Math 553 Partial Differential Equations

Professor Richard S. Laugesen

Section 1.2 and EVANS 3.4
Weak Solutions for Quasilinear Equations

(see also JOHN pp 16-19; Z & T pp 72-85)

The principal goal of this section is to introduce the idea of a "weak" solution. There are many kinds of "weak" solution in PDE theory, but in this first exposure it means a solution of an integrated form of the PDE. This is significant because, generally speaking, an integrated form of a PDE involves fewer derivatives of the unknown u, and hence can be satisfied with a "less smooth" function u.

To keep things simple, the discussion in this section is centered around equations known as "conservation laws". There has been a good deal of research on such equations in the past 30 years, research largely concerned with (potentially) discontinuous or non-smooth solutions. For example, the prototypical conservation law in continuum mechanics arises in the theory of gas dynamics (with u representing the gas density or a component of the fluid velocity) and one is interested in shock waves (solutions that exibit discontinuities in u) and acceleration waves (solutions in which u is continuous but its first derivatives are not). Acceleration wave solutions of Einstein's equations are being proposed currently as alternative explanations for certain astrophysical phenomena (if one works just with smooth solutions, the phenomena are difficult to explain).
The Point: There are areas of physical application where solutions with discontinuities are the norm rather than the exception.

In reading MCOWEN and following my lectures you should come to realize that what we are considering here is the idea of "patching together" solutions that are smooth in their regions of definition and can "fit together" only in a discontinuous way, specified by a jump condition. In constructing such solutions there are two methods:

• EITHER: determine a point at which the projected characteristics for the problem intersect, then find the smooth solutions u_l and u_r to the left and right of this point (using the method of charateristics), then find the shock curve where these two solutions are allowed to join (using the jump and entropy conditions), and conclude that the integrated form of the equation is satisfied;
• OR: determine a region that the projected characteristics do not cover, then find the smooth solutions u_l and u_r to the left and right of this region (using the method of charateristics), then fill in the missing region with a "fan".
• Which of the two methods you apply depends on whether the characteristics are "approaching" each other or "retreating" from each other.

In a more general problem there may be multiple regions to worry about, especially if there are multiple points where the initial data is not smooth.

One of the things you should ask yourself in this section is: how general is the discussion? For example, can the same type of discussion be done if (say) the function G(u) also depended explicitly on x? Etc.

The excerpts from EVANS Sec. 3.4 are included in the lectures for three reasons: to introduce the entropy condition, to give us more complicated examples to understand (e.g. combining a fan and a shock), and to explain the long-time behavior of a solution in both the uniform and integral senses when the initial data is compactly supported.

Finally, Sec. 1.2c explains the origin of the name "conservation law" by means of a particular example involving traffic flow, where the number of cars is the conserved quantity.

For practice try the following problems: MCOWEN p 28 no 2,3,4,5,6,8,9 and JOHN p 19 no 4,6 Note: MCOWEN p 28 no 8 should not involve a shock (why?) but rather a rarefaction wave.