In this chapter we study the Cauchy problem for the nonlinear wave equation
The equation in (0.0.1) is the Euler–Lagrange equation of the least action principle
The equation in (0.0.1) may be regarded as a generalization of equations for harmonic wave maps and arises in a number of different physical contexts, including nematic liquid crystals [54], long waves on a dipole chain in the continuum limit [30,31,77] and in classical field theories and general relativity [30]. Let us take, for example, nematic liquid crystals. We know that the mean orientation of the long molecules in a nematic liquid crystal is described by a director field of unit vectors,
, the unit sphere. Associated with the director field n, there is the well-known Oseen–Franck potential energy density W given by
The positive constants α, β and γ are elastic constants of the liquid crystal. For the special case α = β = γ, the potential energy density reduces to
which is the potential energy density used in harmonic maps into the sphere
. There are many studies on the constrained elliptic system of equations for n derived through variational principles from the potential (1.1.2), and on the parabolic flow associated with it, see [5,15,22,33,41,64] and references therein. In the regime in which inertia effects dominate viscosity, however, the propagation of the orientation waves in the director field may then be modeled by the least action principle [54],
In the special case α = β = γ,this variational principle (1.1.3) yields the equation for harmonic wave maps from (1 + 3)-dimensional Minkowski space into the two sphere, see, for example, [9,57,58]. For planar deformations depending on a single space variable x, the director field has the special form
where the dependent variable
measures the angle of the director field to the x-direction, and ex and ey are the coordinate vectors in the x and y directions, respectively. In this case, the variational principle (1.1.3) reduces to (1.1.1) with the wave speed c given specifically by
The general problem of global existence and uniqueness of solutions to the Cauchy problem of the nonlinear variational wave equation (0.0.1) is open. It has been demonstrated in [29] that (0.0.1) is rich in structural phenomena associated with weak solutions. Writing the highest derivatives of (0.0.1) in conservative form
we see that the strong precompactness in L2 of the derivatives
of a sequence of approximate solutions is essential in establishing the existence of a global weak solution. However, the equation has the phenomenon of persistence of oscillation [20] and annihilation in which a sequence of exact solutions with bounded energy can oscillate forever so that the sequence
is not precompact in L2, but the weak limit of the sequence is still a weak solution. Secondly, the equation has short-time smooth solutions that blow up in finite time. Thirdly, from the study of its asymptotic equation (see further), it is clear that a positive amount of energy of a solution can concentrate in a set of measure zero, and there are multiple choices for the continuation of the solution beyond blow-up time. To put the equation and its particulars into context of nonlinear wave equations under current research, we compare this equation with the equation
where p(·) is a given function, considered by Lax [44]. The derivative
remains bounded for (1.1.5), but we find that it is merely in L2 for (0.0.1). We note interestingly that solutions of (1.1.5) – with a “stronger”
-dependent nonlinearity – are more regular than solutions of (0.0.1) – with an apparently “weaker” u-dependent nonlinearity. (This kind of behavior is well known for nonlinear parabolic partial differential equations.) In each case it appears that singularities develop to the maximum extent permitted by the existence of global weak solutions. For further research into (1.1.5) and its generalizations, we refer the reader to [43] and [50]. Another related equation is
considered by Lindblad [47], who established the global existence of smooth solutions of (1.1.6) with smooth, small, and spherically symmetric initial data in
, where the large-time decay of solutions in high space dimensions is crucial. The multidimensional generalization of (0.0.1),
contains a lower-order term proportional to
, which (1.1.6) lacks. This lower-order term is responsible for the blow-up in the derivatives of u. Finally, we note that (0.0.1) also looks related to the perturbed wave equation
where
satisfies an appropriate convexity condition (for example,
) or some nullity condition. Blow-up for (1.1.8) with a convexity condition has been studied extensively, see [2,28,32,38,40,46,55,60,61] and [62] for more reference. Global existence and uniqueness of solutions to (1.1.8) with a nullity condition depend on the nullity structure and large time decay of solutions of the linear wave equation in higher dimensions (see [42] and references therein). Therefore (0.0.1) with the dependence of c(u) on u and the possibility of sign changes in c′(u) is familiar yet truly different.
