Weak* Solutions II: The Vacuum in Lagrangian Gas Dynamics
(In: SIAM Journal on Mathematical Analysis (2017), 49(3), 1810-1843.)
We develop a framework in which to make sense of solutions containing the vacuum in Lagrangian gas dynamics. At and near vacuum, the specific volume becomes infinite and enclosed vacuums are represented by Dirac masses, so they cannot be treated in the usual weak sense. However, the weak* solutions recently introduced by the authors can be extended to include solutions containing vacuums. We present a definition of these natural vacuum solutions and provide explicit examples which demonstrate some of their features. Our examples are isentropic for clarity, and we briefly discuss the extension to the full system of gas dynamics. We also extend our methods to one-dimensional dynamic elasticity to show that fractures cannot form in an entropy solution.
Key words. gas dynamics, vacuum, conservation laws, elasticity, fracture
AMS subject classifications. 35L67, 35L70, 74B20, 74H20
The oldest and most fundamental system of hyperbolic conservation laws is that of isentropic gas dynamics, which are the simplest analog of Newton’s Law for a continuous medium. The equations can be expressed either in an Eulerian spatial frame, or in a Lagrangian or co-moving material frame. In the Lagrangian frame, the equations are
where is the material variable, is the specific volume, and and are the fluid velocity and pressure, respectively. The system is closed by specifying a constitutive relation , a monotone decreasing function which is integrable as . Alternatively, in an Eulerian frame, the equations are
representing conservation of mass and momentum, respectively, where is the spatial variable, and is the density.
The main effect of nonlinearity in a hyperbolic system is the presence of shock waves, across which the pressure and velocity are discontinuous, and the equations cannot be satisfied in the classical sense. This problem is usually solved by the use of weak solutions, which are defined by multiplying by test functions and integrating by parts. There is now a mature and largely complete theory of weak solutions of systems of conservation laws, provided the data is appropriately small .
Another effect of nonlinearity is the presence of a vacuum, which corresponds to or . The vacuum presents different difficulties depending on the frame: in an Eulerian frame, the equations degenerate and the velocity is underdetermined, while in a Lagrangian frame the vacuum is formally described using a Dirac mass, so the class of weak solutions is not large enough. The goal of this paper is to rigorously justify the use of Dirac masses and thus present a satisfactory notion of solution which includes vacuums in a Lagrangian frame.
In the recent paper , the authors introduced the notion of weak* solution, which we believe holds several advantages over weak solutions. Our approach is natural and general, and allows us to view the system as an evolutionary ODE in Banach space, which in turn confers some regularity. In addition, the “multiplication by test function and integration by parts” step is treated abstractly rather than explicitly, leading to cleaner calculations. Our approach is also general enough to handle certain extensions, including the treatment of vacuums as Dirac masses. In , we also proved that weak solutions are weak* solutions and vice versa, which implies that the well-known uniqueness and regularity results for solutions apply unchanged to weak* solutions.
To define a weak* solution of an abstract system of conservation laws,
we begin with a normed vector space of spatial test functions, and regard the solution as a function taking values in the dual space of . For solutions, we take with , so that , the space of Radon measures. Then if , so is , and so the distributional derivative . We then declare to be a weak* solution if is satisfies the Banach space ODE
where is the appropriate time derivative of . This is the Gelfand weak* derivative, or G-derivative, defined by using the Gelfand weak* integral of functions . The associated spaces are labelled .
In this paper, we extend the ideas of  to include the use of Dirac masses in weak* solutions. The key observation is that in LABEL:psyst, although is no longer bounded, or even a function, the flux vector remains , so that its spatial derivative is a measure, so lives in , and the ODE LABEL:Bode makes sense. Instead of treating the constitutive relation as a pointwise function, we regard it as a map of fields,
and in order to extend weak* solutions, we need only extend this to a map defined on positive Radon measures. Since pressure vanishes at vacuum, this extension is easily accomplished using the Lebesgue decomposition theorem. To avoid unphysical solutions, we impose a condition which we call consistency of the medium, and which states that the density and pressure must vanish whenever a vacuum is present; although this can be regarded as an entropy-type condition, it is distinct from the usual entropy condition which degenerates to an equality at vacuum. We refer to a vacuum weak* solution which satisfies consistency of the medium as a natural vacuum solution. In our framework the entropy and entropy flux are also regarded as maps on which are similarly extended to positive Radon measures. The entropy production is calculated to be a measure which is supported on shocks, and which is required to be negative. This again agrees with the entropy condition for weak solutions.
