In the setting of a distributional product, we consider a Riemann problem for the Hunter-Saxton equation

In the present paper, we investigate the Hunter-Saxton equation

This equation models the propagation of the director field

Liquid crystals are a state of matter intermediate between the crystalline solid state and the liquid state (the water is not an example, since it goes directly from solid state to liquid state). One of the most common phases in which a liquid crystal exists is the so-called nematic phase, in which the liquid crystal is invariant under the transformation

Usually, for a complete description of a liquid crystal, two independent vector fields are needed: one for the fluid flow and another for the orientation of the molecules [

Equation (

Useful and different concepts of weak global solutions were defined by Hunter and Saxton [

For the conservation law

Conditions for the propagation of travelling shock-waves with profiles

In the setting of soliton wave collision, we were able to prove that delta waves under collision behave just as classical soliton collisions (as in the Korteweg-de Vries equation) in models ruled by a singular perturbation of Burgers conservative equation [

Phenomena of gas dynamics known as “infinitely narrow soliton solutions,” discovered by Maslov and his collaborators [

In a Riemann problem for the

Also in the Brio system

In the present paper and within a convenient space of discontinuous functions, we prove that all solutions of the Riemann problem (

Let us summarize the contents of this paper. In Section

Let

Each

Recall that, in the setting of the so-called classical products of distributions, the commutative property is a convention inherent to the definition of such products and the associative property does not hold in general: for instance,

The general

All

In general, the support of the

It is still possible to define many other

Let

Let

for each

The natural injection

Given

We have the following results.

If

Notice that, by a classical solution of (

If

For the proof it is enough to observe that any

Given

As a consequence, an

Let us consider the problem (

Given

if

if

Suppose

From (

Suppose

As a consequence we can conclude that the

Taking

We introduce the following definition for the sake of simplicity.

Let

Given

the speed

Let us suppose that

Supposing

An interesting class of

Let

Then, if

By applying Theorem

Taking

Taking

Taking

In the framework of our distributional products, if

It is interesting to verify (see [

The speeds that we have computed in these examples are independent of

Taking

The author declares that there is no conflict of interests regarding the publication of this paper.

The present research was supported by FCT, PEst-OE/MAT/UI0209/2013.