Despite its simplicity, (0.0.1) is not easy. A study of its geometric optical solutions is helpful and interesting. Look for solutions of the form
where u0 is a constant state and c0 = c(u0) > 0 is its speed, Hunter and Saxton [35] found that u(·, ·) satisfies
up to a scaling factor, assuming that
If u0 is such that
then the equation is
In general, if u0 is such that
the equation is
Another form of the first asymptotic equation is
We shall not study the second and higher asymptotic equations in this short talk.
We shall prove the compactness of the approximate solution sequence by applying Young measure theory [63] and the ideas used by Lions [49] in the proof of the global existence of weak solutions to multidimensional compressible Navier–Stokes equation, and by Joly, Métivier and Rauch [39] in the rigorous justification of weakly nonlinear geometric optics for a semilinear wave equation (see also [69]). For the convenience of the reader, we quote the following lemma from [39] (see also [20,23]) that we use in this chapter.
Lemma 1.3.1.
Let U be an open subset of
, whose boundary has zero Lebesgue measure. Given a bounded family
-valued functions, then there exist a subsequence
and a measurable family of probability measures on
such that for all continuous functions F(y, λ) with
and q < s, there holds
for all
with compact support in the closure of U, where
Moreover,
Physically by approximating directly the Hamiltonian for Euler equations in the shallow water regime, Camassa and Holm [7] derived the following equation
Mathematically, (1.4.13) is obtained and proved to be formally integrable by Fuchssteiner and Fokas [25] as a bi-Hamiltonian generalization of Korteweg–de Vries (KDV) equation. Equation (1.4.13) has several important features that distinguish it from the well-known KDV equation. First, Camassa and Holm discovered that (1.4.13) possesses peaked solutions with a corner at their crest, which is in sharp contrast to the solitary waves for KDV. Second, physical water waves often break down, which can not be predicted by the solutions to the KDV equation.
Formally (1.4.13) is equivalent to
In [11,52], the authors proved the finite time break down of smooth solution to (1.4.14). In particular, McKean gives a necessary and sufficient criterion on the initial data for the finite-time formulation of singularities in a smooth solution to (1.4.14). Furthermore, McKean describes the blow-up process by showing the formation of cusps instead of shocks for compressible fluids. Motivated by the Young measure approach in Section 2 of this chapter (see [72]), Xin and the first author of this chapter (see [67]) proved the global existence of weak solution to (1.4.14) with initial data in
Set
by taking
to the first equation of (1.4.14), we get
Compared this equation with (1.2.10), we find that (1.2.10) can be considered as the first-order approximation to (1.4.14).
Up to now, we already proved the global existence and uniqueness of both dissipative and energy conservative solution to (1.2.10) through Young measure approach and explicit construction of the approximate solutions (see Section 2 of this chapter or [72]). Notice the relations between (1.2.10) and (1.4.14), we can step-by-step follow the method of [67] to prove the global existence of dissipative solution by vanishing viscosity.
Open Problem 1.
The global existence of energy conservative weak solution to (1.2.10) via the vanishing dispersion limit to (1.2.10).
As for the original variational wave equation (0.0.1), up to my recent result [75,76], we only proved the global existence of weak solution to (0.0.1) with monotone wave speed.
Open Problem 2.
The uniqueness of weak solution is open.
And finally we have the third problem.
Open Problem 3.
The global existence and uniqueness of weak solution to (0.0.1) is completely open.
Finally, let us outline the main contents of this chapter. In Section 2 we present some global existence and uniqueness results to the asymptotic equation. In Section 3 we prove some results on invariant region with monotone wave speed and in Section 4 we present one general global existence result to this equation. For some recent progress on this subject see [76].