Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel
CCoagulation-Fragmentation equations withmultiplicative coagulation kernel and constantfragmentation kernel
Hung V. Tran and Truong-Son Van Department of Mathematics, University of Wisconsin Madison,Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706 Department of Mathematical Sciences and Center for NonlinearAnalysis, Carnegie Mellon University, Pittsburgh, PA 15213July 2, 2020
Abstract
We study a critical case of Coagulation-Fragmentation equationswith multiplicative coagulation kernel and constant fragmentation ker-nel. Our method is based on the study of viscosity solutions to a newsingular Hamilton-Jacobi equation, which results from applying theBernstein transform to the original Coagulation-Fragmentation equa-tion. Our results include wellposedness, regularity, and long-time be-haviors of viscosity solutions to the Hamilton-Jacobi equation in certainregimes, which have implications to wellposedness and long-time be-haviors of mass-conserving solutions to the Coagulation-Fragmentationequation.
Key words and phrases. critical Coagulation-Fragmentation equations; singularHamilton-Jacobi equations; regularity; wellposedness; Bernstein functions; long-time be-haviors; gelation; mass-conserving solutions; viscosity solutions..2010
Mathematics subject classification. a r X i v : . [ m a t h . A P ] J u l ontents (1.3) in case m ∈ (0 , C sublinear solutions when m > < m (cid:54) F when 0 < m (cid:54) F in case 0 < m < . . . . . . . . . . . . 213.2.3 Existence of solutions to equation (1.1) for 0 < m (0) < < m <
42A Bernstein functions and transform 44
The Coagulation-Fragmentation equation (C-F) is an integrodifferential equa-tion that finds applications in many different fields, ranging from astronomyto polymerization to the study of animal group sizes. The equation, withpure coagulation, dates back to Smoluchowski [Smo16], when he studied theevolution of number density of particles as they coagulate. Later on, Blatz andTobolsky [BT45] use the full C-F to study polymerization-depolymerizationphenomena. The mathematical studies of this equation did not start until thework of Melzak [Mel57], which was concerned with existence and uniquenessof the solutions for bounded kernels. Since then, although there are still a lotof open questions remained, major advancements have been made by both2nalytic and probabilistic tools. We list here some, but not exhaustive, im-portant works that are relevant to our work. For existence and uniqueness ofsolutions, there are the works of McLeod [McL62], Ball and Carr [BC90], Nor-ris [Nor99], Escobedo, Laurençot, Mischler and Perthame [EMP02; Esc+03].For large time behavior of solutions, there are the works of Aizenman andBak [AB79], Cañizo [Cañ07], Carr [Car92], Menon and Pego [MP04; MP06;MP08], Degond, Liu and Pego [DLP17], Liu, Niethammer and Pego [LNP19],Niethammer and Velázquez [NV13] and Laurençot [Lau19b]. For surveys ofwhat has been done, we refer the readers to two dated by now but still excel-lent surveys by Aldous [Ald99] and da Costa [Cos15] and the new monographsby Banasiak, Lamb, and Laurençot [BLL19].Here, coagulation represents binary merging when two clusters of parti-cles meet, which happens at some pre-determined rates; and fragmentationrepresents binary splitting of a cluster, also at some pre-determined rates.Thus, the C-F describes the evolution of cluster sizes over time given thatthere are only coagulation and fragmentation that govern the dynamics.A particularly interesting phenomenon of the C-F is that given the rightconditions, the solution, while still physical, does not conserve mass at all time.There are two ways that this could happen. One comes from the formationof particles of infinite size; the other comes from the formation of particlesof size zero, both in finite time. The first, called gelation , happens when thecoagulation is strong enough [Esc+03]. The latter, called dust formation ,happens when the fragmentation is strong enough (see Bertoin [Ber06]).Typically, these phenomena happen depending on the relative strengthsbetween the coagulation kernel and fragmentation kernel, not so much on theinitial data. However, there are borderline situations, where it is not veryclear how solutions would behave, hence more careful analysis needs to bedone based on initial data. Both are very interesting and rich phenomena,and have been studied in various contexts.The main goal of this article is to propose a new framework to ana-lyze a borderline situation described by Escobedo, Laurençot, Mischler andPerthame [EMP02; Esc+03], where solutions to the C-F may or may notexhibit gelation, depending on the initial data (as opposed to the type ofkernels). In particular, we analyze the properties of viscosity solutions of anew singular Hamilton-Jacobi equation (H-J), which results from transformingthe C-F equation via the so-called Bernstein transform. This, in our opinion,is natural and elegant since it requires very minimal assumptions.We note that, the Bernstein transform was first used to analyze this type3f equations by Menon and Pego [MP04] (under the name “desingularizedLaplace transform”). This transform is a generalization of the Laplacetransform and has properties that fit well with properties of solutions of C-F.
Let c ( s, t ) (cid:62) s (cid:62) t (cid:62)
0. We write the (continuous) Coagulation-Fragmentation equation asfollowing(1.1) ∂ t c ( s, t ) = Q c ( c ) + Q f ( c ) . Here, the coagulation term Q c and the fragmentation term Q f are given by Q c ( c )( s, t ) = 12 Z s a ( y, s − y ) c ( y, t ) c ( s − y, t ) dy − c ( s, t ) Z ∞ a ( s, y ) c ( y, t ) dy , and Q f ( c )( s, t ) = − c ( s, t ) Z s b ( s − y, y ) dy + Z ∞ b ( s, y ) c ( y + s, t ) dy . In the above, a, b denote the coagulation kernel and fragmentation kernel,respectively, which are nonnegative and symmetric functions defined on(0 , ∞ ) .Throughout this paper, we always assume that a ( s, ˆ s ) = s ˆ s and b ( s, ˆ s ) = 1 for all s, ˆ s > . We take a weak form of the coagulation-fragmentation equation. We saythat c ( s, t ) is a solution to the coagulation-fragmentation equation (1.1) if forevery test function φ ∈ BC ([0 , ∞ )) ∩ Lip([0 , ∞ )) with φ (0) = 0, we have ddt Z ∞ φ ( s ) c ( s, t ) ds = 12 Z ∞ Z ∞ ( φ ( s + ˆ s ) − φ ( s ) − φ (ˆ s )) sc ( s, t )ˆ sc (ˆ s, t ) d ˆ sds − Z ∞ (cid:18)Z s ( φ ( s ) − φ (ˆ s ) − φ ( s − ˆ s )) d ˆ s (cid:19) c ( s, t ) ds . (1.2) 4ere, BC ([0 , ∞ )) is the class of bounded continuous functions on [0 , ∞ ), andLip([0 , ∞ )) is the class of Lipschitz continuous functions on [0 , ∞ ). Considerthe Bernstein transform of c , for ( x, t ) ∈ [0 , ∞ ) , F ( x, t ) def = Z ∞ (1 − e − sx ) c ( s, t ) ds and let φ x ( s ) = 1 − e − sx , we have ∂ t F ( x, t ) = 12 Z ∞ Z ∞ (1 − e − ( s +ˆ s ) x − e − sx − e − ˆ sx ) sc ( s, t )ˆ c (ˆ s, t ) d ˆ sds − Z ∞ Z s (1 − e − sx − e − ( s − ˆ s ) x − e − ˆ sx ) d ˆ s c ( s, t ) ds = − Z ∞ Z ∞ (1 − e − sx )(1 − e − ˆ sx ) sc ( s, t )ˆ sc (ˆ s, t ) d ˆ sds − Z ∞ ( − s − se − sx + 2 x (1 − e − sx )) c ( s, t ) ds = −
12 ( m ( t ) − ∂ x F ( x, t )) + m ( t )2 + ∂ x F ( x, t )2 − F ( x, t ) x = −
12 ( m ( t ) − ∂ x F ( x, t ))( m ( t ) − ∂ x F ( x, t ) + 1) − F ( x, t ) x + m ( t ) . Here, m ( t ) is the total mass (first moment) of all particles at time t (cid:62) m ( t ) = Z ∞ sc ( s, t ) ds . Let us assume that m ( t ) < ∞ for all t (cid:62)
0. The key point is to transform aseemingly hopeless nonlocal equation to a somewhat more tractable nonlinearPDE, which enjoys some major developments in the past few decades. Ifconservation of mass holds, then we can assume m ( t ) = m > t (cid:62) m ∈ (0 , ∞ ). This fact, together with the above computations, leadsto the following PDE for F . ∂ t F + 12 ( ∂ x F − m )( ∂ x F − m −
1) + Fx − m = 0 in (0 , ∞ ) , (1.3a) 0 (cid:54) F ( x, t ) (cid:54) mx on [0 , ∞ ) , (1.3b) F ( x,
0) = F ( x ) on [0 , ∞ ) . (1.3c) 5ne then can study wellposedness and properties of solutions of (1.3) todeduce back information of C-F. Indeed, this is our main goal.Note that the condition (1.3b) implies that F (0 , t ) = 0 and that it comesdirectly from the Bernstein transform. Indeed, as c (cid:62)
0, it is clear that F (cid:62) − e − sx (cid:54) sx for s, x (cid:62) F ( x, t ) = Z ∞ (1 − e − sx ) c ( s, t ) ds (cid:54) Z ∞ sxc ( s, t ) ds = mx . Moreover, the dominated convergence theorem giveslim x →∞ F ( x, t ) x = lim x →∞ Z ∞ − e − sx x c ( s, t ) ds = 0 , which means that F ( x, t ) is sublinear in x . Here, for a given function ψ :[0 , ∞ ) → R , we say that it is sublinear iflim x →∞ ψ ( x ) x = 0 . It is therefore natural to search for solutions of (1.3) that are sublinear in x .It is worth noting that (1.3) is a Hamilton-Jacobi equation with theHamiltonian H ( p, z, x ) = 12 ( p − m )( p − m −
1) + zx − m for all ( p, z, x ) ∈ R × R × (0 , ∞ ) , which is of course singular at x = 0. Besides, H is monotone, but not Lipschitzin z as ∂ z H ( p, z, x ) = 1 x (cid:62) x → ∂ z H ( p, z, x ) = lim x → x = + ∞ . This means that (1.3) does not fall into the classical theory of viscositysolutions to Hamilton-Jacobi equations developed by Crandall and Lions[CL83] (see also Crandall, Evans and Lions [CEL84]). It is thus our purposeto develop a framework to study wellposedness and further properties ofsolutions to (1.3). For a different class of Hamilton-Jacobi equations thatis singular in p (but not in z ), see the radially symmetric setting in Giga,Mitake and Tran [GMT16].We emphasize that for wellposedness and regularity results, we do notneed to impose all the properties of the Bernstein transform of the initial6ata c = c ( · , C ∞ ((0 , ∞ )) (in fact, analytic) function. However, we only assume F to beLipschitz and sublinear for our wellposedness result and more regular for ourregularity results.A more important point is that our assumption on c is minimal. Forexistence and uniqueness results, we do not have any restrictions on momentsof c except finite first moment so that the derivative of the Bernstein transformmakes sense. In particular, we only require m (0) = Z ∞ sc ( s ) ds < ∞ . This also makes physical sense since one often wishes that the initial totalmass to be finite before talking about conservation of mass. Of course, wewill need to put in more conditions for our regularity results.
