GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS

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1 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS HAILIANG LIU AND MAKSYM PRYPOROV Abstract. In this work we construct Gaussian beam approximations to solutions of the strictly hyperbolic system with highly oscillatory initial data. The evolution equations for each Gaussian beam component are derived. Under some regularity assumption of the data we obtain an optimal error estimate between the exact solution and the Gaussian beam superposition in terms of the high frequency parameter ε. The main result is that the relative local error measured in energy norm in the beam approximation decays as ε independent of dimension and presence of caustics, for first order beams. This result is shown to be valid when the gradient of the initial phase may vanish on a set of measure zero.. Introduction In this article we are interested in the accuracy of Gaussian beam approximations to solutions of the symmetric hyperbolic system: (.) A(x) u n t + D j u = 0, x j subject to highly oscillatory initial data, j= (.) u(0, x) = B 0 (x)e is 0(x)/ε, where x, S(x) is a scalar smooth function, B 0 : C m is a smooth vector function, compactly supported in, A(x) is m m symmetric positive definite matrix, and D j are m m symmetric constant coefficient matrices, j =,... n. Symmetric hyperbolic systems represent a wide area of research in PDE theory itself, in particular, the high frequency problem arises in several areas of continuum physics including acoustic waves, and the research in this field can give some insight in the study of some significant physical systems such as the Maxwell system of equations. The symmetry of the hyperbolic system ensures the existence of the orthogonal basis in formed by the associated eigenvectors, and this spectral decomposition is useful in our construction of high frequency approximate solutions. It is well-known that high frequency wave propagation problems create severe numerical challenges that make direct simulations unfeasible, particularly in multidimensional settings. High frequency asymptotic models, such as geometrical optics, can be found in some classical literature (see []). A main drawback of geometrical optics is that the model breaks down at caustics, where rays concentrate and the predicated amplitude becomes unbounded, therefore unphysical. As an alternative one can use the level set method to compute multivalued phases beyond caustics, we refer to [6] for a review of the level set framework for 99 Mathematics Subject Classification. 35A, 35A35, 35Q45. Key words and phrases. Symmetric hyperbolic systems, Gaussian beams.

2 HAILIANG LIU AND MAKSYM PRYPOROV computational high frequency wave propagation. Such a method handles well the crossing of Hamiltonian trajectories, but still fails to give bounded amplitude at caustics. Gaussian beams, as another high frequency asymptotic model, are closely related to geometric optics, yet valid at caustics. The solution is concentrated near a single ray of geometric optics. In Gaussian beams, the phase function is real valued along the central ray, its imaginary part is chosen so that the solution decays exponentially away from the central ray, maintaining a Gaussian shaped profile. More general high frequency solutions can be described by superposition of Gaussian beams. In this paper we are going to use the Gaussian beam approach. This approach has gained considerable attention in recent years from both computational and theoretical points of view. A general overview of the history and the latest development of this method are given in the introduction to [0]. Another related approach is the frozen Gaussian approximation, or the Herman-Kluk formula discovered by several authors in the chemical-physics literature in the eighties. This approach with superposition of beams in phase space is closely related to the Fourier-Integral Operator (FIO) with complex phases. The mathematical analysis of the Herman-Kluk was given only recently; see [7, 6] for semiclassical approximation of the Schrödinger equation, and [] for the frozen Gaussian approximation to linear strictly hyperbolic systems. In this paper we formulate a Gaussian beam superposition in physical space for strictly hyperbolic systems and study the accuracy in terms of the high frequency parameter ε of Gaussian beams. Several such error estimates have been derived in recent years: for the initial data [8], for scalar hyperbolic equations and the Schrödinger equation [8, 9, 0], for the acoustic wave equation with superpositions in phase space [], for the Helmholtz equation with a singular source [], and for the the Schrödinger equation with periodic potentials [7]. The general result is that the error between the exact solution and the Gaussian beam approximation decays as ε N/ for N-th order beams in the appropriate Sobolev norm. For phase space based Gaussian beams with frozen Gaussians, the integral approximation decays as ε N for N-th order beams; see [7, 6, 3]. We note that in the frozen Gaussian beam approximation, the extra order of accuracy is found from a symbolic calculus for FIO with complex quadratic phases, instead of from asymptotic accuracy of each individual beams. The analysis of Gaussian beam superpositions for hyperbolic systems presents a few new challenges compared to the scalar wave equations previously studied in [8, 0]. First, it must be clarified how beams are propagated along each wave field through some field decomposition. Second, the distinction of the eigenvalues of the dispersion matrix is assumed to allow for a correction in the amplitude with uniform estimates. This is similar to the Schrödinger equation with periodic potentials [4, 3] for which no energy band crossing is essentially used in [7] in the accuracy study. During our construction we also prove several minor results: the leading Gaussian beam phase is shown to be stationary (which is related to the Huygens principle) along each wave field, and the momentum does not vanish as long as it is nonzero initially. Another significant improvement is that we can formulate and prove the main result for more general initial phase in the sense that we allow the gradient of the initial phase to vanish on a set of measure zero; this question was considered open in previous works [8, 0] for scalar higher order wave equations. The organization of this paper is as follows: In section, we start with the problem formulation and state the main results, then we proceed with Gaussian beam construction which is new for hyperbolic systems, but quite straightforward and simple for those familiar with the Gaussian beam method. In section 3 we prove our main results for initial phase with