Once we have defined natural vacuum solutions to LABEL:psyst that include Dirac masses which account for vacuums of finite extent, we present a few detailed examples. These are natural vacuum solutions but not weak solutions, and our explicit description of the solutions and calculations of norms clearly demonstrates the advantages gained by treating the test functions and integration by parts abstractly and implicitly in the spaces .
We next describe the straight-forward extension of our results to the full equations of gas dynamics in a Lagrangian frame. We again define an extension of the pressure and specific internal energy to the positive Radon measures, by declaring that the pressure and internal energy vanish at vacuum. We then define a weak* solution and the corresponding entropy condition as would be expected.
As a final application, we extend our results to the equations of one-dimensional elasticity,
where , and are the strain, velocity and stress, respectively; we assume that , with a softening response, . Here we reproduce results of Giesselmann and Tzavaras , in which they introduce so-called slic-solutions to study crack formation and resolve an apparent paradox of nonuniqueness of solutions found in . Following , we study the onset of fracture, which we represent as a Dirac mass in the strain. To do so, we obtain the natural extension of the stress to Dirac masses, namely
Extending the stress and the energy allows us to define weak* solutions, and a brief analysis reveals that weak* solutions admitting a crack are defined if and only if ; however, none of these solutions are entropic. These are the same conclusions as those of , but our results significantly extend the one-dimensional results of , because their analysis applies to the single example of a solution provided in , while ours hold for any crack in a weak* solution. In , slic solutions are obtained as limits of mollified approximations, and their calculation of a single example requires several integrations and error estimates. In contrast, with our approach the mollification and integration by parts is abstract, and we are able to work directly with measures, leading to a direct and exact development without the need for error estimates.
The paper is arranged as follows: in LABEL:sec:prelims, we set notation and recall the definition and properties of weak* integrable functions and the Gelfand integral, developed in our earlier paper . Next we recall the definition of weak* solutions to conservation laws, and specifically to gas dynamics LABEL:psyst, and extend this definition to include vacuums. We derive generalized Rankine-Hugoniot jump conditions and discuss the entropy condition, while showing that it remains an identity at the vacuum. In LABEL:sec:exs we present some detailed examples of natural vacuum solutions which are not weak solutions. LABEL:sec:gd briefly describes the extension of our methods to the full system of gas dynamics, and in LABEL:sec:elast we consider the onset and propagation of fractures in one-dimensional elasticity.
We begin by setting notation and recalling the Gelfand integral and related notions which are necessary to define weak* solutions of systems of conservation laws. For simplicity we work in a single space dimension. We refer the reader to  for a more detailed discussion and proofs of quoted results.
2.1 Banach spaces
Given a vector space with norm , we denote its dual by , and recall
We denote the -fold product by , and equip it with the “Euclidean” norm
It follows that if we define the action of on by
then we can write . In particular, any statements on scalar valued function spaces extend naturally to vector-valued functions .
We recall the hierarchy of spaces that are most useful for us: first, fixing an open bounded , we have the inclusions
Next, any generates a measure , given by
so we regard , the set of Radon measures on ; moreover, we have
Note that for any , , that is is absolutely continuous with respect to Lebesgue measure, and indeed, is the Radon-Nikodym derivative of . On the other hand, by the Lebesgue decomposition theorem, any Radon measure can be uniquely decomposed into absolutely continuous and singular parts,
and moreover . We thus define the map
the Radon-Nikodym derivative of the absolutely continuous part of . It then follows that
so that is projection onto the absolutely continuous part of the measure.
Recall that the Radon measures form the dual of : that is, regarding as the closure of under the sup-norm, we can regard under the action
and it is not difficult to verify that .
We say that has an -valued distributional derivative, written , if, for all , we have
where we recall is dense in , and in this case we define by
Finally, recall that is the set of functions whose distributional derivative is in :
the supremum and sum being taken over finite ordered partitions.
We can combine the above together with product spaces, using inclusions as necessary, to get the following hierarchy of spaces:
where these functions take values in . Moreover, since is arbitrary, these inclusions extend to locally bounded functions on all of ,
2.2 The Gelfand integral
We next recall the definition and calculus of the Gelfand integral, which we need to define weak* solutions. Again we refer the reader to  for more details and proofs of statements. We briefly discuss different ways to integrate functions mapping to an abstract Banach space, namely the Bochner integral and Gelfand integral.
The Bochner integral of is obtained by approximating functions by simple functions. The function is strongly measurable, or Bochner measurable, if is measurable for each measurable . The integral of a simple measurable function is defined in the usual way,
and is Bochner integrable if there is a sequence of simple functions such that the Lebesgue integral as , and in this case we have .
