Hyperbolic PDEs Notes

HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS Syafiq Johar [email protected]

Contents I

Conservation Laws of Hyperbolic Equations

2

1 Linear Theory 1.1 Hyperbolic Systems of First Order Equations . . . . . . 1.2 Symmetric Hyperbolic System with Variable Coefficients 1.3 Vanishing Viscosity Method . . . . . . . . . . . . . . . . 1.4 Hyperbolic Systems with Constant Coefficients . . . . .

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2 2 3 3 3

2 Scalar Conservation Laws

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3 Hyperbolic Systems of Conservation Laws 3.1 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Construction of Shock Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Admissibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 6

4 The Riemann Problem 4.1 Elementary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Construction of Elementary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 General Solution to the Riemann Problem . . . . . . . . . . . . . . . . . . . . . . .

7 8 8 9

5 Functional Analytic Approaches to the Existence Theory 9 5.1 2 × 2 Hyperbolic Systems of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . 10

II

Wave Equations

11

6 Linear Wave Equations

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7 Energy Estimates

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8 Local Existence Results 12 8.1 Quasilinear and Semilinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 12 9 Klainerman-Sobolev Inequality 9.1 Some Tensor Calculus . . . . . . . . . 9.2 Global Existence in Higher Dimensions 9.2.1 Dimension n ≥ 4 . . . . . . . . 9.2.2 Dimension n = 3 . . . . . . . .

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13 13 14 14 15

Part I

Conservation Laws of Hyperbolic Equations 1

Linear Theory

Definition 1.1. Define the following spaces involving time:   Lp (0, T ; X) = u : [0, T ] → X : ||u||Lp (0,T ;X) := 

ˆ 0

! p1

T

||u(t)||pX dt

∈ Rm , Bj (x, t) are m × m matrices, f : Rn × [0, ∞) → Rn and g : Rn → Rm . Definition 1.2 (Hyperbolicity). For each y ∈ Rn , the m × m matrix B(x, t; y) =

n X

yj Bj (x, t)

j=1

is diagonalisable for each x ∈ Rn and t ≥ 0. This implies that for each x, y, t, B(x, t; y) has m real eigenvalues λ1 (x, t; y) ≤ λ2 (x, t; y) ≤ · · · ≤ λm (x, t; y) with corresponding m linearly independent eigenvectors {rj (x, t; y)}m j=1 . If all the eigenvalues are real and distinct i.e. λ1 (x, t; y) < λ2 (x, t; y) < · · · < λm (x, t; y), then we say that the system is strictly hyperbolic. 2

1.2

Symmetric Hyperbolic System with Variable Coefficients

Suppose we have the hyperbolic system as in (1) such that the matrices Bj (x, t) are symmetric for 2 all j, x, t. Suppose further that Bj ∈ C 2 with |Bj | + |Dx,t Bj | + |Dx,t Bj | ≤ C for all j = 1, 2, . . . , n, 1 n n 1 n m g ∈ H (R ; R ) and f ∈ H (R × (0, T ); R ). Definition 1.3. u ∈ L2 (0, T ; H 1 (Rn ; Rm )) with u0 ∈ L2 (0, T ; L2 (Rn ; Rm )) is a weak solution of (1) if hu0 , vi + B[u, v; t] = hf, vi for all v ∈ H 1 (Rn ; Rm ) a.e. 0 ≤ t ≤ T where B is the bilinear form ´ Pn B[u, v; t] = Rn j=1 (Bj uxj ) · v dx and u(0, x) = g.

1.3

Vanishing Viscosity Method

We construct solutions uε = uε (x, t) by the parabolic system uεt − ε∆uε +

n X

Bj uεxj = f

in Rn × (0, T )

(2)

j=1

uε (0, x) = g ε = ηε ∗ g. Theorem 1.4 (Existence of Approximate Solutions). For each ε > 0, there exists a unique uε of (2) st uε ∈ L2 (0, T ; H 3 (Rn ; Rm )) and (uε )0 ∈ L2 (0, T ; H 1 (Rn ; Rm )). Theorem 1.5 (Energy Estimates). There exists C(n, Bj ) such that for each ε > 0, max ||uε (t)||H 1 (Rn ;Rm ) + ||(uε )0 ||L2 (0,T ;L2 (Rn ;Rm )) ≤C(||g||H 1 (Rn ;Rm ) + ||f ||L2 (0,T ;H 1 (Rn ;Rm ))