3 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 3 non-vanishing gradient everywhere in. Finally, in section 4, we extend our results to a more general phase as stated in section. To keep our presentation less lengthy, we consider quite simple hyperbolic systems, although we believe that our results can be generalized to some extent.. Problem formulation and main result Consider the initial value problem (.)-(.). We define the dispersion matrix L(x, k) : n (.) L(x, k) = A (x) D j k j, and introduce the following inner product in : j= u, v A := Au, v. It is known that (see, e.g. [], [6]) L is symmetric with respect to the inner product, A : Lu, v A = u, Lv A. Hence, L has real eigenvalues {λ i (x, k)} m i=, satisfying (.) L(x, k)b i (x, k) = λ i (x, k)b i (x, k), i =,... m, where {b i (x, k)} m i= are eigenvectors, forming an orthonormal basis in l equipped with a weight function A(x), i.e., b i, b j A = δ ij, and λ i (x, k) are scalar smooth functions. We assume that all eigenvalues are simple (i.e., system (.) is strictly hyperbolic) and the following holds: (.3) λ i (x, k) < λ i (x, k) < λ i+ (x, k), i =,..., m, in a neighborhood of any (x, k) in phase space. For the initial data (.) we assume that the amplitude B 0 (x) has compact support in a bounded domain, and the phase S 0 (x) is smooth. Let B 0 (x) have the following eigenvector decomposition (.4) B 0 (x) = a i (x)b i (x, x S 0 (x)), then (.5) a i (x) = B 0 (x), b i (x, x S 0 (x) A, i =,..., m. i= For each wave field associated with b i, we construct a Gaussian beam approximation (.6) u iε GB = A i (t, x; x 0 )e iφ i(t,x;x 0 )/ε, where A i (t, x; x 0 ) and Φ i (t, x; x 0 ) are Gaussian beam amplitudes and phases, respectively, concentrated on a central ray starting from x 0 with p 0 = x S 0 (x 0 ). By the linearity of the hyperbolic system, we then sum the Gaussian beam ansatz (.6) over i =,..., m and x 0 to define the approximate solution (.7) u ε (t, x) = u iε (πε) GBdx n 0, where (πε) n is a normalizing constant which is needed for matching the initial data (.). i=

4 4 HAILIANG LIU AND MAKSYM PRYPOROV Indeed the initial data can be approximated by the same form of the Gaussian beam superposition (.7), (.8) u ε (0, x) = A i (0, x; x (πε) n 0 )e iφ0 (x;x 0 )/ε dx 0, i= where Φ 0 is the initial Gaussian beam phase, which is assumed to be the same for each wave field. By the classical Gaussian beam theory [4], the initial phase can be taken of the form (.9) Φ 0 (x; x 0 ) = S 0 (x 0 ) + x S 0 (x 0 ) (x x 0 ) + (x x 0) ( xs 0 (x 0 ) + ii)(x x 0 ) with coefficients that serve as initial data for ODEs of the Gaussian beam components. The amplitude A i (0, x; x 0 ) are defined later in (4.) using a i (x 0 ) in (.5). We are going to use the following notations in this work. The unmarked norm denotes the usual L -norm. The energy norm E is defined as (.0) u E := Au, u dx. L norm of function f and its derivatives: f C α := max x α f(x). x L matrix norm: A L := sup Av, v R m. v = We can now state the main result. Theorem.. Let be a bounded measurable set, initial amplitude B 0 (x) H ( ), initial phase S 0 (x) C n+4 ( ) and bounded, x S 0 (x) be bounded away from zero on ; eigenvectors b i (x, k) and eigenvalues λ i (x, k) be smooth and bounded functions satisfying (.3), u be the exact solution to (.)-(.) for 0 < t T, and u ε be the first order Gaussian beam superposition (.7). Then (.) u u ε E Cε /, where constant C is independent of ε, but may depend on the finite time T and the data given. An improvement of the above result is that we may allow more general initial phase with possible vanishing phase gradient on a small set. More precisely, we have Corrolary.. Under the assumption of Theorem., if the measure of the set σ := {x, x S 0 (x) = 0} is zero, then the error estimate (.) remains valid if the superposition is over beams issued from points in /Σ, where σ Σ with measure of size ε n. We proceed to construct Gaussian beam asymptotic solutions and obtain the desired error estimate in several steps. First, we present the construction for the Gaussian beam phase components which is a straightforward extension of the Gaussian beam approach developed for hyperbolic and Schrödinger equations, see for example, [0]. While constructing the Gaussian beam amplitude, we address some solvability difficulties and show the way to solve it using the approach developed in [3] and verifying the boundedness of the additional terms. For the error estimate, we rely on the wellposedness argument and prove initial and evolution errors separately. For the initial error, we use some techniques similar to those developed by