The Bochner integral requires strong measurability, which is not always obvious in an abstract Banach space. The Dunford integral is a weak integral, defined using the functionals on . For our purposes it is more convenient to use the Gelfand integral, which is defined for functions which take values in the dual space of a Banach space . The map is weak* measurable if is Lebesgue measurable for all . Two functions and are weak* equivalent if for -almost all . It can be shown that any weak* measurable function is weak* equivalent to a function which is norm-measurable, by which we mean the scalar function is Lebesgue measurable. We will denote the weak* equivalence class of a weak* measurable by , and a norm-measurable representative by , although we will often abuse notation by simply writing when there is no ambiguity.
The Gelfand integral is defined as follows. Suppose that we are given a weak*-measurable function , and suppose also that
For a given Borel set , we define the map by
It is clear that is linear, and if and in , then by the Riesz-Fischer theorem, a subsequence a.e., while also for all . It follows that , so is closed, and further, by the closed graph theorem, it is bounded, so we can write for all . Since integration is a bounded linear operator of into , it follows that the map
is a bounded linear functional on , so defines an element of the dual . This functional is the Gelfand integral of over , and we denote it by . Thus the Gelfand integral over a measurable set is that element of defined by the condition
Again it follows easily that if is Bochner integrable with values in , then it is Gelfand integrable and the integrals coincide.
2.3 Gelfand-Sobolev Spaces
We now describe the valued Gelfand spaces, for . Given a weak* equivalence class of Gelfand integrable functions, set
where is a norm-measurable element of the equivalence class. It follows that is a norm, and we let be the space of equivalence classes of finite norm,
It is not difficult to show that is a Banach space and that the trivial inclusion of the Bochner space in the Gelfand space
is a norm-preserving isomorphism. Moreover, if is norm-measurable, then and
It follows that if is Bochner integrable, then we can calculate the Gelfand integral as a Bochner integral.
Now suppose that , are weak* integrable, so that , . We say that is the Gelfand weak derivative or G-weak derivative of , written or , if
for all and scalar functions .
We now define the space , for , to be the set of weak* equivalence classes with G-weak derivative , with norm
for norm-measurable representatives and .
If in addition, has values in some , then we write , that is we set
Note that we do not assume that is a subspace of , because we use the topology of throughout.
In  we state and prove some basic calculus theorems for the Gelfand integral, and the interested reader is referred there for details. We summarize the main points in the following theorem, which collects parts of Theorems 3.5 and 3.7 of .
If , then it has an absolutely continuous representative , which satisfies
for all , . Moreover, for all strongly integrable, we have the integration by parts formula
3 Weak* solutions
In , the authors introduced the notion of weak* solutions to a general system of hyperbolic conservation laws in one space dimension. Given such a system,
with , , recall that a distributional solution is a locally integrable function satisfying
for all compactly supported test functions , and if in addition is locally bounded, it is a weak solution. We note that the necessity of explicitly multiplying by test function and integrating by parts means that calculations are unwieldy and often error estimates must be employed when analyzing weak solutions.
On the other hand, when considering weak* solutions, we will treat the conservation law LABEL:cl as an ODE in an appropriate Banach space. Indeed, we look at LABEL:cl directly and allow this to act linearly on the Banach space which contains as a dense subspace. That is, for each , we treat and as living in , and we regard LABEL:cl as an ODE in , so that
for appropriately defined time derivative . The critical issue for us is to make sense of the nonlinear flux and its derivative in the space .
We then say that
is a weak* solution of the system LABEL:cl if
and if in , where is the continuous representative of the weak* equivalence class, and where is the G-weak derivative of . Here is understood in the usual sense and we allow any .
In our previous paper , we used , so that , and we took . In that paper we studied the connections between weak* solutions and weak solutions, and proved the following theorem.
Suppose is a weak* solution to the Cauchy problem LABEL:cl, with continuous representative . Then is Hölder continuous as a function into , that is, for . The function is a distributional solution of the Cauchy problem LABEL:cl. In particular, if is locally bounded, that is , then is also a weak solution to the Cauchy problem LABEL:cl.
In the same paper, we showed that a distributional solution with appropriate bounds is also a weak* solution, and in particular weak solutions are weak* solutions. As an immediate consequence, it follows that the global weak solutions generated by Glimm’s method, front tracking, and vanishing viscosity, all of which have uniformly bounded total variation, are all weak* solutions, and the uniqueness and stability results of Bressan et.al. hold unchanged in the framework of weak* solutions.
3.1 Application to Isentropic Gas Dynamics
Because of the flexibilty provided by the choices of growth rate and spaces and , we regard weak* solutions as more general than weak solutions. Indeed, we will generalize weak* solutions to include the vacuum in a Lagrangian frame, in which local boundedness is lost and the specific volume is allowed to be a measure.