0≤t≤T

+ ||f 0 ||L2 (0,T ;L2 (Rn ;Rm )) ). Theorem 1.6. There exists a unique weak solution of (1) that is the limit of uε as ε → 0. Remark 1.1. 1. The vanishing viscosity method also apply to the more general system B0 ut + Pn B u j=1 j xj = f where Bj (x, t) are symmetric for all j = 0, 1, . . . , n. 2. Symmetric hyperbolic systems of the form above generalise to second order hyperbolic PDE of the Pn form vtt − j=1 aij vxi xj = 0 where u = (u1 , u2 , . . . , un+1 ) = (vx1 , . . . , vxn , vt ) and 

a11  .  .. B0 =   a1n 0

1.4

··· .. . ··· ···

a1,n .. . ann 0

 0 ..  . .  0 1



0  .  .. Bj =    0 −a1j

··· .. .

0 .. . 0 −anj

··· ···

 −a1j ..  .  .  −anj  0

Hyperbolic Systems with Constant Coefficients

Suppose we have (1) such that Bj are constant matrices and f ≡ 0. Theorem 1.7. If g ∈ H k (Rn ; Rm ) for k > u ∈ C 1 ([0, ∞); Rm ). Remark 1.2.

n 2

+ m, then the Cauchy problem has a unique solution

1. Note that we do not require symmetry or strict hyperbolicity of the matrices Bj .

2. (Continuous dependence) If gn → g in H k (Rn ; Rm ) for some k > C 1 ([0, ∞); Rm ).

3

n 2

+ m then un (x, t) → u(x, t) in

2

Scalar Conservation Laws

Suppose we have the problem ∂t u + ∇ · f (u) = 0

(3)

u(0, x) = u0 (x) such that f : R → Rd is a given smooth function. Definition 2.1 (Entropy- Entropy Flux). The pair (η(u), q(u)) is called an entropy and entropy flux ´u 0 pair for the equation (3) if q(u) = η (v)f 0 (v) dv. Definition 2.2. A solution u of (3) is called an admissible solution if u ∈ L∞ (R+ × Rd ) and ˆ

∞

ˆ (η(u)∂t φ +

0

Rd

d X

ˆ φ(0, x)η(u0 (x)) dx ≥ 0

qj (u)∂xj φ) dxdt +

(4)

Rd

j=1

for any entropy-entropy flux pair such that η 00 (u) ≥ 0 and any test function φ ∈ C0∞ (R+ × Rd ). Remark 2.1.

1. Condition (4) is equivalent to η(u)t + ∇ · q(u) ≤ 0 in the sense of distributions.

2. If we choose (η(u), q(u)) = (±u, ±f (u)), then (4) implies that u is a weak solution of (3). 3. For every u ¯ ∈ R, The entropy-entropy flux pair (|u − u ¯|, sgn(u − u ¯)(f (u) − f (¯ u))) is called the Kruzkov’s family of entropy-entropy flux pair. Theorem 2.1 (Existence and Uniqueness). If u0 ∈ L∞ (R), then there exists a unique admissible solution of (3) and u(t, ·) ∈ C 0 ([0, ∞); L1loc (Rd ). Theorem 2.2 (Stability in L1 and Monotonicity in L∞ ). Suppose that u0 , v0 ∈ L∞ (Rd ) gives solutions u, v of (3), then there exists constant s(||u0 ||∞ , ||v0 ||∞ ) > 0 such that for all t > 0 and r > 0, we have: ˆ ˆ + (u(t, x) − v(t, x)) dx ≤ (u0 (x) − v0 (x))+ dx |x| 0, we have ||u(t, ·)||Lp (Br ) ≤ ||u0 (·)||Lp (Br+st ) .

4

3

Hyperbolic Systems of Conservation Laws

Suppose we have the problem ut + f (u)x = 0

or ut + A(u)ux = 0

(5)

such that u = (u1 , u2 , . . . , um )> ∈ Rm , f (u) = (f1 (u), f2 (u), . . . , fm (u))> ∈ Rm and A(u) = ∇f (u). The system is strictly hyperbolic if each m × m matrix A(u) has real distinct eigenvalues λ1 (u) < m λ2 (u) < · · · < λm (u) with corresponding right and left eigenvectors {rj (u)}m j=1 and {lj (u)}j=1 respectively. We choose bases so that li · rj = δij . Theorem 3.1 (Invariance under Change of Coordinates). Let u be the smooth solution to (5) in R+ ×R. Assume that Φ : Rm → Rm is a smooth diffeomorphism Ψ. Then, w := Φ(u) solves the strictly hyperbolic system wt + B(w)wx = 0 for B(w) := ∇Φ(Ψ(w))A(Ψ(w))∇Ψ(w). Theorem 3.2 (Dependence of Eigenvalues and Eigenvectors on u). Assume thet the matrix function A(u) is smooth, strictly hyperbolic. Then, the eigenvalues {λj (u)}m j=1 depend smoothly on u and we can select the right and left eigenvectors to depend smoothly on u and satisfy the renormalisation |rj (u)| = |lj (u)| = 1. Theorem 3.3 (Linear Hyperbolic Systems). Suppose that we have the matrix A is a constant matrix with eigenvalues λ1 < λ2 < · · · < λm and eigenvectors {rj }m j=1 . Then the solution of (5) with initial condition u(0, x) = φ(x) is a linear superposition of travelling waves given by u(t, x) =