5 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 5 Tanushev in [8], keeping in mind that here we have to deal with vector valued functions. As for the evolution error estimate, we rely on some phase estimates proved in [0], which is a key technique for the proof. 3. Gaussian beam construction Let P be the differential operator in (.). We look for an approximate solution to (.), which has the form (3.) u ε = (v 0 + εv + + ε l v l (t, x))e iφ(t,x)/ε. Inserting u ε into (.), we obtain: (3.) A (x)p [u ε ] = where (3.3) (3.4) (3.5) c 0 = i(φ t + L(x, x Φ))v 0, ( ) ε c 0 + c + + ε l c l e iφ/ε = 0, c = ( t + L(x, x ))v 0 + i(φ t + L(x, x Φ))v, c l = ( t + L(x, x ))v l + i(φ t + L(x, x Φ))v l, i =,..., l. By geometric optics, the leading term is required to vanish, (3.6) c 0 = i(φ t + L(x, x Φ))v 0 = 0, where L(x, k) is the dispersion matrix defined in (.). We set the leading amplitude as (3.7) v 0 (t, x) = a i (t, x)b i (x, k(t, x)), k(t, x) := x Φ(t, x), to infer from (.) that c 0 = i= ia i (t, x)( t Φ + λ i (x, k(t, x))b i (x, k(t, x)), i= which vanishes as long as Φ solves the Hamilton-Jacobi equation: (3.8) G(t, x) := Φ t + λ(x, x Φ) = 0, for each λ = λ i. From now on we shall supress the index i, since the construction is same for each eigenvalue λ i, i =,... m. 3.. Construction of the Gaussian beam phase. Let ( x(t; x 0 ), p(t; x 0 )) be the phase space trajectory governed by the Hamiltonian in (3.8), then (3.9) x = k λ( x, p), ṗ = x λ( x, p), satisfying x(0, x 0 ) = x 0 and p(0; x 0 ) = x S 0 (x 0 ). Next we introduce an approximation of the phase: (3.0) Φ(t, x; x 0 ) = S(t; x 0 )+p(t; x 0 ) (x x(t; x 0 ))+ (x x(t; x 0)) M(t; x 0 ) (x x(t; x 0 )),

6 6 HAILIANG LIU AND MAKSYM PRYPOROV where S and M are to be chosen so that G(t, x) vanishes on x = x(t; x 0 ) to higher order. From (3.8) and (3.0) we derive: (3.) G(t, x; x 0 ) = Ṡ + ṗ (x x) p x + (x x) Ṁ(x x) x M(x x) + λ. Setting G(t, x; x 0 ) = 0 at x = x, we obtain (3.) Ṡ = p k λ( x, p) λ( x, p). We can actually show that Ṡ = 0 as stated below. Lemma 3.. Let λ(x, k) be an eigenvalue of L(x, k) associated with the eigenvector b(x, k), then (3.3) λ(x, k) = k k λ(x, k) holds for any x, k, and (3.4) λ(x, k) C k, x, if A(x) δ > 0. Proof. Differentiation of (.) with respect to k j leads to the following: (3.5) A (x)d j b(x, k) + L(x, k) k j b(x, k) = j= λ(x, k)b(x, k) + λ(x, k) b(x, k). k j k j Multiplying (3.5) by k j and summing up in j, j =,... m, we obtain (3.6) L(x, k)b(x, k) + k j L(x, k) b(x, k) = k k λ(x, k)b(x, k) + k j λ(x, k) b(x, k). k j k j Hence, (3.7) λ(x, k)b(x, k) = j= j= k j (λ(x, k) L(x, k)) k j b(x, k) + k k λ(x, k)b(x, k). Taking inner product A with b(x, k) and using that matrix L(x, k) is symmetric, we prove (3.3). The estimate (3.4) follows from the relation λ(x, k) = b L(x, k)b(x, k) and the assumption A δ > 0. The identity (3.3) when applied to (3.) yields Ṡ = 0. We observe that x G(t, x, x 0 ) = 0 is equivalent to the p equation in (3.9). Next, we set xg(t, x, x 0 ) = 0, to obtain Ṁ + x(λ(x, k)) = 0, (x,k)=( x,p) which is equivalent to (3.8) Ṁ + K + K M + MK + MK 3 M = 0, where K, K and K 3 are matrices with the correspondent entries: K ij = which are evaluated on the ray trajectory ( x, p). λ, K ij = λ, K 3ij = λ x i x j x i k j k i k j

7 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 7 Using the Taylor expansion, we have (3.9) G = α =3 α! α x G(t, ; x 0 )(x x) α = α =3 α! α x λ(, )(x x) α, which means that G vanishes up to third order on x = x. The heart of the Gaussian beam method is to solve the nonlinear Ricatti equation (3.8). It is known from [4, 5] that if the initial matrix is symmetric and its imaginary part is positive definite, then a global solution M to (3.8) is guaranteed and has the following properties: (i )M = M, and (ii) Im(M) is positive definite for all t > 0. In summary, we obtain evolution equations for the Gaussian beam phase components subject to appropriately chosen initial data: x = k λ( x, p), x t=0 = x 0, ṗ = x λ( x, p), p t=0 = x S 0 (x 0 ), (3.0) Ṡ = 0, S t=0 = S 0 (x 0 ), Ṁ = MK 3 M K M MK K, M t=0 = xs 0 + ii. Note that k λ( x, p) may not be well defined for p = 0; for example, if λ = k, then k λ( x, p) = p. The following result tells that we can construct well-defined beams as long p as p(0; x 0 ) 0. Lemma 3.. If p(0; x 0 ) 0, then (3.) p(t; x 0 ) p(0; x 0 ) e ct, where constant c may depend on T and the data given. Proof. First we show that ṗ c p. Since λ is homogeneous in k of degree, we have λ(x, k) = k λ(x, ω), where ω = k is a directional unit vector. Hence k ṗ = x λ( x, p) = x λ( x, ω) p, which leads to ṗ max xλ( x, ω) p := c p. t T,ω S n Next, we consider d dt ( p e ct ) = (p ṗ + c p )e ct ( c p + c p )e ct = 0. This proves (3.) as claimed. 3.. Construction of the Gaussian beam amplitude. We recall that (3.) c = ( t + L(x, x ))v 0 (t, x) + i( t Φ + L(x, x Φ))v (t, x), where t Φ + L(x, x Φ) = G(t, x) + L(x, k(t, x)) λ(x, k(t, x)), so that we may use G = O((x x) 3 ) when applicable. Here and in what follows, we omit the identity matrix against any scalar quantity unless a distinction is needed. On the ray x = x(t), we require that c = 0, i.e., (3.3) ( t + L( x, x ))v 0 (t, x) x= x + i(l( x, p) λ( x, p))v (t, x) = 0.