Remark . In fact, we are also able to define weak solutions in the measuresense to (1.1) in a similar fashion.For each t (cid:62)
0, let c t ( ds ) be a positive Radon measure in (0 , ∞ ). Then,we say that c t ( ds ) is a weak solution in the measure sense to (1.1) if for everytest function φ ∈ BC ([0 , ∞ )) ∩ Lip([0 , ∞ )) with φ (0) = 0, we have ddt Z ∞ φ ( s ) c t ( ds ) = 12 Z ∞ Z ∞ ( φ ( s + ˆ s ) − φ ( s ) − φ (ˆ s )) s ˆ s c t ( ds ) c t ( d ˆ s ) − Z ∞ (cid:18)Z s ( φ ( s ) − φ (ˆ s ) − φ ( s − ˆ s )) d ˆ s (cid:19) c t ( ds ) . This is clearly a weaker notion of solutions than that in (1.2). Nevertheless,the Bernstein transform of c t ( ds ) and (1.3) still make perfect sense. We willuse this notion of solutions when talking about the existence results for theC-F. In [EMP02; Esc+03], the authors conjectured that in borderline situationswhere coagulation kernel and fragmentation kernel balance each other out,the solution will conserve mass if the initial data have small enough totalmass. Otherwise, for large total mass initial data, gelation will occur. In thepaper by Vigil and Ziff [VZ89], the authors argued that if the zeroth moment7f the solution reaches negative value in finite time, one expects coagulationto dominate, hence gelation will occur.It has been expected by experts in the field that for our specific kernels,the critical initial mass should be m (0) = 1 so that for m (0) >
1, one hasgelation; and for m (0) (cid:54)
1, one has solutions that conserve mass. We givehere a simple reason why such expectation arises.Integrating equation (1.1) and denoting m ( t ) = R ∞ c ( s, t ) ds , the zerothmoment, we get the following equation ddt m ( t ) = 12 m ( t )(1 − m ( t )) . Suppose now m ( t ) = m (0) > m ( t ) will be negative in finite time. On the other hand, m ( t ) remainspositive if 0 (cid:54) m (0) (cid:54)
1. Therefore, by the reasoning above, m (0) = 1 isbelieved to be the critical mass. Our goal is to give results towards resolvingthis conjecture, which will be detailed in the next subsection. In this subsection, we give rigorous statements about our results, which webelieve to be the stepping stones for further investigations in the future, bothin the theory of viscosity solutions and in the theory of C-F.First and foremost, we need to understand the existence and uniquenessof viscosity solutions for equation (1.3).
Theorem 1.2.
Assume that < m (cid:54) . Assume further that F is Lipschitz,sublinear, and (cid:54) F ( x ) (cid:54) mx . Then, (1.3) has a unique Lipschitz, sublinearsolution F . The proof of this theorem is given in Section 2. Theorem 1.2 gives us asimple but important implication about C-F.
Corollary 1.3.
Assume that m (0) = m ∈ (0 , . Then, equation (1.1) hasat most one mass-conserving solution. We believe that the uniqueness result of Corollary 1.3 is new in theliterature although existence results of mass-conserving solutions for (1.1)for the whole range of m (0) ∈ (0 ,
1] are still not yet available. In a recentimportant work, Laurençot [Lau19a] showed existence and uniqueness of8ass-conserving solutions to (1.1) under some additional moment conditionsfor 0 < m (0) <
14 log 2 . In Theorem 1.8 below, we obtain existence (and ofcourse uniqueness) of mass-conserving weak solutions in the measure sense to(1.1) in case that 0 < m (0) < , and c ( · ,
0) has bounded second moment.We note that, in general, if the viscosity solution to the Hamilton-Jacobiequation forms shocks, one cannot have a solution of C-F that conserves massanymore. This is because if there were a solution of C-F that conserves mass,its Bernstein transform would need to solve the Hamilton-Jacobi equationand at the same time would need to be smooth. This cannot be the case ifthere were shocks.It is, therefore, of our interest to study the regularity of the viscositysolutions of the equation (1.3). Moreover, regularity results in the theory ofviscosity solutions are important in their own rights.
Theorem 1.4.
Suppose m > . Assume that F is smooth, sublinear, and (cid:54) F ( x ) (cid:54) mx . Then equation (1.3) does NOT admit a solution F ∈ C ([0 , ∞ ) ) which is sublinear in x . The proof of this theorem is given in Subsection 3.1. Based on ourdiscussion above, Theorem 1.4 implies immediately the following consequence.
Corollary 1.5.
Assume that m (0) = m > . Then, there is no mass-conserving solution to equation (1.1) . A version of Corollary 1.5 already appeared in [BLL19]. We here obtainnon-existence of mass-conserving solutions under the minimal assumption,that is, m (0) >
1. We do not need to assume anything else about othermoments. In particular, we do not need to impose that the zeroth moment,number of clusters, is finite as in [BLL19]. It is also worth noting thatCorollaries 1.3 and 1.5 hold true for mass-conserving weak solutions in themeasure sense to (1.1) as well.To study regularity of F for 0 < m (cid:54)
1, we impose more conditions on F as following. Assume that there exist β ∈ (0 ,
1) and
C > (cid:54) F ( x ) (cid:54) m and F (0) = m , (A1) − C (cid:54) F ( x ) (cid:54) , (A2) − me (cid:54) xF ( x ) (cid:54) k xF k C ,β ([0 , ∞ )) (cid:54) C . (A3) 9he above assumptions hold true when F is the Bernstein transform of c = c ( · , c has m (0) = m and also bounded second moment, thatis, m (0) = Z ∞ s c ( s, ds (cid:54) C .
Indeed, 0 (cid:54) F ( x ) = Z ∞ se − xs c ( s, ds (cid:54) m , and F (0) = m . For second derivative, one has − C (cid:54) F ( x ) = − Z ∞ s e − xs c ( s, ds (cid:54) , and xF ( x ) = − Z ∞ s xe − xs c ( s, ds = − Z ∞ ( sxe − xs ) sc ( s, ds (cid:62) − me . We use the fact that re − r (cid:54) e − for r ≥ x, y ∈ [0 , ∞ ), | xF ( x ) − yF ( y ) | (cid:54) Z ∞ | xe − xs − ye − ys | s c ( s, ds (cid:54) Z ∞ | x − y | s c ( s, ds (cid:54) C | x − y | . In the second inequality above, we use the point that | xe − xs − ye − ys | (cid:54) | x − y | ,which can be derived by the usual mean value theorem and (cid:12)(cid:12)(cid:12) ( ze − zs ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e − zs − zse − zs (cid:12)(cid:12)(cid:12) (cid:54) z (cid:62) . We first show that F is always concave in x provided that (A1)–(A2) holdand 0 < m (cid:54) Lemma 1.6.
Assume (A1)–(A2) , and < m (cid:54) . Assume further that F is sublinear, and (cid:54) F ( x ) (cid:54) mx . Then, the sublinear solution F to theequation (1.3) is concave in x for each t (cid:62) . The concavity of F in the above lemma is rather standard as the Hamilto-nian is convex (in fact quadratic) in p . Of course, we need to be careful withthe singularity of H in x at x = 0, but otherwise, the arguments in the proofof Lemma 1.6 are quite classical. Next, we show that in a smaller range of m (0 < m < ), F ∈ C , ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )) under assumptions(A1)–(A3). It is worth noting that we do not need to put any assumption onthird or higher derivatives of F . 10 heorem 1.7. Assume (A1)–(A3) , and < m < . Assume further that F is bounded, and (cid:54) F ( x ) (cid:54) mx . Then the sublinear solution F to theequation (1.3) is in C , ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )) . Moreover, F satisfiesthat, for ( x, t ) ∈ (0 , ∞ ) , (cid:54) ∂ x F ( x, t ) (cid:54) m and − (cid:54) x∂ x F ( x, t ) (cid:54) . To the best of our knowledge, the regularity result in Theorem 1.7 is newin the literature. The proofs of Lemma 1.6 and Theorem 1.7 are given inSubsection 3.2. Next is our existence result for C-F when 0 < m < . Theorem 1.8.
Assume that F is the Bernstein transform of c = c ( · , ,where c has m (0) = m ∈ (0 , ) and also bounded zeroth and second moments,that is, m (0) = Z ∞ c ( s, ds (cid:54) C and m (0) = Z ∞ s c ( s, ds (cid:54) C .
Then (1.1) has a mass-conserving weak solution in the measure sense.
Of course, this mass-conserving weak solution in the measure sense isunique thanks to Corollary 1.3. The range we get here for 0 < m (0) < is animprovement to the previous range of 0 < m (0) <
14 log 2 obtained in [Lau19a].The proof of Theorem 1.8 is given in Subsection 3.2.3. Basically, under theassumptions of Theorem 1.8, we first need to show that F ∈ C ∞ ((0 , ∞ ) )in Proposition 3.9. Then, we deduce that ( − n +1 ∂ nx F (cid:62) n ∈ N inProposition 3.10. These highly nontrivial regularity results of F , togetherwith a characterization of Bernstein functions (see Appendix A), allow us toobtain Theorem 1.8.We then obtain the following large time behavior result for F in case0 < m <
1. Here, we do not need assumption (A3).
Theorem 1.9.
