The transport of the microlocal energy density along the broken bicharacteristic current at the high frequency limit is proved by the use of Wigner measures. Our approach first consists of explicitly calculating the Wigner measurements under an additional check of the initial data, which allows the solution to be approximated by a superposition of first-order Gaussian beams. We are interested in the high frequency limit of the initial limit value problem (IBVP) for the wave equation.
Wigner measures related to the solution of the wave equation (and hyperbolic problems in general, see e.g. are related to the energy density in the high-frequency limit. More precisely, under suitable hypotheses (see proposition 1.7 in [20 ]), the energy density associated with the solutionCε of the Cauchy problem for the scalar wave equation converges as ε→0 in the sense of measures for. In [4], a weighted integral of Gaussian beams was designed to construct an approximate solution of IBVP (1.1) according to an additional assumption (H5) on the initial data (see p.9).
We simplify the expression of Wigner transforms of such integrals in Section 3, following initial calculations from [45] in the Schr¨odinger case. Then the derivatives of the first-order solution will be approximated by some Gaussian-type integrals. Using (2.3) and the Hamiltonian system corresponding to it (xt0, ξ0t), equation (2.8) at zeroth order gives the following transport equation for the amplitude value on the beam [23].
An alternative method was used in [45], where an integral of the form (3.2) associated with the Wigner transformation for the Schr¨odinger equation with a WKB initial condition was simplified by elementary calculations in an integral overR4n .
Wigner transform for Gaussian integrals
For example, this method was successfully used in [9] to calculate the Wigner measure for smooth data. There, the phase was complex, and its Hessian matrix, restricted to a stationary set, was assumed not to be degenerate in the direction normal to that set. However, in our case the amplitude is not smooth because no such assumption was made onuIε and vIε, and we cannot immediately evaluate the global integral (3.3) using the same techniques.
One possibility to solve this problem would be to resort to the stationary phase theorem with a complex phase depending on parameters for estimation. Although the method encountered difficulties in deriving the exact relation between the Wigner measure of the solution and the initial data, we apply the result of [45] to our problem in Section 3.1 and complete the analysis to obtain the propagation along the flow of the microlocal energy density from uapprε,r0,r∞ asε→0 in Section 3.2. The proof is simple and elementary and the calculations are made in an explicit way.
A Gaussian approximation is then applied to several smooth functions appearing in the integral of the Wigner transform. 2 Proof: [Proof of Lemma 3.2] The matrix M +N has a positive definite real part and is therefore nonsingular. Using the value of the Fourier transform of the Gaussian function (see Theorem 7.6.1 of [21]), it follows that.
Wigner measures for superposed Gaussian beams
Proof: The proof is based on applying Taylor's formula to ρ′fα and ρ′gβ, where ¯ ρ′f and ρ′g are the limits used in the proof of Lemma 3.1 (supported in F and equal to . 1 on suppfε and suppgε ). This leaves us with the computation of the Wigner measure associated with (vt+), where the computations are similar for (v−t). 3.12). In the continuation of this section, we prove the following statement, calculate µtε,k and the limit at ε→0 of the microlocal energy density uapprε,r0,r∞.
If we combine this relation with the symplecticity Fkt, we get the following relation for the matrix Q. Connecting the form of incident and reflected amplitudes in Lemma 2.1 and using C1 smoothness a(k′)onByyields with Lemmas 3.1 and 3.3. What remains is the analysis of the most difficult terms in the amplitude, which include the transported FBI transforms.
On the other hand, since εr±x are the remainder terms in the Taylor expansions of x−−tk(s±√εr, σ±√εδ) of order 2, rx+ε −rxε− is of order√εand that holds also for Rx+ ε ′−Rxε−′, leading to. The inequality (3.15) regarding the distribution Jε,kt(κε, τε) is finally arrived at by plugging the expressions of d0 and γ0. We can now calculate the measure µtε,k from (3.14) using the previous theorem and lemma 4.7.
We define ϕub in (Ω\Ky)×(Rn\{0}) by successive reflection of rays at the boundary. Because only one incident/reflected ray can be inside the domain at a given time∈[−T, T ].
Proof of the main Theorem
By applying this estimate to the difference between the derivatives of the exact and the estimated solutions of the IBVP (1.1a)-(1.1b) with initial conditions (1.1c'), one derives the associated measures. The summation of higher order beams may imply asymptotic formulas for the Wigner transformations and thus for the energy density. For example, higher-order terms in the extension of the Wigner transform have been studied in [14] and [42] for initial WKB data.
The IBVP solution to the wave equation is given by a continuous unitary evolution group in the space H1(Ω, dx)×L2(Ω, dx). For the analysis of uIε−uIε,r0,r∞ in H1(Ω), we begin by evaluating the spatial derivatives of the difference. Since the FBI transformation of a derivative is the derivative of the FBI transformation with (2.13), one must evaluate∂xjuIε−ρTε∗γr0,r∞Tε∂xjuIεkL2(Ω).
Reflected first order and higher order beams
The distinction here is formal, since V can have a complex value from u0, which makes f(u, V(u)) and g(u, V(u)) undefined for u 6= u0. However, at an exact point u0 we can always use the derivative formula of composite functions to obtain a formal expression for the derivatives. We will be superficial in noting the dependence of phase V on its variables.
As a specific case, the phaseψ0 is obtained by setting R = 2 and choosing its initial value on the ray as zero and its initial Hessian matrix on the ray asiId. Let ψref∈ C∞(Rt×Rn . x,C) be the reflected phase associated with ψinc, that is the phase that satisfies. Proof: The strategy of the proof is the following: we consider a phase function θ that satisfies the relationships announced in Lemma 4.1 and we prove that θ satisfies the eikonal equation on the reflected ray up to order R and the correct derivatives have on the moment and point of reflection.
This proves that θ coincides with the reflected phase on the reflected ray up to the orderR. To compare the time and tangential derivatives of θ and ψinc at (T1, xT01), let us introduce aC∞ parametrization of a neighborhoodU ofxT01 in ∂Ω. The reflected amplitudes evaluated on the associated rays satisfy transport equations similar to (2.20) and can be written as
To find the relationship between UkTk and U0Tk fork=±1, we differentiate the equality xTkk=xT0k. Dy,ηxTkk+ ˙xTkk(∂y,ηTk)T =Dy,ηxT0k+ ˙xT0k(∂y,ηTk)T and calculate the derivatives Tk from the conditionaxT0k∈∂Ω.
Approximation operators
Results related to the FBI and the Wigner transforms
Burq, Contrôlabilité exacte des ondes dans les domaines non lisses, (français) [Contrôlabilité exacte des ondes dans les domaines non lisses], Asympt. Burq, Mesures semi-classiques et mesures par défaut, (Français) [Mesures semi-classiques et mesures par défaut], Séminaire Bourbaki, Astérisque. Lebeau, Mesures de défaut de compact´e, application au système de Lame, (français) [Mesures de défauts microlocaux et application au système Lame], Ann.
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Qian, Lagging phase flow and Eulerian Gaussian beams based on the FBI transform for the Schr¨odinger equation, J. Miller, Refraction of high-frequency density waves by sharp interfaces and semiclassical masses at the boundary, J.