8 8 HAILIANG LIU AND MAKSYM PRYPOROV In order for v to exist, it is necessary that (3.4) ( t + L( x, x ))(a(t, x)b(x, p)) x= x, b( x, p) A = 0. For x x(t), we have c = ( t + L(x, x ))v 0 (t, x) + ig(t, x; x 0 )v + i(l(x, k(t, x)) λ(x, k(t, x)))v, where v contains the orthogonal complement of b, satisfying v, b A = 0. We choose (3.5) v = i(l(x, k(t, x)) λ(x, k(t, x))) (( t + L(x, x ))v 0 ( t + L(x, x ))v 0, b A b), which is well defined since the term in the bracket is perpendicular to b against matrix A. Therefore (3.6) where v span{v, b}. c = ( t + L(x, x ))a(t, x)b, b A b + igv, Lemma 3.3. For the first order Gaussian beam construction, a(t, x) = a(t; x 0 ) and satisfies the following evolution equation (3.7) ȧ = a k b D x λ L(x, D x )b, b A x= x, where D x := x + M(t; x 0 ) k. Moreover, (3.8) (3.9) c 0 = ia(t, x 0 )G(t, x; x 0 )b(x, k(t, x)), c = a(t; x 0 )d (x x)b(x, k(x)) + igv, where d C( + x x ), and v span{v, b} with v defined in (3.5), and G = O( x x 3 ). Proof. For the first order Gaussian beams, we look for amplitude of form a(t, x) = a(t; x 0 ), then (3.4) gives (3.30) a t + a t b + L(x, x )b, b A x= x = 0. Note that b = b(x, k(t, x)) with (3.3) k(t, x) = p(t) + M(t)(x x(t)). Let D x denote x with only t fixed, then we have D x = x + x k k = x + M k, then L(x, x )b = L(x, D x )b(x, k(t, x)). Using (3.3) and the ray equation (3.9), we obtain which when evaluated on the ray x = x gives This gives t k(t, x) = x λ + Ṁ(x x) M kλ, t k(t, x(t)) = x λ M k λ = D x λ( x(t), p(t)). t b = k b t k(t, x) = k b D x λ, x = x(t).

9 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 9 These together have justified (3.7). Set f(x, k(x)) = t b + L(x, x )b, b A, then it follows from (3.6) and (3.7) that (3.3) c = a (f(x, k(x)) f( x, p)) b + igv = ad x f(, ) (x x)b + igv, where D x f(, ) is evaluated at the intermediate value between x and x. From the definition of f we have (3.33) f = b A(x) (L(x, D x )b + k b t k) ( n ) = b D j D xj b + A(x) k b ( D x λ + Ṁ(x x)). j= By the product rule we see that D x f C( + x x ) if D i xb and D i xλ are uniformly bounded for i. This bound when inserted into (3.3) gives (3.9). In order to complete the estimate for c in (3.9), we still need to estimate v. Lemma 3.4. Let v be defined in (3.5). If eigenvector b(x, k) Cb simple and assumption (.3) is satisfied, i.e., λ = min λ i λ j > 0, i<j m then sup v C( + x x ), t,x 0 where C depends on Gaussian beam components and λ. and eigenvalue λ is Proof. Since eigenvalue λ is simple, then L(x, k) λ(x, k) is invertable and the following resolvent estimate holds: (L(x, k) λ(x, k)) λ. From (3.5) (3.34) v λ ( t + L(x, x ))v 0 ( t + L(x, x ))v 0, b A b, where v 0 = a(t; x 0 )b(x, k) and hence (3.35) ( t + L(x, x ))v 0 ( t + L(x, x ))v 0, b A b = a( t b t b, b A b + L(x, x )b L(x, x )b, b A b). One can see that t b = k b t k = k b( x λ + Ṁ(x x) M kλ(x, k(x))) C k b ( + x x ). Also L(x, x )b n max j n Dj x b C(n, D j ) x b which implies that the right hand side of (3.35) is bounded in terms of k b, x b, x λ, components of the matrix M and the initial data which completes the proof of the lemma.

10 0 HAILIANG LIU AND MAKSYM PRYPOROV We thus obtain a Gaussian beam approximation for any fixed x 0, (3.36) u εi GB(t, x; x 0 ) = (a i (t; x 0 )b i (x, x Φ i ) + εv(t, i x; x 0 ))e iφ i(t,x;x 0 )/ε. This can be used as a building block for approximating the solution of the initial value problem by the GB superposition over x 0 and i =, m, (3.37) u ε (t, x) = (πε) n/ i= u εi GB(t, x; x 0 )dx 0. Based on our construction, we have the following residual representation (3.38) P (u ε ) = A(x)( (πε) n/ ε c 0i + c i )e iφ i(t,x;x 0 )/ε dx 0, where c 0i and c i can be obtained from (3.8) and (3.9), respectively. The proof of Theorem. is based on the following well-posedness estimate. i= Proposition 3.. (Well-posedness) Let u, u ε be an exact and approximate solution of (.) with initial data u 0 and u ɛ 0, respectively. Then the following error estimate holds: (3.39) u u ε E u 0 u ε 0 E + C T 0 P [u ε ] dt, where C is independent of ε, but may depend on the matrix A. This is a classical result, which can be found, for example, in [5]. The well-posedness estimate tells that the energy norm of the total error is bounded by the sum of initial and evolution error. In the rest of this paper, we are going to estimate both initial error and the evolution error. 4. Error estimates 4.. Initial error estimate. The initial condition is approximated as follows: (4.) u ε 0 = (a (πε) n/ i (x 0 )b i (x, x Φ 0 ) + εv(0, i x; x 0 ))e iφ0 /ε dx 0. i= Here the term v(0, i x; x 0 ) is defined to be consistent with that in (3.5). In other words it is understood to be the limit of v(t, i x; x 0 ) as t 0, therefore we have from previous estimate on v, (4.) max x,x 0 m v(0, i x; x 0 ) C. In this section we state and prove the initial error estimate result. Theorem 4.. Let u ε 0 be defined in (4.), u 0 (x) = a i (x)b i (x, x S 0 (x))e is0(x)/ε. Then the energy norm of the difference u 0 u ε 0 satisfies: (4.3) u 0 u ε 0 E Cε /, i= i=