m X

φi (x − λi t)ri

where

φi (s) = li · φ(s).

i=1

Theorem 3.4 (Wave Interactions). Suppose that A depends on u. Define u1x = li ·ux . We have a system of evolution equations for the scalar components u1x given by X (uix )t + (λi uix )x = (λj − λk )(li [rj , rk ])ujx ukx j>k

where Sijk := (λj − λk )(li [rj , rk ])ujx ukx describes the amount of i-waves produced by the interaction of the j-waves and the k-waves.

3.1

Weak Solutions

Definition 3.1 (Weak Solutions). A measurable function u(x, t) defined on an open set Ω ⊆ R+ × R is a weak solution of (5) if for every φ ∈ Cc∞ (Ω) we have ¨ (uφt + f (u)φx ) dxdt = 0. Ω

Lemma 3.1. Let (um )m≥1 be a uniformly bounded sequence of distributional solution on (5). If um → u and f (um ) → f (u) in L1loc , then the limit function u is also a weak solution. Theorem 3.5 (Rankine-Hugoniot Conditions).  u− u(t, x) = U (t, x) := u+

if x < λt if x > λt

is a weak solution of (5) if and only if the Rankine-Hugoniot equations hold: λ(u+ − u− ) = f (u+ ) − f (u− ). 5

(6)

Theorem 3.6 (Alternative Formulation of Rankine-Hugoniot). Define the averaged Jacobian matrix ˆ

1

∇f (θu + (1 − θ)v) · (u − v) dθ.

A(u, v) := 0

The Rankine-Hugoniot conditions hold if and only if λ(u+ − u− ) = A(u+ , u− )(u+ − u− ). This implies that λ is the eigenvalue of the matrix A(u+ , u− )(u+ − u− ) with eigenvector (u+ − u− ). Definition 3.2 (Points of Approximate Jump). We say that the function u(t, x) has an approximate jump discontinuity at the point (τ, ξ) if there exists vectors u+ 6= u− and a speed λ such that defining U as in (6), there holds ˆ r ˆ r 1 |u(τ + t, ξ + x) − U (t, x)| dxdt = 0. lim 2 r↓0 r −r −r Moreover, we say that u is approximately continuous at the point (τ, ξ) if the above relations hold with u+ = u− and arbitrary λ. Theorem 3.7. Let u be a bounded distributional solution of (5) having an approximate jump at a point (τ, ξ). Then, the Rankine-Hugoniot equations λ(u+ − u− ) = f (u+ ) − f (u− ) hold.

3.2

Construction of Shock Curves

Given u− ∈ Rn , we want to find states u+ ∈ Rn which, sor some speed λ satisfy the Rankine-Hugoniot equations λ(u+ − u− ) = A(u+ , u− )(u+ − u− ). Fix i ∈ {1, . . . , m}. The jump u+ −u− is a right eigenvector of A(u+ , u− ) if and only if it is orthogonal to all left eigenvectors {lj (u+ , u− )}j6=i of A(u+ , u− ) i.e lj (u+ , u− ) · (u+ − u− ) = 0 for all j 6= i. Using implicit function theorem yields that for each i, there exists a curve s 7→ Si (s)(u− ) for points which satisfy the Rankine-Hugoniot condition. At the point u− , this curve has to be perpendicular to all the vectors {lj (u− )}j6=i and therefore, it must be tangent to the i-th eigenvector ri (u− ).

