Dear Mark,

Often hyperbolic conservation laws are limit models, where a process is neglected. The most common case is neglecting viscosity/diffusion, i.e. (LaTeX notation) passing \eps to 0 in the equation

\partial_t u +\partial_x f(u) = \eps \partial_{xx} u (Eq1)

This gives formally

\partial_t u +\partial_x f(u) = 0 (Eq2)

In this respect, admissible shock solution to (Eq2), connecting two states U1 and U2, can be viewed as limits (\eps -> 0) of travelling wave solutions (TW) to (Eq1), connecting the same states. If such a TW exists, then a shock is called "admissible".

This is the classical theory. It provides compressive shocks, where characteristics from the left and right sides of the shock enter into the shock (the speeds are higher/lower than the shock speed).

Undercompressive shocks can be understood similarly, with one remark: the operator on the right of (Eq1) needs not to be a diffusive one. One may consider operators involving third or fourth order spatial derivatives, or mixed derivatives, e.g.

\partial_t u +\partial_x f(u) = \eps \partial_{xx} u + \eps^2 \partial_{xxx} u) (Eq3)

or

\partial_t u +\partial_x f(u) = \eps \partial_{xx} u + \eps^2 \partial_{xxt} u) (Eq4)

These equations model various processes. For example, (Eq4) appears as a model for two-phase flow in porous media where dynamic effects are included in the capillary pressure. Then \eps is the capillary number.

When looking for TW solutions to (Eq3) or (Eq4), the results might differ than those for (Eq1). In other words, given two states U1 and U2, it may be that a TW connecting them exists for (Eq3) and not for (Eq1) and vice-versa. Clearly, the admissibility criterion for shock solutions is changing accordingly, and admissible shocks for (Eq2), understood as limits of TW for, say, (Eq4) may become "undercompressive", i.e. one of the characteristics from left or right does not enter the shock. The shock is hence non-Laxian, and also violates Oleinik's entropy condition.

Such results are provided thoroughly in the book of Ph. LeFloch, focusing mainly on shocks induced by equation (Eq3). Other operators are analyzed in papers by M. Shearer, A. Bertozzi, C. Rohde (I apologize a-priori for missing names of people who made important contributions). For (Eq4) I dare self-advertising :)

Hope this helps.

Good luck,

Sorin

Dear Mark,

Often hyperbolic conservation laws are limit models, where a process is neglected. The most common case is neglecting viscosity/diffusion, i.e. (LaTeX notation) passing \eps to 0 in the equation

\partial_t u +\partial_x f(u) = \eps \partial_{xx} u (Eq1)

This gives formally

\partial_t u +\partial_x f(u) = 0 (Eq2)

In this respect, admissible shock solution to (Eq2), connecting two states U1 and U2, can be viewed as limits (\eps -> 0) of travelling wave solutions (TW) to (Eq1), connecting the same states. If such a TW exists, then a shock is called "admissible".

This is the classical theory. It provides compressive shocks, where characteristics from the left and right sides of the shock enter into the shock (the speeds are higher/lower than the shock speed).

Undercompressive shocks can be understood similarly, with one remark: the operator on the right of (Eq1) needs not to be a diffusive one. One may consider operators involving third or fourth order spatial derivatives, or mixed derivatives, e.g.

\partial_t u +\partial_x f(u) = \eps \partial_{xx} u + \eps^2 \partial_{xxx} u) (Eq3)

or

\partial_t u +\partial_x f(u) = \eps \partial_{xx} u + \eps^2 \partial_{xxt} u) (Eq4)

These equations model various processes. For example, (Eq4) appears as a model for two-phase flow in porous media where dynamic effects are included in the capillary pressure. Then \eps is the capillary number.

When looking for TW solutions to (Eq3) or (Eq4), the results might differ than those for (Eq1). In other words, given two states U1 and U2, it may be that a TW connecting them exists for (Eq3) and not for (Eq1) and vice-versa. Clearly, the admissibility criterion for shock solutions is changing accordingly, and admissible shocks for (Eq2), understood as limits of TW for, say, (Eq4) may become "undercompressive", i.e. one of the characteristics from left or right does not enter the shock. The shock is hence non-Laxian, and also violates Oleinik's entropy condition.

Such results are provided thoroughly in the book of Ph. LeFloch, focusing mainly on shocks induced by equation (Eq3). Other operators are analyzed in papers by M. Shearer, A. Bertozzi, C. Rohde (I apologize a-priori for missing names of people who made important contributions). For (Eq4) I dare self-advertising :)

Hope this helps.

Good luck,

Sorin

Dear Sorin,

Thanks very much for the very helpful answer. The specific problem we're trying to understand is the one studied by Bertozzi et al. in their 1998 PRL [PRL 81, 5189 (1998)]. Their equation of motion is h_t + (h^2 - h^3)_x = -\eps (h^3 h_xxx). I understand the mathematical viewpoint you describe in your answer above. However, it is unclear to me from a physical standpoint why the UC shocks form, and why these solutions seem to be attractive. I imagine that the formation of the UC shocks has something to do with balancing gravity and the Marangoni stress, but is it possible to be more precise?

BTW, could you give me the reference for your Eq. (4)?

Thanks and best regards,

Mark

Hi Sorin,

so would the any shock-rarefaction such as the one arising in

Buckley-Leverett model of two-phase flow be considered undercompressive?

Hi Marc (and Mark :) ),

I'm not sure if I understood you right. If you mean by this that one characteristic enters the shock (this being a "shock" in what you said before) and the other one leaves it (the "rarefaction"), then, indeed, we have an undercompressive shock.

For Buckley-Leverett it is possible to have solutions consisting of a rarefaction wave that ends up precisely with the value where the shock starts (actually this is the "standard" entropy one). I mean, the speed of the endpoint of the RW is exactly the speed of the shock, and the the endpoint of the RW is the (say) left state of the shock. This is not undercompressive, but a standard, compressive shock. So simply the combination RW-shock needs not to be undercompressive.

Is this what you mean?

Have a great weekend and hope to meet you soon,

Sorin

Thanks Sorin.