11 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS where the constant C depends on the data given. We split the proof of the theorem into two parts by estimating u u 0 E and u u ε 0 E, respectively, where u is an intermediate quantity defined by (4.4) u (x) := B (x; x (πε) n/ 0 )e iφ0 (x;x 0 )/ε dx 0, where B (x; x 0 ) = a i (x 0 )b i (x, x S 0 (x)). Lemma 4.. Let u be defined in (4.4), a(x) and b(x, ) H ( ), then (4.5) u u 0 E Cε /, where C depends on A C, b C, B 0 E, and x B 0 E. Proof. First, we rewrite u u 0 = i= (B (x; x (πε) n/ 0 ) B 0 (x))e it x 0 [S0]/ε e x x 0 /ε + B 0 (x)(e it x 0 [S 0]/ε e is 0/ε )e x x 0 /ε dx 0 = I + I, where T x 0 [S 0 ] is the second order Taylor polynomial of S 0 about x 0. Noting that u 0 u E I E + I E. We start with I E. I E = A(x) (B B (πε) n 0 )e it x 0 [S0]/ε e x x 0 /ε dx 0 R n (B B 0 )e it x 0 [S 0]/ε e x x 0 /ε dx 0 dx = (a (πε) n i (x 0 ) a i (x))(a l (x 0) a l (x)) i= l= A(x)b i (x, x S 0 (x)) b l (x, x S 0 (x))e ( x x 0 + x x 0 )/ε dx 0 dx 0dx. Using the orthogonality of vectors b k with respect to the matrix A we derive, I E = (a (πε) n i (x 0 ) a i (x))e x x0 /ε dx 0 dx i= a (πε) n i (x 0 ) a i (x) e x x0 /ε dx 0 e x x0 /ε dx 0 dx i= = a (πε) n/ i (x 0 ) a i (x) e x x0 /ε dx 0 dx. i=

12 HAILIANG LIU AND MAKSYM PRYPOROV Changing variable ξ = x 0 x and by the mean value theorem, we obtain: ε I E = a π n/ i (x + εξ) a i (x) e ξ dξdx = ε π n/ = nε i= x a i (x + θ εξ) ξ e ξ dξdx i= x a i (x) dx =: Cε, i= where we have used the Fubini theorem. Here a careful calculation shows that C depends on (B 0, x B 0 ) E and the bound of A, xa, x b, k b and x S 0. We continue to estimate I E. I E = = A(x) B (πε) n 0 (x)(e it x 0 [S0]/ε e is0/ε )e x x 0 /ε dx 0 R n B 0 (x)(e T x 0 [S 0]/ε e is 0/ε )e x x 0 /ε dx 0 dx a (πε) n i (x) [S0]/ε e is0/ε )e x x 0 /ε dx 0 dx. i= (e it x 0 Simplifying e it x 0 [S0]/ε e is0/ε and using the Hölder inequality, we obtain: I E a (πε) n i (x) R x 0 [S 0 ] e x x 0 /ε dx 0 e R ε n i= R x x 0 /ε dx 0 dx n C a (πε) n/ i (x) x x 0 6 e x x0 /ε dx ε 0 dx, i= where we have used the Talyor remainder so that R x 0 [S 0 ] C x x 0 3 with C depending on S 0 C 3. Making the same change of variables as in the previous step, we have: I E Cε a i (x) ξ 6 e ξ dξdx Hence, which yields Lemma 4.. C B 0 Eε. i= u 0 u E Cε, Lemma 4.. Let u ε 0 be defined in (4.) and u in (4.4). Then (4.6) u ε 0 u E Cε /, where C depends on the matrix A and k b but is independent of ε. Proof. From (4.), u ε 0 = (πε) n/ (a i (x 0 )b i (x, x Φ 0 (x; x 0 )) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0. i=

13 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 3 Then u u ε 0 E = ( A(x) a (πε) n i (x 0 )b i (x, x S 0 (x))e iφ0 (x;x 0 )/ε dx 0 i= ) (a i (x 0 )b i (x, x Φ 0 (x; x 0 )) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 ( i= a i (x 0 )b i (x, x S 0 (x))e iφ0 (x;x 0 )/ε dx 0 i= i= (a i (x 0 )b i (x, x Φ 0 (x; x 0 )) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 )dx. Set K i = b i (x, x S 0 (x)) b i (x, x Φ 0 (x; x 0 )). Using the fact that x S 0 (x) x Φ 0 (x; x 0 ) = x S 0 (x) x S 0 (x 0 ) xs 0 (x 0 )(x x 0 ) ii(x x 0 ) where C depends on S 0 C 3, we obtain x x 0 ( + C x x 0 ), K i C x x 0 ( + x x 0 ), where C = C max k b i (x, ) C b C. Using that each a i (x 0 ) = 0 on \ together with the boundedness of matrix A, we obtain: u u ε 0 E C (πε) n C ( (πε) n a i (x 0 )(K i εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 dx i= ) dx. a i (x 0 )(K i εv(0, i x; x 0 )) e x x 0 /ε dx 0 i= By the Hölder inequality, u u ε 0 E C (πε) n ( m e a i (x 0 )(K i εv(0, i x x x; x 0 )) ) 0 /ε dx 0 i= e x x0 /ε dx 0 dx C ( m e a (πε) n/ i (x 0 )(K i εv(0, i x x x; x 0 )) ) 0 /ε dx 0 dx. i=

