Y_{t}^{2}||_{L^{2}(R^{2})} + ||2Y_{r}^{2}||_{L^{2}(R^{2})} + ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(R^{2})}.

Now we use (13), (14). Then

(18)  ||2Y_{t}^{2}||_{L^{2}(R^{2})} ≤ 2\frac{π}{18}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )||Y_{t}||_{L^{2}(R^{2})} ≤

\frac{π}{9} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )\sqrt{\frac{π(N − N_{∘})}{12}}.

Similarly

(18′)  ||2Y_{r}^{2}||_{L^{2}(R^{2})} ≤ \frac{π}{9} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )\sqrt{\frac{π(N − N_{∘})}{12}}.

Let Ω = {x ∈ R^{2} : |x| ≤ 1}. Then

(19)  ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(R^{2})} ≤ ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(Ω)} + ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(R^{2}\Ω)}

Since (12) holds, we have that there exists constant c_{1} such that |sinh(2Y^{2})| ≤ c_{1}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )^{2}|x|^{2} and

(20)  ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(Ω)} = (∫_{Ω}(\frac{sinh(2Y^{2})}{2r^{2}})^{2}dx)^{\frac{1}{2}} ≤

≤ c_{1}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )^{2}(∫_{Ω}(\frac{|x|^{2}}{2|x|^{2}} )^{2}dx)^{\frac{1}{2}} = \sqrt{2π} \frac{c_{1}}{2} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )^{2}.

On the other hand (here we use (11) and the fact that sinhx increases for every x)

(21)  ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(R^{2}\Ω)} ≤ sinh(\frac{π^{2}}{18} (\sqrt{N} − \sqrt{N_{∘}})^{2})\frac{\sqrt{π}}{2} .

From (19), (20), (21) we get

(22)  ||\frac{sinh(2Y^{2})}{2r^{2}} ||_{L^{2}(R^{2})} ≤ sinh(\frac{π^{2}}{18} (\sqrt{N} − \sqrt{N_{∘}})^{2})\frac{\sqrt{π}}{2} + \sqrt{2π} \frac{c_{1}}{2} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )^{2}

and from (17), (18), (18’), (22)

(22′)  ||f||_{L^{2}(R^{2})} ≤

≤ 2\frac{π}{9} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )\sqrt{\frac{π(N − N_{∘})}{12}} + sinh(\frac{π^{2}}{18} (\sqrt{N} − \sqrt{N_{∘}})^{2})\frac{\sqrt{π}}{2} + \sqrt{2π} \frac{c_{1}}{2} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )^{2},

(23)  ||g_{3}||_{L^{2}(R^{2})} = ||sinhχf||_{L^{2}(R^{2})} ≤ sinh(\frac{π^{2}}{36} (\sqrt{N} − \sqrt{N_{∘}})^{2})||f||_{L^{2}(R^{2})}.

We note that when N_{∘} is close enough to N ||g_{3}||_{L^{1}([0,1]L^{2}(R^{2}))} is close enough to zero . From third equation of (1) we get that χ is solution to the equation

χ_{tt} − Δχ + \frac{sinh(2χ)}{2r^{2}} = f,

i.e.

χ_{tt} − Δχ = −\frac{sinh(2χ)}{2r^{2}} + f.

Then(here we use (11) and the fact that the functions sinhx, coshx are increasing for every x ≥ 0)

(24)  ||g_{1}||_{L^{2}(R^{2})} ≤ cosh(\frac{π^{2}}{36} (\sqrt{N} − \sqrt{N_{∘}})^{2})||f − \frac{sinh2χ}{2r^{2}} ||_{L^{2}(R^{2})}+

cosh(\frac{π^{2}}{36} (\sqrt{N} − \sqrt{N_{∘}})^{2})||\frac{2x_{2}}{r^{2}} χ_{x_{1}} ||_{L^{2}(R^{2})}+

cosh(\frac{π^{2}}{36} (\sqrt{N} − \sqrt{N_{∘}})^{2})||\frac{2x_{1}}{r^{2}} χ_{x_{2}} ||_{L^{2}(R^{2})}+

+||\frac{sinhχ}{r^{2}} ||_{L^{2}(R^{2})} + sinh^{2}(\frac{π^{2}}{36} (\sqrt{N} − \sqrt{N_{∘}})^{2})||\frac{sinhχ}{r^{2}} ||_{L^{2}(R^{2})}.

