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In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, are unique, depend continuously on the initial data, have a high degree of regularity, can be continued in time as long as a sufficiently high regularity norm is under control, and tend to enjoy the same sort of conservation laws that classical solutions do. However, it is a major open problem as to whether these solutions can be extended to be (forward) global in time, because the norms that we know how to control globally in time do not have high enough regularity to be useful for continuing the solution. Also, the theory becomes degenerate in the inviscid limit {\nu \rightarrow 0}.

However, it is possible to construct “weak” solutions which lack many of the desirable features of strong solutions (notably, uniqueness, propagation of regularity, and conservation laws) but can often be constructed globally in time even when one us unable to do so for strong solutions. Broadly speaking, one usually constructs weak solutions by some sort of “compactness method”, which can generally be described as follows.

  1. Construct a sequence of “approximate solutions” to the desired equation, for instance by developing a well-posedness theory for some “regularised” approximation to the original equation. (This theory often follows similar lines to those in the previous set of notes, for instance using such tools as the contraction mapping theorem to construct the approximate solutions.)
  2. Establish some uniform bounds (over appropriate time intervals) on these approximate solutions, even in the limit as an approximation parameter is sent to zero. (Uniformity is key; non-uniform bounds are often easy to obtain if one puts enough “mollification”, “hyper-dissipation”, or “discretisation” in the approximating equation.)
  3. Use some sort of “weak compactness” (e.g., the Banach-Alaoglu theorem, the Arzela-Ascoli theorem, or the Rellich compactness theorem) to extract a subsequence of approximate solutions that converge (in a topology weaker than that associated to the available uniform bounds) to a limit. (Note that there is no reason a priori to expect such limit points to be unique, or to have any regularity properties beyond that implied by the available uniform bounds..)
  4. Show that this limit solves the original equation in a suitable weak sense.

The quality of these weak solutions is very much determined by the type of uniform bounds one can obtain on the approximate solution; the stronger these bounds are, the more properties one can obtain on these weak solutions. For instance, if the approximate solutions enjoy an energy identity leading to uniform energy bounds, then (by using tools such as Fatou’s lemma) one tends to obtain energy inequalities for the resulting weak solution; but if one somehow is able to obtain uniform bounds in a higher regularity norm than the energy then one can often recover the full energy identity. If the uniform bounds are at the regularity level needed to obtain well-posedness, then one generally expects to upgrade the weak solution to a strong solution. (This phenomenon is often formalised through weak-strong uniqueness theorems, which we will discuss later in these notes.) Thus we see that as far as attacking global regularity is concerned, both the theory of strong solutions and the theory of weak solutions encounter essentially the same obstacle, namely the inability to obtain uniform bounds on (exact or approximate) solutions at high regularities (and at arbitrary times).

For simplicity, we will focus our discussion in this notes on finite energy weak solutions on {{\bf R}^d}. There is a completely analogous theory for periodic weak solutions on {{\bf R}^d} (or equivalently, weak solutions on the torus {({\bf R}^d/{\bf Z}^d)} which we will leave to the interested reader.

In recent years, a completely different way to construct weak solutions to the Navier-Stokes or Euler equations has been developed that are not based on the above compactness methods, but instead based on techniques of convex integration. These will be discussed in a later set of notes.

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I’ve just uploaded to the arXiv the paper “Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equation“, submitted to Analysis & PDE.  This paper is concerned with solutions u: I \times {\Bbb R}^d \to {\Bbb C} to the focusing mass-critical NLS equation

i u_t + \Delta u = -|u|^{4/d} u, (1)

where the only regularity we assume on the solution is that the mass M(u(t)) := \int_{{\Bbb R}^d} |u(t,x)|^2\ dx is finite and locally bounded in time.  (For sufficiently strong notions of solution, the mass is in fact conserved, but part of the point with this paper is that mass conservation breaks down when the solution becomes too weak.)  Note that the mass is dimensionless (i.e. scale-invariant) with respect to the natural scale invariance u(t,x) \mapsto \frac{1}{\lambda^{d/2}} u(\frac{t}{\lambda^2}, \frac{x}{\lambda}) for this equation.  For various technical reasons I work in high dimensions d \geq 4 (this in particular allows the nonlinearity in (1) to be locally integrable in space).

In the classical (smooth) category, there is no ambiguity as to what it means for a function u to “solve” an equation such as (1); but once one is in a low regularity class (such as the class of finite mass solutions), there are several competing notions of solution, in particular the notions of a strong solution and a weak solution.  To oversimplify a bit, both strong and weak solutions solve (1) in a distributional sense, but strong solutions are also continuous in time (in the space L^2({\Bbb R}^d) of functions of finite mass).   A canonical example here is given by the pseudoconformally transformed soliton blowup solution

\displaystyle u(t,x) := \frac{1}{|t|^{d/2}} e^{-i/t} e^{i|x|^2/4t} Q(x/t) (2)

to (1), where Q is a solution to the ground state equation \Delta Q + |Q|^{4/d} Q = Q.  This solution is a strong solution on (say) the time interval (-\infty,0), but cannot be continued as a strong solution beyond time zero due to the discontinuity at t=0.  Nevertheless, it can be continued as a weak solution by extending by zero at t=0 and at t>0 (or alternatively, one could extend for t>0 using (2); thus there is no uniqueness for the initial value problem in the weak solution class. Note this example also shows that weak solutions need not conserve mass; all the mass in (1) concentrates into the spatial origin as t \to 0 and disappears in the limit t=0).

There is a slightly stronger notion than a strong solution, which I call a Strichartz-class solution, in which one adds an additional regularity assumption u \in L^2_{t,loc} L^{2d/(d-2)}_x.  This assumption is natural from the point of view of Strichartz estimates, which are a major tool in the analysis of such equations.

There is a vast theory for the initial value problem for these sorts of equations, but basically one has the following situation: in the category of Strichartz class solutions, one has local existence and uniqueness, but not global existence (as the example (2) already shows); at the other extreme, in the category of weak solutions, one has global existence, but not uniqueness (as (2) again shows).

(This contrast between strong and weak solutions shows up in many other PDE as well.  For instance, global existence of smooth solutions to the Navier-Stokes equation is one of the Clay Millennium problems that I have blogged about before, but global existence of weak solutions is quite easy with today’s technology and was first done by Leray back in 1933.)

In this paper, I introduce a new solution class, which I call the semi-Strichartz class; rather than being continuous in time, it varies right-continuously (in both the mass space and the Strichartz space) in time in the future of the initial time t_0, and left-continuously in the past of t_0.  With this tweak of the definition, it turns out that one has both global existence and uniqueness in this class.  (For instance, if one started with the initial data u(-1) given by (2) at time t=-1, the unique global semi-Strichartz solution from this initial data would be given by (2) for negative times and by zero for non-negative times.)  This notion of solution is analogous (but much, much simpler than) the notion of Ricci flow with surgery used by Hamilton and Perelman; basically, every time a singularity develops, the semi-Strichartz solution removes the portion of mass that was becoming discontinuous, leaving only the non-singular portion of the solution to continue onwards in time.

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