3.3

Admissibility Conditions

The concept of weak solutions is not enough to ensure uniqueness for a Cauchy problem. Therefore, to achieve uniqueness, the notion of weak solution must be supplemented with admissibility conditions. Admissibility Condition 1: Vanishing Viscosity A weak solution u of (5) is admissible in the vanishing viscosity sense if there exists a sequence of smooth solutions uε to uεt + f (uε )x = εuεxx which converge to u in L1loc as ε ↓ 0. Admissibility Condition 2: Entropy Condition Definition 3.3. A continuously differentiable function η : Rn → R is an entropy for the system (5) with entropy flux q : Rn → R if for all u ∈ Rn there holds Dη(u) · Df (u) = Dq(u). A consequence of the above is that if u(t, x) is a C 1 solution of (5), then η(u)t + q(u)x = 0. However, for discontinuous u, the η(u) may not be conserved. A weak solution u of (5) is entropy admissible if η(u)t + q(u)x ≤ 0 in the sense of distributions for every entropy-entropy flux pair (η, q) for (5) st η is convex i.e ∇2 η(u) ≥ 0.

6

Admissibility Condition 3: Stability Condition/Liu Condition 1. For the scalar case, stability implies that for any two states u− and u+ , if we insert an arbitrary intermediate state u∗ in the interval [u− , u+ ] (or [u+ , u− ]), we want the [speed of jump behind] > + ∗ (u∗ ) (u− ) ≥ f (uu+)−f for all u∗ ∈ [u− , u+ ] (or [u+ , u− ]). Thus, we [speed of jump ahead] i.e. f (uu∗)−f −u− −u∗ have stability condition for scalar case: • If u− < u+ , then the graph f should remain above the secant line. • If u− > u+ , then the graph f should remain below the secant line. 2. For vector-valued case, notice that the condition above is equivalent to saying that the speed of the original shock (u− , u+ ) should not be greater than the speed of any intermediate shock (u− , u∗ ), so we can generalise this to the Liu admissibility condition. Let u+ = Si (σ)(u− ) for some σ ∈ R. The shock with left and right states u− and u+ respectively satisfies the Liu admissibility condition provided that its speed is less or equal to the speed of every smaller shock joining u− with an intermediate state u∗ = Si (s)(u− ) where s ∈ [0, σ]. Admissibility Condition 4: Lax Condition A shock of the i-th family connecting the states u− and u+ and travelling with speed λ = λi (u− , u+ ) satisfies the Lax admissibility condition if λi (u− ) ≥ λi (u− , u+ ) ≥ λi (u+ ). i.e. the Lax condition requires that the i-characteristic lines run into the shock from both sides.

4

The Riemann Problem

Consider the Riemann problem given by ut + f (u)x = 0

 u− with u(0, x) = u0 (x) = u+

if x < 0

(7)

if x > 0.

This provides the basic building block towards the solution of the Cauchy problem with more general initial data. The Riemann problem yield precisely the weak solutions which are invariant with respect to the scaling (t, x) 7→ (θt, θx) for all θ > 0. Example 4.1. Consider the Riemann problem for a linear system (A is a constant n × n matrix)  u− if x < 0 ut + Aux = 0 with u(0, x) = u0 (x) = u+ if x > 0.

(8)

The solution is constructed as follows: write u + −u− as a linear combination of the eigenvectors of A Pn P (i.e. u+ − u− = j=1 cj rj ) and define the intermediate states ωi = u− + j≤i cj rj for i = 1, 2, . . . , n. Then the solution takes the form  −  for x/t < λ1  ω0 = u u(t, x) =

ωi    ωn = u+

for λi < x/t < λi+1 for x/t > λn .

7

4.1

Elementary Waves

Assumption (Lax). Each i-th characteristic field is either genuinely nonlinear (i.e. Dλi (u) · ri (u) > 0 for all u) or linearly degenerate (i.e. Dλi (u) · ri (u) ≡ 0 for all u). In other words, we have either the character speed λi (u) to be strictly increasing or constant along the integral curve of the eigenvectors ri (u). This assumption ensures that the solution of the Riemann problem is a superposition of n elementary waves: shocks, rarefactions and contact discontinuities. 4.1.1

Construction of Elementary Waves

Fix a state u0 ∈ Rn and an index i ∈ {1, 2, . . . , n}. Let ri (u) be an i-eigenvector of the matrix A(u) = Df (u). The integral curve of the vector field ri through the point u0 (i.e. s 7→ Ri (s)(u0 ) obtained by du solving dσ = ri (u)) is called the i-rarefaction curve through u0 . We always orientate the curve Ri so that λi (u) increases with the parameter s. Next, recall from Section 3.2 that we had the i-shock curve through u0 given by s 7→ Si (s)(u0 ). The curves Ri and Si have a second order contact point at u0 as they are both tangent to ri (u) at u0 . We define the elementary waves as follows: 1. Centred Rarefaction Wave. Let the i-th field be genuinely nonlinear and assume that u+ lies on the positive i-rarefaction curve through u− i.e. u+ = Ri (σ)(u− ) for some σ > 0. For each s ∈ [0, σ], define the characteristic speed λi (s) = λi (Ri (s)(u− ). Thus, the function

u(t, x) =

 −   u

if x/t < λi (u− )