Assume (A1)–(A2) . Let < m < , F be sublinear, and (cid:54) F ( x ) (cid:54) mx . Let F be the Lipschitz, sublinear solution to equation (1.3) .Then (1.4) lim t →∞ F ( x, t ) = mx locally uniformly on [0 , ∞ ) . t → ∞ , all the solutions (mass-conserving or not) will turn to dusts (particles of size zero) if their initialtotal mass is less than 1. To see this, we note that, if F ∞ ( x ) = lim t →∞ F ( x, t )is a Bernstein transform, then for some measure µ ∞ , F ∞ ( x ) = Z ∞ (1 − e − sx ) µ ∞ ( ds ) = mx . Differentiating in x , it is necessary that Z ∞ se − sx µ ∞ ( ds ) = m , which implies sµ ∞ ( ds ) = mδ ( ds ).To avoid any confusion, we conclude the introduction by emphasizing thefollowing points. • While the viscosity solution to the Hamilton-Jacobi equation (1.3) itselfdoes not correspond to any extension of weak solutions to the C-F, ifthe viscosity solution F is smooth (i.e., a smooth classical solution)and ( − n +1 ∂ nx F (cid:62) , ∞ ) for all n ∈ N , it would correspondto a mass-conserving weak solution in the measure sense to the C-F.Therefore, regularity of the viscosity solution will imply whether onecould have a mass-conserving weak solution in the measure sense to theC-F or not. This is, obviously, an extremely hard and central issue inthe theory of viscosity solutions. • Here, we achieve uniqueness of mass-conserving weak solutions to the C-F for 0 < m (0) (cid:54)
1. We show existence of such mass-conserving weaksolutions for 0 < m (0) < , and of course, the range (cid:54) m (0) (cid:54) • To obtain a classical mass-conserving solution for equation (1.1) incase 0 < m (0) < , one needs to show that the mass-conserving weaksolution in the measure sense actually admits a density, which requiresmore properties from the corresponding Bernstein function. This hasbeen done by Degond, Liu and Pego [DLP17] in a different setting, butremains a hard problem here and will be addressed in future works.12 Wellposedness of (1.3) in case m ∈ (0 , We first prove the existence and uniqueness of viscosity solutions to (1.3). Inthis section, we always assume that conditions of Theorem 1.2 are in force. (1.3)
We search for sublinear solutions to (1.3) which satisfy (1.3b), that is,0 (cid:54) F ( x, t ) (cid:54) mx for all ( x, t ) ∈ [0 , ∞ ) . Since (1.3) is singular at x = 0, we cut off its singularity by introducing asequence of function { φ n } where φ n ( x ) = max (cid:26) n , x (cid:27) for all x ∈ [0 , ∞ ) . By the classical theory of viscosity solutions, we have that for each n ∈ N ,the equation(2.1) ∂ t F + ( ∂ x F − m )( ∂ x F − m −
1) + Fφ n ( x ) − m = 0 in (0 , ∞ ) ,F ( x,
0) = F ( x ) on [0 , ∞ ) ,F (0 , t ) = 0 on [0 , ∞ ) , has a unique sublinear viscosity solution F n . In fact, the sublinearity of F n can be seen through the fact that F ( x ) − Ct (cid:54) F n ( x, t ) (cid:54) F ( x ) + Ct for all ( x, t ) ∈ [0 , ∞ ) , as F ( x ) − Ct, F ( x ) + Ct are a subsolution and a supersolution to (2.1),respectively, for some C > C + 12 ( ∂ x F ( x ) − m )( ∂ x F ( x ) − m −
1) + F ( x ) + Ctφ n ( x ) − m (cid:62) − C + 12 ( ∂ x F ( x ) − m )( ∂ x F ( x ) − m −
1) + F ( x ) − Ctφ n − m (cid:54) C (cid:62) m + sup x ∈ (0 , ∞ ) | ( ∂ x F ( x ) − m )( ∂ x F ( x ) − m − | . emma 2.1. For each n ∈ N , let F n be the viscosity solution to equation (2.1) .Then, we have that (2.2) F n +1 (cid:54) F n for all n ∈ N .Proof. To see this, we note that φ n (cid:62) φ n +1 . Therefore F n φ n (cid:54) F n φ n +1 , which implies that F n is a supersolution to equation (2.1) with φ n +1 . Thus,(2.2) follows. Lemma 2.2.
For each n ∈ N , let F n be the viscosity solution to equation (2.1) .Then, { F n } is equi-Lipschitz, that is, there exists a constant C > so thatfor every n ∈ N , (2.3) | F n ( x , t ) − F n ( x , t ) | (cid:54) C ( | t − t | + | x − x | ) , for every t , t , x , x ∈ [0 , ∞ ) .Proof. We achieve global Lipschitz property in time using the solutions tothe approximation problems. We note that equation (2.1) obeys the classicaltheory of viscosity solutions so the comparison principle holds.For each n ∈ N , we have that φ − ≡ φ + = mx + n is a supersolution to equation (2.1). To see the subsolution, we have that12 m ( m + 1) − m = m ( m − (cid:54) . To see the supersolution, we have that mx + n φ n ( x ) − m = nx if x (cid:62) n ,nmx + 1 − m if x (cid:54) n , which is always nonnegative. On the other hand, as shown just before Lemma2.1, we also have that F ( x ) − Ct and F ( x ) + Ct are a subsolution and asupersolution to (2.1), respectively. Therefore, G − ( x, t ) def = max { , F ( x ) − Ct }
14s also a subsolution, and G + ( x, t ) def = min { mx + n , F ( x ) + Ct } is also asupersolution to (2.1). And so, by the comparison principle,(2.4) G − ( x, t ) (cid:54) F n ( x, t ) (cid:54) G + ( x, t ) . Thus, for t > | F n ( x, t ) − F n ( x, | (cid:54) Ct .
By the L ∞ -contractive property of solutions to Hamilton-Jacobi equations(which follows from the comparison principle itself), for every t , t ∈ [0 , ∞ )with t > t ,(2.5) sup x | F n ( x, t ) − F n ( x, t ) | (cid:54) sup x | F n ( x, t − t ) − F n ( x, | (cid:54) C | t − t | . This is equivalent to the fact that(2.6) | ∂ t F n ( x, t ) | (cid:54) C in the viscosity sense. Therefore, rearranging equation (2.1) and using triangleinequality, estimates (2.4) and (2.6), we have | ( ∂ x F n − m )( ∂ x F n − m − | = 2 (cid:12)(cid:12)(cid:12)(cid:12) − ∂ t F n + m − F n φ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) C in the viscosity sense. Therefore, there exists a constant C > n ∈ N ) so that | ∂ x F n | (cid:54) C in the viscosity sense, which is equivalent to(2.7) | F n ( x , t ) − F n ( x , t ) | (cid:54) C | x − x | for every x , x ∈ (0 , ∞ ). Combining estimates (2.5) and (2.7), we get thedesired inequality (2.3). Lemma 2.3.
There exists a function F so that { F n } converges to F locallyuniformly on [0 , ∞ ) , and F is sublinear, uniformly Lipschitz with the sameLipschitz constant as in Lemma 2.2. Furthermore, F is a viscosity solutionto equation (1.3) .Proof. The locally uniform convergence follows from Lemmas 2.1 and 2.2. It isclear from the convergence and (2.4) that F is sublinear, and 0 (cid:54) F ( x, t ) (cid:54) mx for all ( x, t ) ∈ [0 , ∞ ) . The fact that F is a viscosity solution to (1.3) followsdirectly from the definition and the facts that { F n } converges to F locallyuniformly and { φ n } converges to x uniformly.15 .2 Uniqueness of solutions to (1.3) Lemma 2.4 (Comparison Principle) . Let u be a sublinear viscosity subsolutionand v be a sublinear viscosity supersolution to equation (1.3) , respectively.Then u (cid:54) v .Proof. We have that for every n ∈ N , u is a subsolution, and v n def = v + n is asupersolution to equation (2.1), respectively. The subsolution is clear to see.To check the supersolution property, we note that, since m (cid:54) v + n φ n − m = vx + nx − m (cid:62) vx − m, for x (cid:62) n ,nv + 1 − m (cid:62) (cid:62) vx − m, for x < n . Therefore, ∂ t v n + 12 ( ∂ x v n − m )( ∂ x v n − m −
1) + v n φ n ( x ) − m (cid:62) ∂ t v + 12 ( ∂ x v − m )( ∂ x v − m −
1) + vx − m (cid:62) u (cid:54) v n . But as v n → v locally uniformly as n → ∞ , we then conclude u (cid:54) v , as desired.Let us now give the proof of Theorem 1.2. Proof of Theorem 1.2.
By Lemma 2.3, (1.3) admits a solution F , whichis Lipschitz on [0 , ∞ ) , and is sublinear in x . Lemma 2.4 then yields theuniqueness of F .Corollary 1.3 then follows immediately. Proof of Corollary 1.3.
Let c be a mass-conserving solution to (1.1) with m = m (0) ∈ (0 , F, F be the Bernstein transforms of c, c = c ( · , F is a solution to (3.1), F is sublinear in x , and F ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) ). In particular, F is the unique sublinear viscositysolution to (3.1). This gives the uniqueness of c .16 Regularity results C sublinear solutions when m > We first show the impossibility of C sublinear solutions when m >
1. It isimportant to note that the sublinear requirement is used crucially here as(1.3) admits special solutions ψ ( x, t ) = mx and ψ ( x, t ) = ( m − x for all( x, t ) ∈ [0 , ∞ ) , which are both linear in x . Proof of Theorem 1.4.
We proceed by contradiction and suppose thatsuch a solution F exists. Then, F (0 , t ) = 0 and ∂ t F (0 , t ) = 0 . Let x → + in (1.3) and use the fact that ∂ x F (0 , t ) = lim x → + F ( x, t ) − F (0 , t ) x = lim x → + F ( x, t ) x to yield 12 ( ∂ x F (0 , t ) − m )( ∂ x F (0 , t ) − m −
1) + ∂ x F (0 , t ) − m = 0 . Thus, either ∂ x F (0 , t ) = m or ∂ x F (0 , t ) = m −
1. In other words, ∂ x F (0 , t ) (cid:62) m − >
0. Now, fix σ ∈ (0 , m − x of F , for a fixed t >
0, there exists x t > ϕ ( t ) def = max x ∈ [0 , ∞ ) ( F ( x, t ) − σx ) = F ( x t , t ) − σx t > . The computations from here to the end of this proof are all justified in theviscosity sense. Observe that, at x = x t , ∂ x F ( x t , t ) = σ and F ( x t , t ) /x t > σ .Therefore, ∂ t F ( x t , t ) (cid:54) −
12 ( σ − m )( σ − m − − ( σ − m ) = −
12 ( σ − m )( σ − m +1) def = − c . Furthermore, ϕ ( t ) = lim s → + ϕ ( t ) − ϕ ( t − s ) s = lim s → + [ F ( x t , t ) − σx t ] − [ F ( x t − s , t − s ) − σx t − s ] s lim s → + [ F ( x t , t ) − σx t ] − [ F ( x t , t − s ) − σx t ] s = ∂ t F ( x t , t ) (cid:54) − c < . Therefore, there exists
T > ϕ ( T ) <
0, which is a contradiction.
Proof of Corollary 1.5.