14 4 HAILIANG LIU AND MAKSYM PRYPOROV Going further, u u ε 0 E Cε n/ ( + ε = I + I. a i (x 0 ) K i e x x 0 /ε dx 0 dx i= v(0, i x; x 0 ) ) e x x 0 /ε dx 0 dx i= Applying the change of variable for fixed x 0 ξ = x x 0, ε dx = (ε) n/ dξ we have K i C( ε ξ + ε ξ ), hence I C a i (x 0 ) (ε ξ + ε ξ 4 )e ξ dξdx 0 Cε B 0 E. As for I, using (4.) we have i= I ε max x,x 0 Cε, m v(0, i x; x 0 ) i= which produces an additional rate of convergence. Therefore, we recover the needed order of convergence for u u ε 0 E. Combining both lemmas and using the triangle inequality we finish the proof of Theorem Evolution error estimate. From the residual representation (3.38) we have P (u ε ) ( I 0i + I i ), i= where I li := εl A(x)c (πε) n/ li e iφi/ε dx 0 is vector-valued. Since the estimate for each wave field is similar, we thus omit the index i using only I l (t, x; x 0 ) in the sequel. We follow the idea in [0] to complete the estimate of P (u ε ). Let denote quantities defined on the ray radiating from x 0 such as x, c l and Φ. Then we can represent the L norm of I l by I l = I l (t, x; x 0 ) I l (t, x; x 0)dx R n = J l (t, x, x 0, x 0)dx 0 dx 0dx,

15 where GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 5 (4.7) J l = ε n+l (π) A(x)c l(t, x; x n 0 ) A(x)c l (t, x, x 0)e iψ(t,x;x0,x 0 )/ε with (4.8) ψ(t, x, x 0, x 0) = Φ(t, x; x 0 ) Φ(t, x; x 0). The rest of this section is to establish the following (4.9) J l dx 0 dx 0dx Cε. With this estimate we have I l Cε, leading to the desired estimate P (u ε ) Cε /, which when combined with the initial error obtained in Theorem 4. and the wellposedness inequality (3.39) gives the main result (.) stated in Theorem.. In order to estimate (4.9), we note that hence Iψ = IΦ + IΦ δ ( x x + x x ), (4.0) J l Cε n+l c l (t, x; x 0 ) c l (t, x, x 0) e δ ε ( x x + x x ), with C = (π) n A, and l = 0,. Let ρ j (x, x 0, x 0) C be a partition of unity such that {, x x η x x η, (4.) ρ = 0, x x η x x η, and ρ + ρ =. Moreover, let J l = ρ J l (t, x, x 0, x 0), J l = ρ J l (t, x, x 0, x 0), so that J l (t, x, x 0, x 0) = J l + J l. We first estimate c 0 : using (3.) with (3.3) and (3.9), we have (4.) G(t, x; x 0 ) = λ(x, k) λ( x, p) x λ( x, p) (x x) k λ( x, p)m(x x) + (x x) Ṁ(x x), with Ṁ = xλ( x, p). Also from (3.4) and k = p + M(x x), we thus obtain (4.3) (4.4) c 0 = agb C( + x x ), x x η, c 0 = agb C x x 3, x x η, provided η is sufficiently small. As for c, if x x η, we use (3.3) of the form c = a(f(x, k(x)) f( x, p))b + igv. Note that both (3.33) and Lemma 3.4 imply that f + v C( + x x ), hence (4.5) (4.6) c C( + x x )( + x x ), x x η, c C x x, x x η,

16 4.3. Estimate of J l. Denote s = x x, s = x x, 6 HAILIANG LIU AND MAKSYM PRYPOROV where we have used (3.9) and Lemma 3.4 to infer (4.6). then from (4.7) using (4.3) and (4.5) it follows that J l Cρ ε n+l ( + s)( + s )( + s )( + (s ) )e δ ε (s +(s ) ). Using the estimate ( q ) q/c (4.7) s q e cs p/ e cs /, e with c = δ, we have ε ( + s)( + s )e δ ε s C( + ε / + ε + ε 3/ )e δ 4ε s 4Ce δ 4ε s. Hence Jl Cε n+l e δ 4ε (s +(s ) ) Cε n+l e δ 4ε s e η δ ε, where we have assumed s > η due to the definition of ρ, we thus obtain an exponential decay J0 dx 0 dx 0dx n Cεl K0 e η δ ε Cε r r Estimate of J. For x x η, both (4.4) and (4.6) imply that c l C x x 3 l, then from (4.0) it follows that Jl dx Cε n+l ρ c l (t, x; x 0 ) c l (t, x, x 0) e δ ε ( x x + x x ) dx R n Cε n+l x x 3 l x x 3 l e δ ε ( x x + x x ) dx R n Cε n+ δ 4ε ( x x + x x ) dx. e Using the identity (4.8) x x + x x = x we obtain Hence, (4.9) Jl dx Cε n+ e x + x δ x+ x x ε Jl dx 0 dx 0dx n Cε + + x x, dxe δ 8ε x x. e δ 8ε x x dx 0 dx 0. In order to obtain (4.9), we need to recover an extra ε n from the integral on the right hand side, which is difficult when x x is small. Following [0], we split the set into { } D (t, θ) = (x 0, x 0) : x x θ x 0 x 0,

