Microlocal Analysis and Nonlinear Waves

Let Ω ⊆ R ³ be an open subset and P be a second order strictly hyperbolic differential operator in Ω. Let S _j , 1 ≤ j ≤ n, be smooth characteristic hypersurfaces for P simply tangent along a line L.

We consider local solutions to second order partial differential equations of the form Pu = f(x, u), for which u is smooth on the complement of a characteristic surface with a cusp singularity. If P is strictly hyperbolic and u is assumed to be regular in the past with respect to differentiation by a natural family of smooth vector fields, then u is regular in the future, and “conormal” with respect to a larger family of vector fields which are nonsmooth at the singularity of the cusp. If P is a Tricomi operator associated with the cusp, and the natural initial data (Dirichlet or Cauchy) are conormal with respect to a hyperplane, then u is again shown to be conormal with respect to the cusp.

We study the regularity of the solution of a compressible Euler system with a Cauchy data which is conormal with respect to the origin. We prove that the light cone issued from the origin is the union of a smooth hypersurface and a smooth curve.

I will speak today about a classical fluid dynamical problem involving a free surface, the problem known as water waves. The results I want to describe are simply the derivation of a formalism that arises in posing the problem. This consists in writing the equations of motion as a Hamiltonian system. In doing this, several interesting questions arise, which are associated with Laplace’s equation on plane domains. I am not reporting on analytical results, rather this talk should be taken as a description of a point of view, or a set of coordinates that have a certain elegance.

We say that an evolution equation has an infinite gain of regularity if its solutions are C ^∞ for t > 0, for initial data with only a finite amount of smoothness. An equation need not be hypoelliptic for this to happen provided the initial data vanish at spatial infinity. For instance, for the Schrödinger equation in R ⁿ , this is clear from the explicit solution formula if the initial data decay faster than any polynomial. For the Korteweg-deVries equation, T. Kato [4], motivated by work of A. Cohen, showed that the solutions are C ^∞ for any data in L ² with a weight function 1 + e ^σx . While the proof of Kato appears to depend on special a priori estimates, some of its mystery has been resolved by the recent results of finite regularity for various other nonlinear dispersive equations due to Constantin and Saut [1], Ponce [5] and others [3]. However, all of them require growth conditions on the nonlinear term.

This paper is one of a series devoted to the Cauchy problem for an equation of the form with small Cauchy data Here □ = _t ² − Δ is the wave operator in R ¹⁺ⁿ, with variables denoted by t = x ₀ and x = (x1,..., x _n ), G is a C ^∞ function vanishing of second order at the origin, u′ and u″ denote all first and second order derivates of u, and u _j ∈ C ₀ ^∞ . General results have been obtained with simple proofs based on the idea of Klainerman [13] to use enrgy integral estimates for all equations obtained from (1.1) by multiplication with any product Ẑ ^I of │I│ vector fields ∂/∂x _j , j = 0, …, n, the infinitesimal generators of the Lorents group which commute with □, and the radial vector field (We shall use the notation Z ^I for products of the vector fields (1.3), (1.4) only; note that these preserve homogeneity.) However, in low dimension the results are not always optimal when G depends on u itself.

Weakly nonlinear wave interactions are resonant or nonresonant. The linearized dispersion relation of the wave motion determines the resonant interactions. Resonant interactions cause significant changes in the wave-field. The evolution of the wave-field is determined using weakly nonlinear asymptotics. Quadratically nonlinear resonant interactions of dispersive waves satisfy the three wave resonance condition. The wave amplitudes solve the three wave resonant interaction equations. The phase velocity of hyperbolic waves is independent of frequency. As a result, hyperbolic waves participate in many resonant interactions. The amplitude of a single hyperbolic wave satisfies the inviscid Burgers equation. Harmonic resonance causes wave-form distortion and shock formation. The amplitudes of several interacting hyperbolic waves solve a system of integro-differential equations. The interaction of three oblique hyperbolic planar waves can generate a countably infinite family of new waves. Weak resonance of nonplanar hyperbolic waves also generates infinitely many new waves.

In this note we construct an example of a solution to a first order semilinear hyperbolic system which has for t > 0 three incoming high frequency wave trains and for t > 0 has outgoing oscillations propagating in directions which are dense on the unit circle in R ². Section 1 is devoted to describing the mechanism which can produce such a phenomenon. The next two sections complete the construction of such solutions. The last section discusses some special considerations for the case of quadratic nonlinearities. We would like to acknowledge the help of our friends, Jean Fresnel with §2 and Jon Hunter with §4.

In a unified and simple way we get lower bounds of the life-span of classical solutions to the Cauchy problem with small initial data for fully nonlinear wave equations of the general form □u = F(u, Du, D _x Du) for the space dimension n ≥ 3.

In this paper we analyze the stronger singularities of solutions \( u \in H_{\text{loc}}^s(\Omega ),s < \frac{n}{2} \), for semilinear wave equations, using the microlocal multiplication of the solutions, the fact of the loss in smoothness of the products in H ^s , s < n/2, and the trick of reducing the loss in smoothness of the products.

In this note we consider the problem of associating to a given geometry, in the form of a <Emphasis FontCategory=“NonProportional”>C</Emphasis>^∞ variety containing possibly singular submanifolds, spaces of finitely regular conormal functions. For non-linear problems it is highly desirable that the bounded elements in these spaces form algebras and that they have appropriate solvability properties for certain linear differential operators. This leads to the general approach discussed here, mixing microlocalization and blow-up techniques.

This paper is concerned with the construction of a quasimode for the Laplace operator in a bounded domain Ω in R ⁿ , n ≥ 2, with a Dirichlet (Neumann) boundary condition. The quasimode is associated either with a closed gliding ray on the boundary or with a closed broken ray in T*Ω. The frequency set of the quasimode consists of the conic hull of the union of the bicharacteristics of the cosphere bundle S*Ω issuing from a family of invariant tori of the billiard ball map. To construct a quasimode near a gliding ray we find a global symplectic normal form for a pair of glancing hypersurfaces.

We will describe a pointwise decay estimate for solutions u(t, x) of the inhomogeneous Klein-Gordon equation on R ⁺ × R ³, with zero initial data . The desired bound takes the form , with the norm ||| • ||| to be made precise later on.

We consider solutions, u ∈ H _loc ^s (Ω̄), s > (dim Ω + 1)/2, satisfying on Ω = R _t × ω, where ω ⊂ R _z ⁿ is an open set with smooth boundary; P is a second-order differential operator with C ^∞ coefficients on R _{(t, z)} ⁿ⁺¹ , noncharacteristic with respect to bΩ and strictly hyperbolic with respect to the planes t = c; and f ∈ C ^∞. Interactions between singularity-bearing bicharacteristics taking place in the interior of Ω in t > t̄ can produce anomalous singularities in u, that is, singularities not present in the function u satisfying Pu = 0, u│ _{b Ω} = u│ _{b Ω} in t < t̄ These singularities can have strength at most ~ 3s - n (Beals [1]) and are generated by processes, crossing and self-spreading, that have been well-understood for some time (Beals [2], [3]). In this paper we shall describe propagation and interaction at the boundary, where generalized bicharacteristics [6], which typically contain segments of reflecting, grazing, or gliding rays, carry singularities.