Ri (s)(u− ) if x/t = λi (s) ∈ [λi (u− ), λi (u+ )]    + u if x/t > λi (u+ )

is a piecewise smooth solution of the Riemann problem, continuous for t > 0. 2. Shock. Assume again that the i-th family is genuinely nonlinear and the state u+ is connected to the right of u− by an i-shock i.e. u+ = Si (σ)(u− ). Then, letting λ := λi (u+ , u− ) be the Rankine-Hugoniot speed of the shock, the function  u− if x < λt u(t, x) = (9) u+ if x > λt is a piecewise constant solution to the Riemann problem. If σ < 0, the solution is entropy admissible in the sense of Lax (and it is violated if σ > 0). 3. Contact Discontinuities. Assume that the i-th field is linearly degenerate and that the state u+ lies on the i-th rarefaction curve through u− i.e. u+ = Ri (σ)(u− ) for some σ. By assumption, the i-th character speed is constant along this curve. Choosing λ = λ(u− ), the piecewise constant function (9) is the solution to our Riemann problem. In summary, for a fixed left state u− ∈ Rn and i ∈ {1, 2, . . . , n} define the mixed curve  R (σ)(u− ) if σ ≥ 0 i Ψi (σ)(u− ) = Si (σ)(u− ) if σ < 0. In the case where u+ = Ψ(σ)(u− ) for some σ, the Riemann problem can be solved by elementary waves.

8

4.1.2

General Solution to the Riemann Problem

The general solution can be obtained by by finding intermediate states ω0 = u− , ω1 , . . . , ωn = u+ such that each pair of adjacent states ωi , ωi+1 can be connected by an elementary wave i.e. ωi+1 = Ψi (σi )(ωi ). Therefore, for u+ is sufficiently close to u− , there exists {σj }nj=1 such that u+ = Ψn (σn )◦· · ·◦Ψ1 (σ1 )(u− ). The complete solution is obtained by piecing together the solutions of the Riemann problems  ω if x < 0 i ut + f (u)x = 0 with u(0, x) = ωi+1 if x > 0.

5

Functional Analytic Approaches to the Existence Theory

Definition 5.1 (Young Measures). Let Ω ⊂ Rn and K ⊂ Rm be bounded and open subsets and uk : Ω → Rm be measurable such that uk (y) ∈ K almost everywhere. Then there exists a family of probability measures on Rm given by {νy }y∈Ω such that ¯ for all y ∈ Ω. 1. supp(νy ) ⊂ K k kj 2. ∀ f ∈ C(Rm ; R), ∃ {ukj }∞ j=1 ⊂ {u } s.t. w*-limf (u ) = hνy (λ), f (λ)i =

´

f (λ) dνy (λ).

3. ukj → u a.e. ⇔ νy (λ) = δu(y) (λ) (i.e. the Dirac mass). Remark 5.1. 1. The deviation between the weak and srong convergence is measured by the spreading of the support of νy i.e. ||f (w*-lim uk ) − w*-limf (uk )||L∞ ≤ C supy (diam(supp(νy ))). 2. The Young measure family {νy }y∈Ω can be thought of as the limiting probability distribution of the values of {uk (y)} near the point y as k → ∞. Theorem 5.1 (Weak Continuity of 2 × 2 Determinants). Let Ω ⊂ R+ × R be bounded and open and uk : Ω → R4 be measurable. Assume also that: 1. w*-limk→∞ uk = u in L2 (Ω; R4 ). ∂uk

∂uk

∂uk

∂uk

−1 (Ω). 2. Both ∂t1 + ∂x2 and ∂t3 + ∂x4 are compact in Hloc uk uk u u 1 2 2 Then, 1k * in D0 . u3 uk4 u3 u4

Theorem 5.2 (Div-Curl Lemma). Let Ω ⊂ R+ × R be bounded and open and p, q > 1 such that 1 1 k p n k q n p + q = 1. Assume that v ∈ L (Ω; R ) and w ∈ L (Ω; R ) such that 1. v k * v in Lp (Ω; Rn ) and wk * w in Lq (Ω; Rn ). 2. div v k are compact in W −1,p (Ω; R) and curl wk are compact in W −1,q (Ω; R). Then, v k · wk * v · w in D0 . Theorem 5.3 (Compensated Compact Embedding Lemma). Let Ω ⊂ Rn be bounded and open. Then, for any 1 < q ≤ p < r < ∞, we have −1,q −1,r −1,p (compact set of Wloc (Ω)) ∩ (bounded set of Wloc (Ω)) ⊂ (compact set of Wloc (Ω)).