17 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 7 which corresponds to the non-caustic region of the solution, and the set associated with the caustic region { } D (t, θ) = (x 0, x 0) : x x < θ x 0 x 0. For the former we have e δ D 8ε x x dx 0 dx 0 Changing to spherical coordinates, we obtain δθ 8ε x 0 x 0 dx 0 dx 0 C as needed. To estimate J l D e D e 0 Cε n δθ 8ε x 0 x 0 dx 0 dx 0. s n e δθ 8ε s ds 0 e δθ 8ε s ds Cε n restricted on D, we need the following result on phase estimate. Lemma 4.3. (Phase estimate) For (x 0, x 0) D, it holds (4.0) x ψ(t, x, x 0, x 0) C(θ, η) x 0 x 0, where C(θ, η) is independent of x and positive if θ and η are sufficiently small. The proof of this result is due to [0], where the non-squeezing lemma is crucial. Since all requirements for the non-squeezing argument are satisfied by the construction of Gaussian beam solutions in present work, we therefore omit details of the proof. To continue, we note that the phase estimate ensures that for (x 0, x 0) D, x 0 x 0, x ψ(t, x, x 0, x 0) 0. Therefore, in order to estimate Jl D we shall use the following nonstationary phase lemma. Lemma 4.4. (Non-stationary phase lemma) Suppose that u(x, ξ) C0 (Ω Z) where Ω and Z are compact sets and ψ(x; ξ) C (O) for some open neighborhood O of Ω Z. If x ψ never vanishes in O, then for any K = 0,,..., Ω u(x; ξ)e iψ(x;ξ)/ε dx C K ε K where C K is a constant independent of ξ. K β = Ω β x u(x; ξ) x ψ(x; ξ) K β e Iψ(x;ξ)/ε dx, Using the non-stationary lemma, (4.7), (4.0) and the lower bound for ψ in (4.0), we obtain for (x 0, x 0) D, K Jl dx L CεK n+l R R l β δ e ε ( x x + x x ) dx n n x ψ K β C K β = where we have used the notation β = ε K n+l inf x x ψ K β δ L l β e ε ( x x + x x ) dx, L l β := β x [ρ A(x)c l (t, x, x 0 ) A(x)c l (t, x, x 0)].

18 8 HAILIANG LIU AND MAKSYM PRYPOROV We claim the following estimate for L l β, (4.) L l β C Therefore, Jl dx C Using (4.8) we have Hence, K β = β + β = β ε K n+l inf x x ψ K β e δ ε ( x x + x x ) dx K ε K n β /+ C inf x x ψ K β β = Jl dx C C K β = K β = x x (3 l β ) + x x (3 l β ) +. β + β = β e δ x x (3 l β ) + x x (3 l β ) + 4ε ( x x + x x ) dx. ε K n β /+ ( e δ 4ε inf x x ψ K β ε K n/ β /+ inf x x ψ δ e 8ε x x. K β ) x x+ x + x x dx Jl dx 0 dx 0dx n Cε D e δ 8ε x x D K β = inf x ψ/ ε K β dx 0dx 0. The last estimate together with (4.9) yields: Jl D Cε n Cε n Cε n Cε n e δ D 8ε x x min e δ 8ε x x D K β = e δ 8ε x x K β = [, K β = ] inf x ψ/ dx ε K β 0 dx 0 [ ] min, inf x ψ/ dx ε K β 0 dx 0 K β = + inf x ψ/ ε K β dx 0dx 0 + (C(θ, η) x 0 x 0 / ε) K β dx 0dx 0,

19 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS 9 where we have used the inequality min{, b } +b for any b > 0. Taking K = n + and changing variable ξ = x 0 x 0, we compute ε Jl D Cε n + ( x 0 x 0 / ε) n+ dx 0dx 0 Cε dξ = Cε. 0 + ξn+ which gives (4.9) when restricted to the caustic region. This completes the proof of (4.9), except the claim (4.), which we show below. We assume smoothness and boundedness of any component contributing to β x [ρ A(x)c l (t, x, x 0 )A(x)c l (t, x, x 0)]. Note that the typical term in L 0 β has form β x [ρ A(x)b A(x)b gg (x x) α (x x ) α ], where g is a third order partial derivative of λ and α is a multiindex, α = 3. For the sake of brevity, we denote h := ρ A(x)b A(x)b gg. Hence L 0 β C β x [h(x x) α (x x ) α ] = C = C β + β = β β x h β + β = β β + β = β β x h β x [(x x) α (x x ) α ] (x x) (α β ) + (x x ) (α β ) +. In the worst case, i.e., when β = 0 we obtain the lowest power of (x x)(x x ) and since x is near the ray, then the higher order terms are controlled by lower order terms, and (4.) is satisfied for l = 0. As for l = case, we use (3.3) to only take care of the lower order term, L β C x β [ρ A(x)aD x f(, ) (x x)b A(x)aD x f(, ) (x x )b], so that (4.) follows for l = too. 5. Extensions to more general initial phase Our GB construction and the error estimates have been carried out for the case that x S 0 (x) 0, x. In this section, we show that this restriction can be relaxed so that a stronger result as stated in Corollary. can be proved. Set σ = {x, x S 0 (x) = 0}. Since σ has measure zero, i.e., µ(σ) = 0, then set σ can be covered by a union of open sets Σ such that µ(σ) ɛ n for any ɛ > 0. For each x 0 \Σ, we can construct a single Gaussian beam as illustrated in section. The superposition of these beams, (5.) u ε (t, x) = (a (πε) n i (x 0 )b k (x 0, x Φ i ) + εv(t, i x; x 0 ))e iφ i(t,x;x 0 )/ε dx 0 \Σ i=