Since χ_{x_{1}} = 2Y Y_{x_{1}},

|\frac{2x_{2}}{r^{2}} χ_{x_{1}} | ≤ 2\frac{|x_{2}|}{r^{2}}|Y ||Y_{x_{1}}| ≤

(from (12))

≤ \frac{π}{9} \frac{|x_{2}||x|}{r^{2}}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )|Y_{x_{i}}|,

(25)  ||\frac{2x_{2}}{r^{2}} χ_{x_{1}} ||_{L^{2}(R^{2})} ≤ \frac{π}{9} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )||Y_{x_{1}}||_{L^{2}(R^{2})} ≤

(here we use (16))

≤ \frac{π}{9} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )\sqrt{\frac{π(N − N_{∘})}{12}}.

Similarly

(26).  ||\frac{2x_{1}}{r^{2}} χ_{x_{2}} ||_{L^{2}(R^{2})} ≤ \frac{π}{9} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )\sqrt{\frac{π(N − N_{∘})}{12}}

From (17), (22’), (22), (24), (25), (26) we get

(27)  ||g_{1}||_{L^{2}(R^{2})} ≤ C_{1},

where C_{1} is close enough to zero when N_{∘} is close enough to N. Similarly,

(28)  ||g_{2}||_{L^{2}(R^{2})} ≤ C_{2},

where C_{2} is close enough to zero when N_{∘} is close enough to N. From (23), (27), (28) we have

||g||_{L^{2}(R^{2})} ≤ C,

where C is close enough to zero when N_{∘} is close enough to N. From here the second inequality of (5) is hold for every R > 0 when N_{∘} is close enough to N. Sinse

Y (0, x) = 0, χ(0, x) = 0, χ_{t}(0, x) = 2Y (0, x)Y_{t}(0, x) = 0,

χ_{x_{i}}(0, x) = 2Y (0, x)Y_{x_{i}}(0, x) = 0, i = 1, 2,
u_{1t}(0, x) = coshχ(0, x)cosϕ_{1}χ_{t}(0, x) = 0,
u_{2t}(0, x) = coshχ(0, x)sinϕ_{1}χ_{t}(0, x) = 0,
u_{3t}(0, x) = sinhχχ_{t}(0, x) = 0,

u_{1x_{1}}(0, x) = coshχ(0, x)cosϕ_{1}χ_{x_{1}}(0, x) + sinhχ(0, x)sinϕ_{1} \frac{x_{2}}{r^{2}} = 0,

u_{2x_{1}}(0, x) = coshχ(0, x)sinϕ_{1}χ_{x_{1}}(0, x) − sinhχ(0, x)cosϕ_{1} \frac{x_{2}}{r^{2}} = 0,

u_{3x_{1}}(0, x) = sinhχ(0, x)χ_{x_{1}}(0, x) = 0,

u_{1x_{2}}(0, x) = coshχ(0, x)cosϕ_{1}χ_{x_{2}}(0, x) − sinhχ(0, x)sinϕ_{1} \frac{x_{1}}{r^{2}} = 0,

u_{2x_{2}}(0, x) = coshχ(0, x)sinϕ_{1}χ_{x_{2}}(0, x) + sinhχ(0, x)cosϕ_{1} \frac{x_{1}}{r^{2}} = 0,

u_{3x_{2}}(0, x) = sinhχ(0, x)χ_{x_{2}}(0, x) = 0,

we have

E(0, u) = 0,

i.e. the first inequality of (5) holds for every δ > 0. From (6) we get that

(29)  ||∂_{t}u||_{L^{2}(R^{2})}^{2} ≤ ϵ ∀ t ∈ [0, 1].