9

5.1

2 × 2 Hyperbolic Systems of Conservation Laws

Consider the problem ut + f (u)x = 0 for u ∈ R2

with

u(0, x) = u0 (x).

(10)

Definition 5.2 (Riemann Invariant). We say wi : R2 → R is an i-th Riemann invariant provided that Dwi (u) is parallel to lj (u) for u ∈ R2 and j 6= i. Equivalently, we have Dwi (u) · ri (u) = 0 for i = 1, 2. Theorem 5.4. Assume there exists a strictly convex entropy η∗ (u) and a globally defined Riemann invariant w = (w1 , w2 ) : R2 → R2 . If u ∈ C 1 , then (10) becomes ∂t wj + λj (u)∂x wj = 0 for j = 1, 2. Theorem 5.5 (Entropy Equation). Recall the definition of entropy-entropy flux pair from Definition 3.3. Using the Riemann invariant, we have the following relations: qwj = λj ηwj

and ηw1 w2 −

λ1 w2 λ2 w1 ηw + ηw = 0. λ2 − λ1 1 λ2 − λ1 2

Theorem 5.6 (Dissipation Estimate for Vanishing Viscosity Solutions). Suppose uε is the viscosity √ approximation of (10) as defined in Section 1.3. Then, || εuεx ||L2 ≤ C where C is independent of ε. Theorem 5.7 (H −1 Compactness). For any entropy-entropy flux pair (η, q) ∈ C 2 , η(uε )t + q(uε )x is −1 compact in Hloc .

10

Part II

Wave Equations 6

Linear Wave Equations

Lemma 6.1 (Gronwall’s Inequality). Let E, A and b be non-negative functions defined on [0, T ] with A ´t ´t increasing. If E(t) ≤ A(t) + 0 b(τ )E(τ ) dτ for t ∈ [0, T ], then E(t) ≤ A(t) exp( 0 b(τ ) dτ ) for t ∈ [0, T ]. Consider the linear wave operator on R1+n defined as  := ∂t2 −∆. Given functions f and g, the problem u = 0

on

[0, ∞) × Rn

with

u(0, x) = f and ∂t u(0, x) = g

(11)

has been well understood with some results as follows: 1. (Uniqueness). The problem (11) has at most one solution u ∈ C 2 ([0, ∞) × Rn ). 2. (Existence). If f ∈ C [n/2]+2 (Rn ) and g ∈ C [n/2]+1 (Rn ), then (11) has a unique solution and for the cases n = 1, 2, 3, the solutions are given explicitly by the D’Alembert formula:   ˆ x+t 1 u(t, x) = f (x + t) + f (x − t) + g(τ ) dτ for n = 1. 2 x−t ! ˆ ˆ f (x + ty) g(x + ty) t t p p dy + dy for n = 2. u(t, x) = ∂t 2 2π |y| 0 such that (14) has a unique solution u ∈ C 2 ([0, T ] × Rn ). Moreover, u ∈ L∞ ([0, T ], H s+1 ) ∩ C 0,1 ([0, T ], H s ). Theorem 8.3 (Semilinear Equation). Suppose that g jk = δjk in (14) i.e. u = F (u, ∂u) with F smooth and F (0, 0) = 0. If g, h ∈ C0∞ (Rn ), then there exists T > 0 such that the semilinear equation has a unique solution such that u ∈ C ∞ ([0, T ] × Rn ). Theorem 8.4 (Continuation Principle). Assume that u is a solution of (14) with f, g ∈ C0∞ (Rn ). Let T ∗ := sup{T > 0 : (14) has a solution u ∈ C ∞ ([0, T ] × Rn }. If T ∗ < ∞, then X |∂ α u(t, x)| ∈ / L∞ ([0, T ∗ ) × Rn ). |α|≤ n+6 2