Assume by contradiction that there exists a mass-conserving solution c to (1.1) with m = m (0) >
1. Let
F, F be the Bernsteintransforms of c, c = c ( · , F is a solution to (3.1), F issublinear in x , and F ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) ). This of course contradictsTheorem 1.4. The proof is complete. < m (cid:54) In the case 0 < m (cid:54)
1, a central topic we set out to study is when is it thatclassical solutions to the equation (1.3) exist for all time. This is not a simpletask as viscosity solutions to Hamilton-Jacobi equations are Lipschitz, butmight not be C in general.To do this, we study another regularized version of equation (1.3) by addinga viscosity term and then study the vanishing viscosity limit. Specifically, for ε >
0, we consider(3.1) ∂ t F + ( ∂ x F − m )( ∂ x F − m −
1) + Fx − m = εa ( x ) ∂ xx F ,F ( x,
0) = F ( x ) ,F (0 , t ) = 0 . In this section, we use assumptions (A1)–(A3) whenever needed.We give ourselves some freedom of choices for the nonnegative function a ( x ). This freedom gives us some flexibility in proving bounds. F when < m (cid:54) F is sublinear, and 0 (cid:54) F ( x ) (cid:54) mx . For each ε >
0, let F ε be the classical solution to equation (3.1)corresponding to a ≡
1. By regularity theory for parabolic equations, F ε ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )) (for example, see Ladyženskaja, Solonnikov,Ural’ceva [LSU68], Lieberman [Lie96], Krylov [Kry96]). Here, C ([0 , ∞ ) × (0 , ∞ )) is the space of functions which are C in x and C in t on [0 , ∞ ) × (0 , ∞ ). 18 emma 3.1. Assume (A1)–(A2) . Assume further that F is sublinear and (cid:54) F ( x ) (cid:54) mx . For each ε > , let F ε be the classical solution to equa-tion (3.1) corresponding to a ≡ . Then, (3.2) 0 (cid:54) ∂ x F ε (cid:54) m . Proof.
Firstly, as 0 (cid:54) F ε ( x, t ) (cid:54) mx for each t (cid:62)
0, we imply that(3.3) 0 (cid:54) ∂ x F ε (0 , t ) (cid:54) m . Differentiate (3.1) to get L ε [ ∂ x F ε ] + ∂ x F ε x − F ε x ! = 0 , where L ε [ φ ] def = ∂ t φ + ∂ x F ε ∂ x φ − ( m + 12 ) ∂ x φ − ε∂ x φ is a linear parabolic operator.By Taylor’s expansion, for each ( x, t ) ∈ (0 , ∞ ) , there exists α = α ( x, t ) ∈ (0 ,
1) so that 0 = F ε (0 , t ) = F ε ( x, t ) − x∂ x F ε ( αx, t ) . Thus, L ε [ ∂ x F ε ] + ∂ x F ε ( x, t ) − ∂ x F ε ( αx, t ) x = 0 . We only show here that ∂ x F ε (cid:54) m by the usual maximum principle. Thelower bound can be done in a similar manner. Suppose that for some T > x (cid:62) [0 , ∞ ) × [0 ,T ] ∂ x F ε = ∂ x F ε ( x , T ) . Thanks to (3.3), we only need to consider the case that x >
0. At this point ∂ x F ε ( x , T ) (cid:62) ∂ x F ε ( αx , T ), and so L ε [ ∂ x F ε ]( x , T ) (cid:54)
0. By repeating theproof of the maximum principle for a linear parabolic operator, we obtainthe desired conclusion that ∂ x F ε (cid:54) m . Remark . In the use of the maximum principle, to keep the presentationclean, it is typically the case that one assumes that maximum points of abounded continuous function ( ∂ x F ε in the above proof) occur. To justify thispoint rigorously, one can consider maximum of ∂ x F ε ( x, t ) − δx on [0 , ∞ ) , for δ >
0, and let δ → + . 19 emma 3.3. Let F ε be the classical solution to equation (3.1) with a ≡ .Then, (3.4) ∂ x F ε (0 , t ) (cid:54) for all t (cid:62) . Proof. As F ε (0 , t ) = 0 for all t (cid:62) ∂ t F ε (0 , t ) = 0 andlim x → + F ε ( x, t ) x = ∂ x F ε (0 , t ) . Let x → + in (3.1) and use the above to get(3.5) 12 ( ∂ x F ε (0 , t ) − m )( ∂ x F ε (0 , x ) − m + 1) = ε∂ x F ε (0 , t ) , which, together with (3.2), yields (3.4).We are now ready to prove that F ε is concave in x . Lemma 3.4.
Assume (A1)–(A2) . Assume further that F is sublinear and (cid:54) F ( x ) (cid:54) mx . For each ε > , let F ε be the classical solution to equa-tion (3.1) corresponding to a ≡ . Then, for ( x, t ) ∈ (0 , ∞ ) , (3.6) ∂ x F ε (cid:54) . Proof.
We proceed by the maximum principle. Differentiating (3.1) twice in x , we get(3.7) L ε [ ∂ x F ε ] + ( ∂ x F ε ) + (cid:18) ∂ x F ε x + 2( F ε − x∂ x F ε ) x (cid:19) = 0 . Recall that L ε [ φ ] = ∂ t φ + ∂ x F ε ∂ x φ − ( m + 12 ) ∂ x φ − ε∂ x φ . By Taylor’s expansion, for each ( x, t ) ∈ (0 , ∞ ) , there exists θ = θ ( x, t ) ∈ (0 , F ε (0 , t ) = F ε ( x, t ) − x∂ x F ε ( x, t ) + x ∂ x F ε ( θx, t ) . This implies ∂ x F ε x + 2( F ε − x∂ x F ε ) x = ∂ x F ε ( x, t ) − ∂ x F ε ( θx, t ) x , L ε [ ∂ x F ε ] + ( ∂ x F ε ) + ∂ x F ε ( x, t ) − ∂ x F ε ( θx, t ) x = 0 . Let us now show that ∂ x F ε (cid:54) T >
0, there exists x (cid:62) [0 , ∞ ) × [0 ,T ] ∂ x F ε = ∂ x F ε ( x , T ) . Thanks to (3.4), we might assume further that x >
0. By the maximumprinciple, L ε [ ∂ x F ε ]( x , T ) (cid:62) ∂ x F ε ( x , T ) − ∂ x F ε ( θx , T ) (cid:62) , which yields ( ∂ x F ε ( x , T )) (cid:54) ⇒ ∂ x F ε ( x , T ) = 0 . This implies ∂ x F ε (cid:54)
0, as desired.Then, Lemma 1.6 is an immediate consequence of Lemmas 3.1 and 3.4. F in case < m < Suppose 0 < m < . Here, we always assume (A1)–(A3), and F is sublinearand 0 (cid:54) F ( x ) (cid:54) mx . Let a ∈ C ∞ ([0 , ∞ )) be a nondecreasing and concavefunction such that(3.8) a ( x ) = x , x ∈ [0 , , , x ∈ [3 , ∞ ) . For each ε >
0, let F ε be the viscosity solution to equation (3.1) correspondingto the above a . It is worth noting that in this case, (3.1) is a degenerateparabolic equation, and one needs to be careful with regularity of F ε at x = 0.Of course, F ε ∈ C ∞ ((0 , ∞ ) ), but boundary regularity is not obvious. In thefollowing, we study further properties of F ε by using the specific structure ofthe equation. Lemma 3.5.
For each ε > , let F ε be the viscosity solution to equation (3.1) with a defined as in (3.8) . Then, F ε is concave in x and (cid:54) ∂ x F ε (cid:54) m in (0 , ∞ ) . roof. For each δ >
0, consider(3.9) ∂ t F + ( ∂ x F − m )( ∂ x F − m −
1) + Fx − m = ( εa ( x ) + δ ) ∂ xx F ,F ( x,
0) = F ( x ) ,F (0 , t ) = 0 . Let F ε,δ be the unique solution to the above. Then, F ε,δ ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )).By repeating the proof of Lemma 3.1, we obtain that 0 (cid:54) ∂ x F ε,δ (cid:54) m . Ina similar fashion, ∂ x F ε,δ (0 , t ) (cid:54) t (cid:62) F ε,δ is concave in x . Indeed, replicating the proof of Lemma 3.4, we find that for some T > x > [0 , ∞ ) × [0 ,T ] ∂ x F ε,δ = ∂ x F ε,δ ( x , T ) . The maximum principle then gives us that (cid:16) ∂ x F ε,δ ( x , T ) (cid:17) (cid:54) εa ( x ) ∂ x F ε,δ ( x , T ) . Note that a ( x ) (cid:54) a is chosen to be concave. Therefore, ∂ x F ε,δ ( x , T ) (cid:54)
0. Let δ → + to get the desired results. Lemma 3.6.
For each ε > , let F ε be the viscosity solution to equation (3.1) with a defined as in (3.8) . Then, F ε ∈ C ([0 , ∞ ) ) and ∂ x F ε (0 , t ) = m . In other words, for t (cid:62) , (3.10) lim x → + x∂ x F ε ( x, t ) = 0 . Proof.
By Lemma 3.5, x ∂ x F ε ( x, t ) is decreasing in (0 , ∞ ) and 0 (cid:54) ∂ x F ε ( x, t ) (cid:54) m , and so, lim x → + ∂ x F ε ( x, t ) exists. By the L’Hopital rule, ∂ x F ε (0 , t ) = lim x → + F ε ( x, t ) − F ε (0 , t ) x = lim x → + ∂ x F ε ( x, t ) , which means that x F ε ( x, t ) is C on [0 , ∞ ) for each fixed t (cid:62)
0. Besides, bythe results of Daskalopoulos and Hamilton [DH98], Koch [Koc98], Feehan and22op [FP13], we yield further that, for each
T > F ε ∈ C βs ([0 , ∞ ) × [0 , T ]),and k F ε k C βs (cid:54) C k F k C βs for some constant C = C ( ε, T ) >
0. Here, k f k C βs def = k f k C βs + k ∂ x f k C βs + k ∂ t f k C βs + k x∂ x f k C βs , and k f k C βs def = k f k L ∞ ([0 , ∞ ) × [0 ,T ]) + sup ( x ,t ) =( x ,t )( x ,t ) , ( x ,t ) ∈ [0 , ∞ ) × [0 ,T ] | f ( x , t ) − f ( x , t ) | s (( x , t ) , ( x , t )) β . The distance s is defined as: For ( x , t ) , ( x , t ) ∈ [0 , ∞ ) , s (( x , t ) , ( x , t )) def = | x − x |√ x + √ x + q | t − t | . Let us show now that in fact ∂ x F ε (0 , t ) = m for all t (cid:62)
0. For any0 < b < b , denote by G ( x ) def = Z b b F ε ( x, t ) dt . Integrate (3.1) with respect to t ∈ [ b , b ] and let x → + to yieldlim x → + εx∂ x G ( x ) = 12 Z b b ( ∂ x F ε (0 , t ) − m )( ∂ x F ε (0 , t ) − m + 1) dt (cid:54) . Suppose by contradiction that the right hand side above is negative, which isdenoted by − C <
0. Then,lim x → + x∂ x G ( x ) = − Cε < . Thus, by the L’Hopital rule, − Cε = lim x → + x∂ x G ( x ) = lim x → + ∂ x G ( x )1 /x = lim x → ∂ x G ( x )log x . However, note that | ∂ x G ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b ∂ x F ε ( x, t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) m ( b − b ) = C , x → ∂ x G ( x )log x = 0 , which is a contradiction. Thus, we always have lim ε → + εx∂ x G ( x ) = 0 for any0 < b < b and, therefore, ∂ x F ε (0 , t ) = m for all t (cid:62)
0. This gives us (3.10)and also that lim x → + ∂ t F ε ( x, t ) = 0 = ∂ t F ε (0 , t ) . The proof is complete.
Lemma 3.7.