On the other hand

||∂_{t}u||_{L^{2}(R^{2})}^{2} = ||∂_{t}u_{1}||_{L^{2}(R^{2})}^{2} + ||∂_{t}u_{2}||_{L^{2}(R^{2})}^{2} − ||∂_{t}u_{3}||_{L^{2}(R^{2})}^{2} =
= ||coshχcosϕ_{1}χ_{t}||_{L^{2}(R^{2})}^{2} + ||coshχsinϕ_{1}χ_{t}||_{L^{2}(R^{2})}^{2} − ||sinhχχ_{t}||_{L^{2}(R^{2})}^{2} =
= ∫_{R^{2}} χ_{t}^{2}(cosh^{2}χ − sinh^{2}χ)dx = ∫_{R^{2}} χ_{t}^{2}dx = ||χ_{t}||_{L^{2}(R^{2})}^{2}.

Therefore, using (29), we get

||χ_{t}||_{L^{2}(R^{2})} ≤ ϵ^{\frac{1}{2}}

or

2||Y Y_{t}||_{L^{2}(R^{2})} ≤ ϵ^{\frac{1}{2}} .

From here

(30)  2 ∫_{R^{2}} ψY Y_{t}dx ≤ ||ψ||_{L^{2}(R^{2})}ϵ^{\frac{1}{2}}

for any function ψ ∈ L^{2}(R^{2}). Let

(31)  B := a_{1}^{4}a_{2}^{2}a_{4}^{2} \frac{π^{5}}{18.126^{2}} (N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )(N^{\frac{7}{4}} − N∘^{\frac{7}{4}} )^{2}(\sqrt{N} − \sqrt{N_{∘}})^{2}

and ϵ = \frac{B}{2} ,

ψ ≡ ψ_{N}(ξ) = H(A_{N})\frac{1}{|ξ|^{\frac{1}{4}}} .

For t = 1 and x ∈ A_{N} we have

(32)  Y ≥ a_{1}a_{4} \frac{π}{6} (\sqrt{N} − \sqrt{N_{∘}}) > 0,
(33)  Y_{t} ≥ a_{1}a_{2} \frac{π}{18}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} ) > 0.
(34)  ∫_{A_{N}} ψdx = \frac{π}{21}(N^{\frac{7}{4}} − N∘^{\frac{7}{4}} ),
(35)  ||ψ||_{L^{2}(A_{N})} = \sqrt{\frac{π}{18}}(\sqrt{N^{\frac{3}{2}} − N_{∘}^{\frac{3}{2}}} ).

From (30), (32), (33), (34), (35) we have for t = 1

a_{1}^{2}a_{2}a_{4} \frac{π^{\frac{5}{2}}}{126\sqrt{18}}(N^{\frac{3}{2}} − N∘^{\frac{3}{2}} )^{\frac{1}{2}} (N^{\frac{7}{4}} − N∘^{\frac{7}{4}} )(\sqrt{N} − \sqrt{N_{∘}}) ≤ ϵ^{\frac{1}{2}} ,

i. e. ϵ ≥ B which is contradiction with ϵ = \frac{B}{2} . Therefore the solution map is not uniformly continuous.

References

[1] D’Ancona, P., V. Georgiev. On the continuity of the solution operator the wave map system. Preprint

[2] D’Ancona, P., V. Georgiev. On Lipshitz continuity of the solution map for two dimensional wave maps. Preprint.

[3] Shatah, J., M. Struwe. Geometric wave equation. Courant lecture notes in mathematics 2(1998).

[4] Struwe, M. Radial symmetric wave maps from 1+2 - dimensional Minkowski space to the sphere, preprint 2000.

[5] Klainerman, S., S. Selberg. Remarks on the optimal regularity of equations of wave maps type, CPDE, 22(1997), 901-918.

[6] Tatary, D. Local and global results for wave maps I, CPDE, 23(1998), 1781-1793. Appeared on 2003-10-10.