12

9 9.1

Klainerman-Sobolev Inequality Some Tensor Calculus

Definition 9.1 (Vector Field and Tangent Space). A vector field X in R1+n is the first order differential operator of the form X = X µ ∂µ . The collection of all such vector fields is the tangent space, T R1+n . Definition 9.2 (Lie Bracket). For any two vector fields X = X µ ∂µ and Y = Y µ ∂µ , the Lie bracket is the vector field [X, Y ] := XY − Y X = (X(Y µ )) − Y (X µ ))∂µ . Definition 9.3 (Minkowski Metric). Let mµν = diag(−1, 1, . . . , 1) be an (1 + n) × (1 + n) diagonal matrix. We define the Minkowski metric on R1+n by m : T R1+n × R1+n → R by m(X, Y ) = mµν X µ Y ν for all X, Y ∈ T R1+n . Definition 9.4. A vector field X in (R1+n , m) is called spacelike, timelike or null if m(X, X) > 0, m(X, X) < 0 or m(X, X) = 0 respectively. Definition 9.5. In (R1+n , m), we define the D’Alembertian as  := mµν ∂µ ∂ν where (mµν ) = (mµν )−1 . Definition 9.6 (Energy-Momentum Tensor). The energy-momentum tensor associated to u = 0 is the symmetric 2-tensor Q[u]µν := ∂µ u∂ν u − 21 mµν (mρσ ∂ρ ∂σ u). For any vector fields X, Y , we have 1 Q[u](X, Y ) = (Xu)(Y u) − m(X, Y )m(∂u, ∂u). 2 The divergence of the energy-momentum tensor is div(Q[u])ρ = (u)∂ρ u and the divergence of the 1-form Pµ := Q[u]µν X ν is 1 div(P ) = (u)Xu + Q[u]µρ (∂µ Xρ + ∂ρ Xµ ). 2 The term ∂µ Xν + ∂ν Xµ :=

(X)

(15)

πµν is called the deformation tensor of X with respect to m.

Theorem 9.1. Let u ∈ C 2 (R1+n ) that vanishes for large |x| at each t. Then, for any vector field X and t0 < t1 , we have: ˆ ˆ ¨ Q[u](X, ∂t ) dx = Q[u](X, ∂t ) dx + div(P ) dtdx. t=t1

[t0 ,t1 ]×Rn

t=t0

Remark 9.1. By choosing X suitably, many useful energy estimates can be derived from Theorem 9.1. If X is chosen such that (X) πµν = 0 or (X) πµν = f mµν , then the identity (15) is simpler. Definition 9.7 (Killing Vector Field). A vector field X is called Killing if

(X)

πµν = ∂µ Xν + ∂ν Xµ = 0.

Proposition 9.1 (Killing Vector Field Representation). Any Killing vector field in (R1+n , m) can be written as a linear combination of the vector fields {∂µ }nµ=0 ∪ {Ωµν }nµ,ν=0 where Ωµν = (mρµ xν − mρν xµ )∂ρ

where

0 ≤ µ < ν ≤ n.

Remark 9.2. Since (mµν ) = diag(−1, 1, . . . , 1) the set {Ωµν } consists of Ω0i = xi ∂t + t∂i for i = 1, 2, . . . , n and Ωij = xj ∂i − xi ∂j for 1 ≤ i < j ≤ n. Corollary 9.1. Let u ∈ C 2 (R1+n ) that vanishes for large |x| at each t. Then, for any Killing vector field X and t0 < t1 , we have: ˆ ˆ ¨ Q[u](X, ∂t ) dx = Q[u](X, ∂t ) dx + (u)Xu dtdx. t=t1

[t0 ,t1 ]×Rn

t=t0

13

Definition 9.8 (Conformal Killing Vector Field). A vector field is called conformal Killing if f mµν for some function f .

(X)

πµν =

Proposition 9.2 (Conformal Killing Vector Field Representation). If X is conformal Killing, it has to be a linear combination of 1. Killing vector fields (with f = 0). Pn 2. L0 = µ=1 xµ ∂µ = t∂t + r∂r (with f = 2). 3. Kµ = 2mµν xν xρ ∂ρ − mην xη xν ∂µ (with f = 4mνµ xν ). Note that: K0 = −(t2 + r2 )∂t − 2tr∂r . Corollary 9.2. Let u ∈ C 2 (R1+n ) that vanishes for large |x| at each t. Then, for any conformal Killing vector field X and t0 < t1 , we have:     ˆ ˆ ¨ n−1 2 n−1 ˜ ˜ f u − f u dtdx Q(X, ∂t ) dx = Q(X, ∂t ) dx + u Xu + 4 8 t=t1 t=t0 [t0 ,t1 ]×Rn
˜ f u∂t u − 12 u2 ∂t f . where Q(X, ∂t ) := Q[u](X, ∂t ) + n−1 4 Theorem 9.2 (Klainerman-Sobolev Inequality). Let u ∈ C ∞ ([0, ∞) × Rn ) vanish when |x| is large. Then, for t > 0 and x ∈ Rn , we have for Γ = (Γ0 , Γ1 , . . . , Γm ) = (∂0 , . . . , ∂n , Ω0,1 , . . . , Ωn−1,n , L0 ) (where m + 1 is the number of such vector fields i.e. m + 1 = 12 (n2 + 3n + 4)), the inequality X (1 + t + |x|)n−1 (1 + |t − |x||)|u(t, x)|2 ≤ C(n) ||Γα u(t, ·)||2L2 |α|≤ n+2 2 αm 0 α1 where we denote for a multi-index α, Γα = Γα 0 Γ1 . . . Γm . Note: Γi commute well with each other.