For each ε > , let F ε be the viscosity solution to equation (3.1) with a defined as in (3.8) . Then, for ε > sufficiently small, (3.11) x∂ x F ε (cid:62) − in (0 , ∞ ) . Proof.
We break the proof into a few steps as following.
Step 1.
Again, differentiating (3.1) twice in x , we get(3.12) (cid:18) ∂ t ∂ x F ε + (cid:20) ∂ x F ε − ( m + 12 ) (cid:21) ∂ x F ε (cid:19) + ( ∂ x F ε ) + ∂ x F ε x − ∂ x F ε x + 2 F ε x = ε (cid:18) a ∂ x F ε + 2 a ∂ x F ε + a∂ x F ε (cid:19) . Let G ε def = x∂ x F ε . By concavity of F ε in x (Lemma 3.5) and the proof of Lemma 3.6, G ε ∈C βs ([0 , ∞ ) × [0 , T ]) for each T > G ε (cid:54)
0, and G ε (0 , t ) = 0 for all t (cid:62) − (cid:54) − me (cid:54) xF ( x ) = G ε ( x, (cid:54) x (cid:62) . For t (cid:62)
0, denote by α ( t ) def = inf [0 , ∞ ) × [0 ,t ] G ε . Surely, α : [0 , ∞ ) → ( −∞ ,
0] is decreasing and bounded, and α (0) ∈ [ − , α is continuous. Fix T >
0. For s, t ∈ [0 , T ], we use theproperty G ε ∈ C βs ([0 , ∞ ) × [0 , T ]) to see that, for each x > | G ε ( x, s ) − G ε ( x, t ) | (cid:54) C | s − t | β/ , C = C ( ε, T ) >
0. Therefore, for s, t ∈ [0 , T ],(3.13) | α ( s ) − α ( t ) | (cid:54) C | s − t | β/ . Thus, α is locally Hölder continuous, and hence, is continuous on [0 , ∞ ). Itis of our goal now to show that α ( t ) (cid:62) − t ∈ [0 , ∞ ). Step 2.
Fix
T > α ( T ) < α (0). Suppose that there exists( x , t ) ∈ (0 , ∞ ) × (0 , T ] such thatmin [0 , ∞ ) × [0 ,T ] G ε ( x, t ) = G ε ( x , t ) = α ( T ) < x , t ),0 (cid:62) ∂ t G ε = x ∂ t ∂ x F ε , and(3.14) 0 = ∂ x G ε = x ∂ x F ε + ∂ x F ε ⇐⇒ ∂ x F ε = − x ∂ x F ε , and, therefore,(3.15) 0 (cid:54) ∂ x G ε ⇐⇒ x ∂ x F ε (cid:62) − x ∂ x F ε = 2 ∂ x F ε . Multiplying equation (3.12) by x and use estimate (3.15) to evaluate at( x , t ), we obtain α ( T ) + α ( T ) (cid:18) m + 32 − ∂ x F ε (cid:19) + 2( F ε − x ∂ x F ε ) x (cid:62) εα ( T ) (cid:18) a ( x ) x − a ( x ) + a ( x ) x (cid:19) (cid:62) εα ( T ) . The last inequality follows since α ( T ) (cid:54) a ,2 a ( x ) x − a ( x ) + a ( x ) x (cid:54) a ( x ) x (cid:54) . Therefore, rearranging terms, we have α ( T ) + Aα ( T ) + B (cid:62) , where A = m + 32 − ε − ∂ x F ε ( x , t ) , B = 2( F ε ( x , t ) − x ∂ x F ε ( x , t )) x . We have that, since 0 (cid:54) ∂ x F ε (cid:54) m and F ε is concave in x , for κ = m − ∂ x F ε ( x , t ), 0 (cid:54) κ (cid:54) m and 0 (cid:54) B (cid:54) κ . Therefore, 32 + κ − ε = A (cid:54) m + 32 − ε < , and 0 (cid:54) B (cid:54) κ (cid:54) m . As 0 < m < , obviously 0 < κ (cid:54) m < . For ε > A − B (cid:62) (cid:18)
32 + κ − ε (cid:19) − κ (cid:62)
94 + κ − κ − ε = (cid:18) − κ (cid:19) (cid:18) − κ (cid:19) − ε (cid:62) (cid:18) − κ (cid:19) + 4 (cid:18) − m (cid:19) − ε > (cid:18) − κ (cid:19) > . From the quadratic formula and the above estimates, we find that either α ( T ) (cid:54) − A − √ A − B , or α ( T ) (cid:62) − A + √ A − B . It is worth noting that, for ε > − A − √ A − B (cid:54) − A − (cid:16) − κ (cid:17) − ε (cid:54) − − m , and − A + √ A − B (cid:62) − A + (cid:16) − κ (cid:17) − − κ + ε (cid:62) − − m .
26e then deduce that, for each T (cid:62)
0, either(3.16) α ( T ) (cid:54) − − m , or(3.17) α ( T ) (cid:62) − − m . Surely, − − m < − − m and there is a gap of size − m between these twonumbers. Step 3.
We show that, for small enough ε >
0, only (3.17) holds for all T (cid:62)
0. Assume by contradiction that this is not the case, then there exists
T > α ( T ) (cid:54) − − m < − − m < α (0) , By the continuity of α , there exists T ε ∈ (0 , T ) so that − − m < α ( T ε ) = min [0 ,T ε ] α < − − m , which is a contradiction with the conclusion of Step 2 above.Thus, for small enough ε > x∂ x F ε ( x, t ) (cid:62) − − m > − x, t ) ∈ (0 , ∞ ) , as desired. Remark . In the use of the maximum principle in the above proof, tokeep the presentation clean, we assume that minimum points of G ε , which iscontinuous and bounded, exist on [0 , ∞ ) × [0 , T ] for T >
0. To justify thispoint rigorously, one can consider minimum of G ε ( x, t ) + δx , for δ >
0, andlet δ → + . Let us supply the details here.Pick T > α ( T ) = min [0 ,T ] α < α (0) . k ∈ N sufficiently large, we choose δ k ∈ (0 , k ) sufficiently small suchthat α ( T ) (cid:54) min [0 , ∞ ) × [0 ,T ] ( G ε ( x, t ) + δ k x ) = G ε ( x k , t k ) + δ k x k (cid:54) α ( T ) + 1 k < α (0) , for some ( x k , t k ) ∈ (0 , ∞ ) × (0 , T ]. In particular, δ k x k (cid:54) k . Let α k = G ε ( x k , t k ) ∈ (cid:18) α ( T ) , α ( T ) + 1 k (cid:19) . We use the maximum principle at ( x k , t k ) and perform careful computationsto deduce that α k + α k (cid:18) m + 32 − ∂ x F ε (cid:19) + 2( F ε − x k ∂ x F ε ) x k + δ k x k (cid:18) m + 12 + 2 εa ( x k ) − ∂ x F ε (cid:19) (cid:62) εα k (cid:18) a ( x k ) x k − a ( x k ) + a ( x k ) x k (cid:19) (cid:62) εα k . Let k → ∞ and argue in a similar way as in Step 2 of the above proof toyield that either α ( T ) (cid:54) − − m , or α ( T ) (cid:62) − − m, from which the proof follows. As this is of course tedious and distracting, weintentionally avoid putting it in the above already technical proof.We are now ready to prove one of our main regularity results that F ∈ C , ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )) when 0 < m < . Proof of Theorem 1.7.
From Lemma 3.5, Lemma 3.6 and Lemma 3.7, wehave that | ∂ x F ε | (cid:54) m , | x∂ x F ε | (cid:54) | ∂ t F ε | (cid:54) C . Thus, by the Arzelà-Ascoli theorem, there exists F in C ([0 , ∞ ) ) and a subsequence { ε i } → i →∞ F ε i = F .
By stability of viscosity solutions, F solves equation (1.3).28ow, fix x >
0. For x > x , by Lemmas 3.5 and 3.7, − x (cid:54) ∂ x F ε ( x ) (cid:54) . Letting x , x > x , we have(3.18) (cid:12)(cid:12)(cid:12)(cid:12) ∂ x F ε ( x , t ) − ∂ x F ε ( x , t ) x − x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) R x x ∂ x F ε ( x, t ) dxx − x (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) x . Thus, there exist constants
C > z >
0, such that for x > x and0 < z < z , we can uniformly bound the double difference quotient (cid:12)(cid:12)(cid:12)(cid:12) F ε ( x + 2 z, t ) − F ε ( x + z, t ) + F ε ( x, t ) z (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) Cx . Letting ε to 0, we get (cid:12)(cid:12)(cid:12)(cid:12) F ( x + 2 z, t ) − F ( x + z, t ) + F ( x, t ) z (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) Cx . This implies F is C , in x on [ x , ∞ ) × (0 , ∞ ) for all x >
0, which yieldsthat F is locally C , in x in (0 , ∞ ) . It is clear then that F is concave and F inherits estimate (3.11) from F ε , that is, − (cid:54) x∂ x F ( x, t ) (cid:54) x, t ) ∈ (0 , ∞ ) . On the other hand, differentiating equation (1.3) in x , we have ∂ t U + ∂ x U (cid:18) U − m − (cid:19) + Ux − Fx = 0in the viscosity sense, where U = ∂ x F .Now, letting x > x , by the obtained estimates on F ,0 (cid:54) U ( x, t ) (cid:54) m, (cid:54) F ( x, t ) x (cid:54) m and − x (cid:54) ∂ x U ( x, t ) (cid:54) . Therefore, there exists C = C ( x ) such that for x > x , | ∂ t U ( x, t ) | = | ∂ tx F ( x, t ) | (cid:54) C in the viscosity sense. In a similar way, differentiate the equation with respectto t to deduce that for x > x and t >
0, there exists C = C ( x ) such that | ∂ t F ( x, t ) | (cid:54) C .