We work with the system of gas dynamics in a Lagrangian frame, namely
in which the pressure is specified as a function of specific volume by a constitutive relation of the form
satisfying the appropriate properties: the most common such constitutive law is that of an ideal gas, for which , .
It follows immediately that as long as remains , then a weak* solution can be defined as above. However, we want to allow solutions which include vacuums, which are represented by Dirac masses in a Lagrangian frame. To do so, we simply allow the specific volume to be a Radon measure, which includes all Dirac masses. We note that the velocity remains , even when is unbounded and includes Dirac masses. We thus extend the target set to include Dirac masses in the first component, while still requiring that the vector of conserved quantities remain in the set . In order for this extension to make sense, we must extend the constitutive relation so that the pressure is defined for any specific volume, which can now be a positive Radon measure.
The constitutive relation expresses the thermodynamic pressure in terms of the specific volume, as . This extends naturally to a map of functions,
where is the domain of , and allows us to close LABEL:psys. We now wish to extend this constitutive map to be defined on Radon measures, and use this to define vacuum solutions of LABEL:psys, which will include Dirac masses which represent vacuums.
Recalling the Lebesgue decomposition, in the notation of LABEL:Pi, LABEL:iota, we write the measure as
so that for any Borel set ,
where is the Radon-Nikodym derivative of the absolutely continuous part of . Since the Lebesgue decomposition is unique, we extend the constitutive function to be defined on positive measures by
since pressure vanishes at vacuum. That is, we declare that the singular part of the specific volume makes no contribution to the pressure.
When generalizing the specific volume to a positive measure, we use the following notation: given and referring to LABEL:Pi, LABEL:iota, we write
so that , with and . It then follows that the (generalized) pressure is
so that, as expected, the generalized pressure is the composition of the pressure function with the Radon-Nikodym derivative of the absolutely continuous part of the measure .
As a first attempt at defining a solution with vacuum, we again take to be the set of continuous test functions, , and we set
where denotes Radon measures that are (strictly) positive on all open sets, so that
A vacuum weak* solution of the -system LABEL:psys is a pair
where denotes the G-weak derivative. When solving a Cauchy problem, the Cauchy data must be taken on in the space by the time-continuous representative , that is
3.2 Properties of Solutions with Vacuum
As in the general case of weak* solutions, we immediately observe that vacuum weak* solutions have some implicit regularity: first, the solutions have an absolutely continuous representative . Also, since the flux has a distributional derivative in , both and are functions (of material variable ) for all .
3.2.1 Evolution of Atomic Measures
Next, recalling that is a material rather than spatial variable, we show that vacuums are stationary in a Lagrangian frame.
A nontrivial continuous Dirac measure is stationary: that is, a measure
with and , is continuous on the interval if and only if is continuous and is constant on .
Recalling that , it follows easily that for , , and , ,
where is the indicator function on .
It follows immediately that if is continuous on , then so is the stationary measure , for any .
Similarly, for , , we have
both terms being non-negative. Now if is continuous at , then
so, since , LABEL:mudiff implies both
It follows that, given any , there exists such that for all , and moreover
Since is an arbitrary point in , is continuous on . Finally, let be the maximal interval for which for all . If , find another so that for to obtain a contradiction; this implies . Similarly, and the result follows.
Note that in other topologies such as the Wasserstein distance used in mass transfer problems, continuity need not imply that singular measures are stationary.
3.2.2 Evolution of Unbounded Maps
We next show that integrable functions which are unbounded blow up on stationary sets, consistent with vacuums being stationary in a material coordinate.
To this end, let and , so that the curve
and let be such that
Also suppose that the function is continuous at each point of the set , that the possibly infinite one-sided limits exist for each , and that for some , the map
Denote the sets on which is unbounded by
We first show that is almost uniformly unbounded on the set , in the sense of .
The sets , , and are measurable, and for any , there are measurable sets , with
such that for every , there exists such that
For , the functions
are defined and continuous on all of .
so we can write this as
and continuity of yields measurability of .
Now take any . By assumption on for every , so we can write
By Zakon , there exist measurable sets , with , such that for every there exists such that
and the proof follows.
We next show that if the discontinuity is non-stationary, then is bounded almost everywhere along .
Let and suppose that the curve given in LABEL:Ccurve2 satisfies on . If satisfies the conditions LABEL:fconds given above, then
The same conclusion holds if on .
We shall obtain a contradiction by constructing a sequence of test functions for which one side of the integration by parts formula LABEL:w*intbyparts is unbounded, while the other remains bounded.
Without loss of generality, we assume that that and on . According to LABEL:lem:essunifconv, there exists a set with , such that for every there exists such that
Let be a monotone function such that for , for , for , and elsewhere. For each , define