Lemma 9.1 (Commutator Relations). For the vector fields ∂µ , Ωµν , L0 , we have commutator relations [∂µ , ∂ν ] = 0

[∂ρ , Ωµν ] = (mσµ δρν − mσν δρµ )∂σ

[∂µ , L0 ] = ∂µ

[Ωµν , Ωρσ ] = mσµ Ωρν − mρµ Ωσν + mρν Ωσµ − mσν Ωρµ [Ωµν , L0 ] = 0

[, ∂µ ] = 0

[, Ωµν ] = 0

[, L0 ] = 2.

Consequently, for any multi-index α, there exists constants cαβ such that: Γα = in particular [Γ, ] = c  for some constant c.

9.2 9.2.1

P

|β|≤|α| cαβ Γ

β

, and

Global Existence in Higher Dimensions Dimension n ≥ 4

Consider the problem u = F (∂u)

on

[0, ∞) × Rn

with

u(0, x) = εf and ∂t u(0, x) = εg

(16)

where n ≥ 4, ε > 0 is a number and F : R1+n → R is a given C ∞ function which vanishes to the second order at the origin i.e. F (0) = 0 and DF (0) = 0. Theorem 9.3. Suppose that f, g ∈ Cc∞ (Rn ) for the problem (16), then there exists ε0 > 0 such that (16) has a unique solution u ∈ C ∞ ([0, ∞) × Rn ) for any 0 < ε < ε0 . Remark 9.3. For n = 2, 3, global existence can be guaranteed if F satisfy a stronger condition i.e. Dk F (0) = 0 for all 0 ≤ k ≤ 5 − n. 14

9.2.2

Dimension n = 3

Consider the problem φi =

n X

i Γijk (φ)Bjk (Dφj , Dφk )

[0, ∞) × R3

on

with u(0, x) = εf and ∂t u(0, x) = εg

(17)

j,k=1 i with φ, f, g ∈ C0∞ (R3 ; Rn ), Γijk : Rn → R are smooth functions and Bjk (Dφj , Dφk ) takes the form i Bjk (Dφj , Dφk ) = aijk Q0 (Dφj , Dφk ) +

X

j k bi;µν jk Qµν (Dφ , Dφ )

0≤µ 0 such that (17) has a global smooth solution provided ε < ε0 . Definition 9.9. We define Pn 1. ∂r = 1r i=1 xi ∂i . 2. L = ∂t + ∂r and L = ∂t + ∂r . 3. Let the tensor of projection to a sphere centered at (t, 0) be Πij = δ ij − ∂ri ∂rj where ∂ri means the i i-th component of ∂r i.e. ∂ri = xr . Then, for f ∈ C ∞ (R1+n ), the spatial angular component of ∂i f / )i = δil Πlj ∂j f . is given by (∇f Lemma 9.2. For Q = Q0 , Qµν , we have ¯ + |∂φ||∂ψ| ¯ |Q(Dφ, Dψ)| < C(|∂φ||∂ψ| / i or L. Note that ∇ / i = ∂i − where ∂¯ = ∇

xi r ∂r

for i = 1, 2, 3.

˜ = Q0 or Qµν , we have Lemma 9.3. Let Γ be defined as in Theorem 9.2. Then, letting Q ˜ ΓQ(Dφ, Dψ) = Q(DΓφ, Dψ) + Q(Dφ, DΓψ) + Q(Dφ, Dψ.) Lemma 9.4. For any δ > 0, there exists C(δ) > 0 such that ¨ [0,T ]×R3

¯ |∂φ| dxdt (1 + |t − r|)1+δ

ˆ

! 21 ≤C

||(φ(0, ·), ∂t φ(0, ·))||H˙ 1 (R3 )×L2 (R3 ) +

!

T

||φ(t, ·)||L2 (R3 ) dt . 0

Lemma 9.5. If φ is a function compactly supported in {(t, x) : |x| ≤ t + R}, then X n+1 1 ¯ ≤ C(n, R) (1 + t + r) 2 (1 + |t − |x||)− 2 |∂φ| ||∂Γα φ(t, ·)||L2 (Rn ) . |α|≤ n+4 2

15