Therefore, F ∈ C , ((0 , ∞ ) ), and F is concave in x . A similar argument(but easier) as that in the proof of Lemma 3.6 shows that F ∈ C ([0 , ∞ ) × (0 , ∞ )). 29 .2.3 Existence of solutions to equation (1.1) for < m (0) < We now prove the existence of mass-conserving weak solutions in the measuresense to equation (1.1) when 0 < m = m (0) < . Therefore, in thissubsection, we will always assume F is the Bernstein transform of c = c ( · , c has m (0) = m ∈ (0 , ) and also bounded second moment, that is, m (0) = Z ∞ s c ( s, ds (cid:54) C .
Our goal is to show, via a combination of the maximum principle andlocalizations around the characteristics (see Evans [Eva10, Chapter 3]), that F is a Bernstein function (see Appendix A) and, therefore, has a representationas a Bernstein transform of a measure.By Theorem 1.7, we already have that F ∈ C , ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )). Let us now use this result to yield further that F ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) ). Proposition 3.9.
Assume all the assumptions in Theorem 1.8. Then F ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) ) .Proof. We proceed by using characteristics and earlier results. Denote by X ( x, t ) the characteristic starting from x , that is, X ( x,
0) = x . Set P ( x, t ) = ∂ x F ( X ( x, t ) , t )), and Z ( t ) = F ( X ( x, t ) , t ) for all t (cid:62)
0. When there is noconfusion, we just write X ( t ) , P ( t ) , Z ( t ) instead of X ( x, t ) , P ( x, t ) , Z ( x, t ),respectively. Then, X (0) = x , P (0) = ∂ x F ( x ), Z (0) = F ( x ). We have thefollowing Hamiltonian system ˙ X = ∂ p H ( P ( t ) , Z ( t ) , X ( t )) = P ( t ) − (cid:16) m + (cid:17) , ˙ P = − ∂ x H − ( ∂ z H ) P = Z ( t ) X ( t ) − P ( t ) X ( t ) , ˙ Z = P · ∂ p H − H = P ( t ) − Z ( t ) X ( t ) + m (1 − m )2 . Note first that F ∈ C , ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )), and also 0 (cid:54) ∂ x F (cid:54) m thanks to Theorem 1.7. Therefore,(3.19) − (cid:54) − (cid:18) m + 12 (cid:19) (cid:54) ˙ X (cid:54) − . Besides, the concavity of F in x yields further that˙ P = Z ( t ) X ( t ) − P ( t ) X ( t ) = 1 X ( t ) F ( X ( t ) , t ) X ( t ) − ∂ x F ( X ( t ) , t ) ! (cid:62) . { X ( x, · ) } x ∈ (0 , ∞ ) are well-ordered in (0 , ∞ ) , andnone of these two intersect. Assume otherwise that X ( x, t ) = X ( y, t ) > x = y and t >
0. As F ∈ C , ((0 , ∞ ) ) ∩ C ([0 , ∞ ) × (0 , ∞ )), ∂ x F ( X ( x, t ) , t ) is uniquely defined, and therefore, P ( x, t ) = P ( y, t ) = ∂ x F ( X ( x, t ) , t ) and Z ( x, t ) = Z ( y, t ) = F ( X ( x, t ) , t ) . Figure 1: CharacteristicsHence, (
X, P, Z )( x, t ) = ( X, P, Z )( y, t ), and this contradicts the unique-ness of solutions to the Hamiltonian system on [0 , t ] as we reverse the time.Next, for each t >
0, let l ( t ) > X ( l ( t ) , t ) = 0. Thisis possible because of (3.19). As F is smooth, X, P, Z are smooth in x .Thanks to our Hamiltonian system and the well-ordered of { X ( x, · ) } x ∈ (0 , ∞ ) ,the map x X ( x, t ) is a smooth bijection from ( l ( t ) , ∞ ) to (0 , ∞ ). Let X − ( · , t ) : (0 , ∞ ) → ( l ( t ) , ∞ ) be the inverse of X ( · , t ).Let us show further that X ( · , t ) is a smooth diffeomorphism. It is enoughto show that X ( · , t ) : ( l ( t ) + n − , n ) → ( X ( l ( t ) + n − , t ) , X ( n, t )) is a smoothdiffeomorphism for each n ∈ N sufficiently large. Let O = n ( X ( x, s ) , s ) : x ∈ ( l ( t ) + n − , n ) , s ∈ [0 , t ] o . Thanks to Theorem 1.7, there exists
C > − C (cid:54) ∂ x F ( x, s ) (cid:54) O
31n the viscosity sense. We differentiate the first equation in the Hamiltoniansystem with respect to x and use the fact that P ( x, s ) = ∂ x F ( X ( x, s ) , s ) toyield that ∂ x ˙ X ( x, s ) = ∂ x P ( x, s ) = ∂ x F ( X ( x, s ) , s ) · ∂ x X ( x, s ) (cid:62) − C∂ x X ( x, s ) . Thus, ∂ x X ( x, s ) satisfies a differential inequality, and in particular, s e Cs ∂ x X ( x, s ) is nondecreasing on [0 , t ] . It is then clear that ∂ x X ( x, s ) > x ∈ ( l ( t ) + n − , n ) , s ∈ [0 , t ]. By theinverse function theorem, X − ( · , t ) is then smooth, and F ( x, t ) = Z ( X − ( x, t ) , t )is smooth as Z is also smooth.Let us finally use the property ˙ P (cid:62) F ∈ C ([0 , ∞ ) ). Weonly need to show that ∂ x F is continuous at (0 , ε >
0, we areable to find r > F ( x ) ∈ [ m − ε, m ] for all x ∈ [0 , r ]. Let V r = { ( y, s ) ∈ [0 , ∞ ) : y = X ( x, s ) for some x ∈ [0 , r ] and s (cid:62) } . Then, as ˙ P (cid:62)
0, we see that ∂ x F ( y, s ) ∈ [ m − ε, m ] for all ( y, s ) ∈ V r . Theproof is complete.It is worth noting that in this problem, for the characteristics, only thecondition for t = 0 is in use. The boundary condition for x = 0, though stillsatisfied, is not being used (ineffective).Now that we have F ∈ C ∞ ((0 , ∞ ) ) ∩ C ([0 , ∞ ) ), we continue to provethe last requirement to have that F is a Bernstein function. Proposition 3.10.
Assume all the assumptions in Theorem 1.8. Then, ( − n +1 ∂ nx F (cid:62) in (0 , ∞ ) for all n ∈ N . Of course, we verified the above claim already when n = 1. A maindifficulty to achieve this result is that ∂ nx F might be singular at x = 0, andthus, we do not have much knowledge on the boundary behavior there. Thisis also clear in view of the method of characteristics as described above. Hereis a way to fix this issue, which is motivated by Lemma 3.6.32 emma 3.11. We have that, for all t (cid:62) , lim x → + x∂ x F ( x, t ) = 0 . Proof.
Let Q = ∂ x F . Differentiate (1.3) with respect to x twice, we get(3.20) ∂ t Q − (cid:18) m + 12 − ∂ x F (cid:19) ∂ x Q = − Q − Qx + 2 x∂ x F − Fx . A very important point here is that (3.20) has the same characteristics X ( x, t )as in Proposition 3.9. Recall that˙ X = − (cid:18) m + 12 (cid:19) + ∂ x F ( X ( t ) , t ) , and (3.19) holds. Let R ( t ) = Q ( X ( t ) , t ), then˙ R = − R − RX + 2 XP − ZX . Since − (cid:54) x∂ x F (cid:54)
0, we infer that R (cid:54)
0, 1 + RX (cid:62)
0, and(3.21) ˙ R = − R − RX + 2 XP − ZX (cid:62) XP − ZX = 2 X (cid:18) P − ZX (cid:19) . This differential inequality about R will be used to give us the desired result.Note that F ∈ C ([0 , ∞ ) ), and for each t (cid:62) x → + ∂ x F ( x, t ) − F ( x, t ) x ! = 0 . So, for fixed
T >
0, there exists a modulus of continuity ω : (0 , ∞ ) → [0 , ∞ )with lim r → + ω ( r ) = 0 such that for all r > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ x F ( x, t ) − F ( x, t ) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) ω ( r ) for all ( x, t ) ∈ (0 , r ] × [0 , T ] . Fix r > X ( x, · ), which reaches 0 in finitetime, take s (cid:62) < X ( x, s ) (cid:54) r . For s (cid:62) s , we use this in(3.21) to get that ˙ R ( s ) (cid:62) − ω ( r ) X ( s ) . t (cid:62) s , R ( t ) (cid:62) R ( s ) − ω ( r ) Z ts X ( s ) ds (cid:62) R ( s ) − ω ( r ) Z ts − X ( s ) X ( s ) ds = R ( s ) − ω ( r ) X ( t ) − X ( s ) ! . Thus, X ( t ) R ( t ) (cid:62) X ( t ) R ( s ) − ω ( r ) . Besides, X ( t ) R ( t ) (cid:54) X ( t ) (cid:54) r and t ∈ [0 , T ],(3.22) | X ( t ) R ( t ) | (cid:54) CX ( t ) + 4 ω ( r ) , where C = max x ∈ [0 ,r ] | F ( x ) | + max t ∈ [0 ,T ] | ∂ x F ( r, t ) | . Let X ( t ) → + and r → + in this order in the above to get the conclusion. Lemma 3.12.
Fix n ∈ N with n (cid:62) , and R > . Then, there exists aconstant C = C ( n, R ) > such that (3.23) k x n − ∂ nx F ( x, t ) k L ∞ ((0 ,R ) ) (cid:54) C .
Proof.
The proof is rather tedious with a lot of terms appearing in thedifferentiations. We prove by induction with respect to j = n in (3.23). Thebase case j = 2 was already done by Theorem 1.7. Assume that (3.23) holdstrue for j = n − (cid:62)
2, and we now show that it is also true for j = n . Step 1.