20 0 HAILIANG LIU AND MAKSYM PRYPOROV can thus be used as our approximate solution. The initial Gaussian beam approximation then takes the form (5.) u ε 0 = (a (πε) n/ i (x 0 )b i (x, x Φ 0 ) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0, \Σ i= which approximates the given initial data u 0 (x) = a k (x)b i (x, x S 0 (x))e is0(x)/ε. i= We are now ready to prove Corollary.. The wellposedness estimate u u ε E u 0 u ε 0 E + T 0 P [u ε ] dt again tells that we need to bound both initial and evolution error. Since the exclusion of set Σ from set will not affect the estimate of P (u ε ), hence we have P (u ε ) Cε /. To bound the initial error, we can use the same technique as in the proof of Theorem 4.. That is, we use the triangle inequality (5.3) u 0 u ε 0 E u 0 u E + u u ε 0 E, where u is introduced in (4.4), (5.4) u := a (πε) n/ i (x 0 )b i (x, x S 0 (x))e iφ0 (x;x 0 )/ε dx 0. i= It was shown in Lemma 4. that u 0 u E Cε. We next estimate u u ε 0 E. Using the fact that a i (x 0 ) = 0 on \, and for constant C depending on A L we have u u ε 0 E C a (πε) n i (x 0 )b i (x, x S 0 (x))e iφ0 (x;x 0 )/ε dx 0 i= (a i (x 0 )b i (x, x Φ 0 ) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 dx \Σ i= C (πε) n \Σ i= (a i (x 0 )(b i (x, x S 0 (x)) b i (x, x Φ 0 )) εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 dx + C (a (πε) n i (x 0 )b i (x, x S 0 (x)) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 dx = I + I. Σ i=

21 GAUSSIAN BEAM METHODS FOR STRICTLY HYPERBOLIC SYSTEMS For I we can repeat the proof of Lemma 4. to obtain the same result. For I, we proceed to obtain I Cε n dx 0 (a i (x 0 )b i (x, x S 0 (x)) + εv(0, i x; x 0 ))e iφ0 (x;x 0 )/ε dx 0 C Σ Cε n, Σ Σ i= (a i (x 0 )b i (x, x S 0 (x)) + εv(0, i x; x 0 ))e x x 0 /ε dxdx 0 i= where, as before, we have used Hölder inequality and the Fubini theorem. All these estimates when inserted into (5.3) yield the desired initial error u 0 u ε 0 E Cε. References [] S. Bougacha and J. L. Akian and R. Alexandre. Gaussian beams summation for the wave equation in a convex domain. Commun. Math. Sci., 7(4): , 009. [] A. Bensoussan, J-L. Lions, G. Papanicolaou. Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam, 978 [3] M. Dimassi, J.-C. Guillot and J. Ralston. Gaussian beam construction for adiabatic perturbations. Mathematical Physics, Analysis and Geometry, 9:87 0, 006. [4] J.-C. Guillot J. Ralston, E. Trubovitz. Semi-classical methods in solid state physics. Commun. Math. Phys. 6: 40 45, 988. [5] F. John. Partial Differential Equations. Applied Mathematical Sciences, Springer-Verlag 99. [6] H. Liu, S. Osher and R. Tsai. Multi-valued solution and level set methods in computational high frequency wave propagation. Commu. Comp. Phys. (5): , 006. [7] H. Liu and M. Pryporov. Error Estimates of the Bloch Band-Based Gaussian Beam Superposition for the Schrödinger Equation. Multiscale Model. Sim., submitted 03. [8] H. Liu and J. Ralston. Recovery of high frequency wave fields for the acoustic wave equation. Multiscale Model. Sim. 8(): , 009. [9] H. Liu and J. Ralston. Recovery of high frequency wave fields from phase space based measurements. Multiscale Model. Sim., 8(): 6 644, 00. [0] H. Liu, O. Runborg, and N. M. Tanushev. Error estimates for Gaussian beam superpositions. Math. Comp., 8:99 95, 03. [] H. Liu, J. Ralston, O. Runborg, and N. M. Tanushev. Gaussian beam methods for the Helmholtz equation. SIAM Appl. Math., submitted 03. [] J. Lu and X. Yang. Frozen Gaussian approximation for general linear strictly hyperbolic systems: formulation and eulerian methods. Multiscale Modeling & Simulation, 0():45 47, 0. [3] J. Lu and X. Yang. Convergence of frozen Gaussian approximation for high frequency wave propagation. Comm. Pure Appl. Math., 65: , 0. [4] J. Ralston. Gaussian beams and the propagation of singularities. In Studies in partial differential equations, volume 3 of MAA Stud. Math., pages Math. Assoc. America, Washington, DC, 98. [5] J. Ralston. Gaussian beams. available online from ralston/pub/gaussnotes.pdf, 005. [6] D. Robert. On the Herman-Kluk semiclassical approximation. Rev. Math. Phys., (0):3, 00. [7] V. Rousse and T. Swart. A mathematical justification for the Herman Kluk propagator. Comm. Math. Phys., 86():75 750, 009. [8] N. M.Tanushev. Superpositions and higher order Gaussian beams. Comm. Math. Sci., 6(): , 008. Department of Mathematics, Iowa State University, Ames, Iowa address: hliu@iastate.edu; pryporov@iastate.edu