Differentiate (1.3) with respect to x by n times, we get ∂ t ∂ nx F − (cid:18) m + 12 (cid:19) ∂ n +1 x F + 12 ∂ nx (cid:16) ( ∂ x F ) (cid:17) + ∂ nx (cid:18) Fx (cid:19) = 0 . Let Q = ∂ nx F . Then(3.24) ∂ t Q − (cid:18) m + 12 − ∂ x F (cid:19) ∂ x Q = f ( x, t ) , where the source term f is f ( x, t ) = − n ( ∂ x F ) Q − Qx − n − X k =2 n !( ∂ k +1 x F )( ∂ n +1 − kx F ) k !( n − k )! − n − X k =0 ( − n − k n !( ∂ kx F ) k ! x n − k +1 . X ( x, t ) as in Proposition 3.9˙ X = − (cid:18) m + 12 (cid:19) + ∂ x F ( X ( t ) , t ) , and (3.19) holds. Thanks to Lemma 3.11 and (3.22), for fixed T >
0, we areable to find a modulus of continuity ω : (0 , ∞ ) → [0 , ∞ ) with lim r → + ω ( r ) = 0such that (cid:12)(cid:12)(cid:12) x∂ x F ( x, t ) (cid:12)(cid:12)(cid:12) (cid:54) ω ( r ) for all ( x, t ) ∈ (0 , r ] × [0 , T ] . Let R ( t ) = Q ( X ( t ) , t ) and fix r >
0. As X ( t ) reaches 0 in finite time, we canpick s (cid:62) X ( s ) (cid:54) r . Surely, s = 0in case X (0) = x (cid:54) r . Without loss of generality, we assume that for some t (cid:62) s , X ( t ) >
0, and(3.25) M def = X ( t ) n − | R ( t ) | = max s ∈ [ s ,t ] X ( s ) n − | R ( s ) | > . Step 2.
It is our goal to bound X ( t ) n − R ( t ) uniformly in x . Again, withoutloss of generality, we may assume that R ( s ) does not change sign for s ∈ ( s , t ](otherwise, change s to be a bigger constant such that R ( s ) = 0 and R ( s )does not change sign for s ∈ ( s , t ]). Let us note right away that − QX = − RX is a good term and needs not to be controlled. Indeed, if R > s , t ),then − RX (cid:54) | R ( t ) | = R ( t ) = R ( s ) + Z ts f ( X ( s ) , s ) ds (cid:54) R ( s ) + Z ts − n ( ∂ x F ) R ds + Z ts − n − X k =2 n !( ∂ k +1 x F )( ∂ n +1 − kx F ) k !( n − k )! − n − X k =0 ( − n − k n !( ∂ kx F ) k ! X n − k +1 ! ds . A similar claim holds in case
R < s , t ). A key point that we need herein order to bound the above complicated sum is that, for i (cid:62)
2, by (3.19)(3.27) Z ts X ( s ) i ds (cid:54) Z ts − X ( s ) X ( s ) i ds (cid:54) i − X ( t ) i − − r i − ! . This, together with the induction hypothesis, gives us that(3.28) X ( t ) n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ts − n − X k =2 n !( ∂ k +1 x F )( ∂ n +1 − kx F ) k !( n − k )! − n − X k =0 ( − n − k n !( ∂ kx F ) k ! X n − k +1 ! ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) C . R . As − ω ( r ) (cid:54) x∂ x F (cid:54) , r ] × [0 , T ], one has(3.29) n (cid:12)(cid:12)(cid:12)(cid:12)Z ts ( ∂ x F ) R ds (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) Z ts nω ( r ) MX ( s ) n ds (cid:54) nω ( r ) Mn − X ( t ) n − − r n − ! (cid:54) ω ( r ) MX ( t ) n − (cid:54) M X ( t ) n − for r > M (cid:54) C + M , which yields that M (cid:54) C . By definition of M , X ( t ) and R ( t ), we reach thedesired result.We are now ready to prove Proposition 3.10 by induction. Our idea hereis to use the maximum principle for x k − ∂ kx F for k (cid:62) x k − ∂ kx F is unclear as x → + , weneed to use localizations around characteristics to take care of this issue. Proof of Proposition 3.10.
Let us show that ( − n +1 ∂ nx F (cid:62) , ∞ ) by induction. By Theorem 1.7, this is true for n = 2 already. Assume thatthis is true for all n (cid:54) k − k (cid:62)
3. We now show that this is truefor n = k . Let us just deal with the case that k is even as the other case canbe done analogously. Step 1.
Differentiate (1.3) with respect to x by k times, we get(3.30) ∂ t ∂ kx F − (cid:18) m + 12 (cid:19) ∂ k +1 x F + 12 ∂ kx (cid:16) ( ∂ x F ) (cid:17) + ∂ kx (cid:18) Fx (cid:19) = 0 . Let W ( x, t ) = x k − ∂ kx F , and we aim at deriving a PDE for W . As always,the last term on the left hand side above is not so easy to deal with. Thefollowing is a new insight to handle this term thanks to Lemma 3.12, x k − ∂ kx (cid:18) Fx (cid:19) = x k − ∂ kx (cid:18)Z ∂ x F ( rx, t ) dr (cid:19) = x k − Z r k ∂ k +1 x F ( rx, t ) dr = 1 x Z x z k ∂ k +1 x F ( z, t ) dz W ( x, t ) x − kx Z x W ( z, t ) dz . We used integration by parts in the last equality above. Multiply (3.30) by x k − and use the above identity, we arrive at(3.31) ∂ t W − (cid:18) m + 12 − ∂ x F (cid:19) (cid:18) ∂ x W − ( k − Wx (cid:19) + Wx − kx Z x W ( z, t ) dz = − k ( ∂ x F ) W − x k − k − X i =2 k !( ∂ i +1 x F )( ∂ k +1 − ix F ) i !( k − i )! . Again, this equation has the same characteristics X ( x, t ) as in Proposition 3.9,˙ X = − (cid:18) m + 12 (cid:19) + ∂ x F ( X ( t ) , t )and (3.19) holds. This clear localization of characteristics is very important. Step 2.
We now need to show that W (cid:54) , ∞ ) . Assume bycontradiction that there exists ( x , T ) ∈ (0 , ∞ ) such that W ( x , T ) >
0. Ofcourse, x = X ( z, T ) for some z > x .For the initial condition of W , it is not hard to see that W (0 ,
0) = 0and W ( x, (cid:54) x ∈ [0 , ∞ ). Choose z , z very close to z such that z < z < z , and define a new initial condition f W ( · , , ∞ ), such that f W ( x,
0) = W ( x,
0) for x ∈ [ z , z ] , f W ( x, (cid:54) W ( x,
0) for x / ∈ [ z , z ] . Let f W be the solution to (3.31) corresponding to this new initial condition f W ( · , f W ( x , T ) = W ( x , T ). In fact, we can choose f W ( · ,
0) to be as negative as we wish outsideof [ z , z ]. For our purpose, we choose z , z , and f W ( · ,
0) so that(3.32) f W ( x, t ) < f W ( X ( z, t ) , t ) for all x ∈ (cid:18) , x (cid:21) ∪ [ z + 1 , ∞ ) , t ∈ [0 , T ] . Now, slightly abusing the notations, let us assume that W satisfies (3.32)as well (in other words, write W in place of f W for simplicity). For each t ∈ [0 , T ], by (3.32), there exists x t ∈ (cid:16) x , z + 1 (cid:17) so that ξ ( t ) def = max x ∈ [0 , ∞ ) W ( x, t ) = W ( x t , t ) .
37e use the maximum principle in (3.31) to get an estimate for ξ . Noticethat, as k is even, ( ∂ i +1 x F )( ∂ k +1 − ix F ) (cid:62) (cid:54) i (cid:54) k − x t , t ), we have ∂ x W ( x t , t ) = 0, and1 x t Z x t W ( z, t ) dz (cid:54) W ( x t , t ) . Therefore, ξ ( t ) + ξ ( t ) x t (cid:18) ( k − (cid:18) m − − ∂ x F (cid:19) + kx t ∂ x F ( x t , t ) (cid:19) (cid:54) . Note that x t ∈ (cid:16) x , z + 1 (cid:17) , and (cid:12)(cid:12)(cid:12)(cid:12) ( k − (cid:18) m − − ∂ x F (cid:19) + kx t ∂ x F ( x t , t ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) k . As ξ (0) (cid:54)
0, by the usual differential inequality, we get that ξ ( t ) (cid:54) t ∈ [0 , T ]. In particular, 0 (cid:62) ξ ( T ) (cid:62) W ( x , T ) >
0, which is absurd. Theproof is complete.
Proof of Theorem 1.8.
The result follows immediately by combining Propo-sitions 3.9, 3.10 and Theorem A.2.
In this section, we study the equilibria of equation (3.1) in the case 0 < m (cid:54) ∂ x F − m )( ∂ x F − m −
1) + Fx − m = 0 . Let us emphasize again that we search for Lipschitz, sublinear viscositysolution F which satisfies 0 (cid:54) F ( x ) (cid:54) mx for x ∈ [0 , ∞ ). Lemma 4.1.
Suppose < m < . Let F be a Lipschitz, sublinear viscositysolution to equation (4.1) which satisfies (cid:54) F ( x ) (cid:54) mx for x ∈ [0 , ∞ ) .Then, there exists a constant C > so that all the local minimums of F belongto [0 , C ] . roof. By contradiction, if there exists a sequence of local minimums { x n } →∞ of F , then by the supersolution test, we have12 m ( m + 1) + F ( x n ) x n − m (cid:62) . This means, for n ∈ N , F ( x n ) x n (cid:62) m (1 − m ) > , which is a contradiction as F ( x n ) /x n → Proposition 4.2.
Suppose < m < . Then equation (4.1) has no Lipschitz,sublinear viscosity solution F which satisfies (cid:54) F ( x ) (cid:54) mx for x ∈ [0 , ∞ ) .Proof. Suppose by contradiction that F is a Lipschitz, sublinear solution toequation (4.1) and 0 (cid:54) F ( x ) (cid:54) mx for x ∈ [0 , ∞ ). By Lemma 4.1, thereexists a C > F ( x ) is monotone on [ C, ∞ ), i.e., for a.e. x ∈ [ C, ∞ ),either F ( x ) (cid:62) F ( x ) (cid:54) . Let us consider two cases in the following.
Case 1. F ( x ) (cid:62) x (cid:62) C . Since F ( x ) (cid:54) mx , we have12 ( F ( x ) − m )( F ( x ) − m −
1) = m − F ( x ) x (cid:62) . Thus, either F ( x ) (cid:54) m or F ( x ) (cid:62) m + 1. We claim that F ( x ) (cid:54) m fora.e. x (cid:62) C by changing C to be a bigger value if needed. Indeed, assumeotherwise, that this is not the case. Since F ( x ) (cid:54) mx , we cannot have that F ( x ) (cid:62) m + 1 for a.e. x > C . Then, we can find x > x > C such that F ( x ) (cid:54) m , and F ( x ) (cid:62) m + 1. Let φ ( x ) = ( m + ) x for x ∈ [ x , x ] be atest function, and let x ∈ [ x , x ] be a minimum point of F − φ on [ x , x ].As F ( x ) (cid:54) m < φ ( x ) = m + 12 = φ ( x ) < m + 1 (cid:54) F ( x ) , it is clear that x = x and x = x . In other words, x ∈ ( x , x ), and one isable to use the viscosity supersolution test to yield that0 (cid:54) (cid:18) m + 12 − m (cid:19) (cid:18) m + 12 − m − (cid:19) + F ( x ) x − m (cid:54) − , (cid:54) F ( x ) (cid:54) m for a.e. x (cid:62) C .
In particular, for a.e. x (cid:62) C ,12 ( F ( x ) − m )( F ( x ) − m − (cid:54)
12 (0 − m )(0 − m −
1) = 12 m ( m + 1) , which implies F ( x ) x (cid:62) m − m ( m + 1) = 12 m (1 − m ) > . But this means that F is not sublinear. Case 2. F ( x ) (cid:54) x (cid:62) C . Then F is decreasing on [ C, ∞ ) andthere exists α (cid:62) α = lim x →∞ F ( x ). Consequently,lim x →∞ (cid:18)
12 ( F ( x ) − m )( F ( x ) − m − − m (cid:19) = 0 . On the other hand, as F (cid:62) { y n } → ∞ suchthat F ( y n ) →
0. Let x = y n in the above and let n → ∞ to deduce that0 = 12 (0 − m )(0 − m − − m = 12 m ( m − < , which is absurd.Therefore, in all cases, we are led to contradictions. The proof is complete. Proposition 4.3.
Let m = 1 . Then equation (4.1) admits a Lipschitz,sublinear viscosity solution F which satisfies (cid:54) F ( x ) (cid:54) mx for x ∈ [0 , ∞ ) .Proof. Let G = 1 − ∂ x F . Then the equilibrium equation reads as(4.2) 12 G ( G + 1) − x Z x G = 0 . This is the same equation studied in the work of Degond, Liu and Pego [DLP17,Section 5], of which the solution must satisfy the transcendental equation(4.3) G ( x )(1 − G ( x )) = Cx ,
C >
0. Let us recall a quick proof of (4.3). Multiply (4.2)by x , then differentiate the result with respect to x to imply12 G ( G + 1) + 12 x (2 G∂ x G + ∂ x G ) − G = 0 , which means that 1 x = 3 ∂ x G − G + ∂ x GG .
Integrate the above to yield (4.3). Therefore, we can pick C = 1 in (4.3) and G to be a Bernstein function taking the form(4.4) G ( x ) = Z ∞ (1 − e − sx ) γ ( s ) e − s/ ds , where Z ∞ γ ( s ) e − s/ ds = 1 . See [DLP17, Section 5] for further details on the derivation of (4.4). Thisimplies that(4.5) ∂ x F ( x ) = 1 − Z ∞ (1 − e − sx ) γ ( s ) e − s/ ds (cid:62) , and that lim x →∞ F ( x ) x = 0 . Furthermore, by successive differentiations, we can also see that ∂ x F iscompletely monotone, that is, ( − n +1 ∂ nx F (cid:62) n ∈ N , which meansthat F is a Bernstein function. Remark . From the above proposition, it is actually not hard to see that,for m = 1, equation (4.1) admits a family of Lipschitz, sublinear viscositysolution { F λ } λ> which satisfies 0 (cid:54) F λ ( x ) (cid:54) x for x ∈ [0 , ∞ ). Indeed, take F as in the above proof, and denote by F λ ( x ) = λF (cid:18) xλ (cid:19) for all x ∈ [0 , ∞ ) . Then, G λ ( x ) = 1 − ∂ x F λ ( x ) = 1 − ∂ x F (cid:18) xλ (cid:19) = G (cid:18) xλ (cid:19) , G λ ( x )(1 − G λ ( x )) = xλ . This implies that (4.3) is satisfied with C = λ . Hence, F λ is a solution to(4.1) for each λ > { F λ } λ> to (4.1) makes the studyof large time behavior of the viscosity solution to (1.3) for m = 1 quitedifficult. < m < In this section, we study the large time behavior of the viscosity solution toequation (1.3) for 0 < m <
1. Our goal is to prove Theorem 1.9.From Proposition 4.2, one cannot expect a sublinear equilibrium, that is,a Lipschitz sublinear solution to (4.1). However, it is very interesting thatthe solution to equation (1.3) still converges to the linear function mx locallyuniformly as t → ∞ . This implies that, even if we have a mass-conservingsolution at all time, the sizes of particles decrease until they become dust attime infinity.To prove the theorem, we need the following results. Lemma 5.1.
Let ¯ F be a viscosity supersolution to equation (4.1) that satisfiesthe following (5.1) ¯ F is concave , lim inf x →∞ ¯ F ( x ) x > , (cid:54) ¯ F ( x ) (cid:54) mx . Then, ¯ F ( x ) = mx .Proof. First, observe that x ∂ x ¯ F ( x ) is decreasing whenever ∂ x ¯ F ( x ) isdefined. By the requirement thatlim inf x →∞ ¯ F ( x ) x > , we have that ∂ x ¯ F ( x ) (cid:62)
0. As ¯ F is differentiable almost everywhere, pick { x n } → ∞ so that F is differentiable at x n for all n ∈ N . Denote0 < α def = lim n →∞ ∂ x ¯ F ( x n ) = lim x →∞ ¯ F ( x n ) x n (cid:54) m . x n → ∞ in the equation (4.1), we get0 (cid:54)
12 ( α − m )( α − m + 1) (cid:54) . Therefore, it is necessary that α = m and ¯ F ( x ) = mx for all x ∈ [0 , ∞ ).We immediately have the following consequence. Corollary 5.2.
Let ¯ F be a viscosity solution to equation (4.1) satisfying (5.1) . Then ¯ F ( x ) = mx for x ∈ [0 , ∞ ) . Lemma 5.3.
Let F be the Lipschitz, sublinear viscosity solution to equa-tion (1.3) . Then, locally uniformly for x ∈ [0 , ∞ ) , (5.2) lim inf t →∞ F ( x, t ) (cid:62) m (1 − m ) x . Proof.
We construct a sublinear subsolution to the equation (1.3) so that theinequality (5.2) holds. Define, for ( x, t ) ∈ [0 , ∞ ) , ϕ ( x, t ) def = min (cid:26) m (1 − m ) x, m (1 − m ) t (cid:27) . To see that ϕ is a subsolution to (1.3), we first note that m (1 − m ) x is asubsolution. Furthermore, ϕ ( x, t ) = m (1 − m ) x , x < t , m (1 − m ) t , x (cid:62) t . So, for x > t , ∂ t ϕ + 12 ( ∂ x ϕ − m )( ∂ x ϕ − m −
1) + ϕx − m (cid:54) m (1 − m ) + 12 m ( m −
1) + 14 m (1 − m ) = 0 . Since equation (1.3) has a convex Hamiltonian, minimum of two subsolutionsis a subsolution (see Tran [Tra19, Chapter 2] and the references therein).Note that this property is not true for general Hamiltonians.By the comparison principle, we have that F (cid:62) ϕ . Letting t → ∞ , weobtain (5.2) locally uniformly for x ∈ [0 , ∞ ).43 roof of Theorem 1.9. By Lemma 5.3, locally uniformly for x ∈ (0 , ∞ ), m (cid:62) lim inf t →∞ F ( x, t ) x (cid:62) m (1 − m ) > . Let G ( x ) def = lim inf t →∞ F ( x, t ) for all x ∈ [0 , ∞ ) . This function is well-defined since F is globally Lipschitz on [0 , ∞ ) and0 (cid:54) F ( x, t ) (cid:54) mx . By stability of viscosity solutions, G is a supersolution toequation 12 ( ∂ x G − m )( ∂ x G − m −
1) + Gx − m (cid:62) , ∞ ). As x F ( x, t ) is concave for every t (cid:62) G is concave. Moreover,0 (cid:54) G (cid:54) mx and G ( x ) (cid:62) m (1 − m ) x for all x ∈ [0 , ∞ ) . By Lemma 5.1, G ( x ) = mx for x ∈ [0 , ∞ ). We use this and the fact that F ( x, t ) (cid:54) mx for all ( x, t ) ∈ [0 , ∞ ) to conclude that, locally uniformly for x ∈ [0 , ∞ ), lim t →∞ F ( x, t ) = G ( x ) = mx , as desired.It is worth noting that (5.2) is only useful for 0 < m <
1, and is meaninglesswhen m = 1. Large time behavior of F in case m = 1 remains an openproblem. A Bernstein functions and transform
In this appendix, we record a representation theorem of Bernstein functions,which is important for the inference of existence of solutions to equation (1.1)from smooth solution of equation (1.3).
Definition A.1.
A function f : (0 , ∞ ) → [0 , ∞ ) is a Bernstein function if f ∈ C ∞ ((0 , ∞ )) and, for n ∈ N ,( − n +1 d n dx n f (cid:62) . heorem A.2. A function f : (0 , ∞ ) → [0 , ∞ ) is a Bernstein function ifand only if it can be written as (A.1) f ( x ) = a x + a ∞ + Z (0 , ∞ ) (1 − e − sx ) µ ( ds ) , x ∈ (0 , ∞ ) , where a , a ∞ (cid:62) and µ is a measure such that Z (0 , ∞ ) min { , s } µ ( ds ) < ∞ . In other words, a Bernstein function is a Bernstein transform on theextended real line [0 , ∞ ]. The proof of this theorem and more beautifulproperties of Bernstein functions and transform could be found in the bookby Schilling, Song, and Vondraček [SSV12].Next, consider f : [0 , ∞ ) → [0 , ∞ ) which is a Bernstein function suchthat f (0) = 0 and f is sublinear. By Theorem A.2, f has the representationformula (A.1). Firstly, let x → + to get that a ∞ = lim x → + f ( x ) = 0 . Secondly, divide (A.1) by x , let x → ∞ and use the sublinearity of F to yieldfurther that a = lim x →∞ f ( x ) x = 0 . Thus, under two additional conditions that f (0) = 0 and f is sublinear, weget that a = a ∞ = 0, and therefore, f ( x ) = Z (0 , ∞ ) (1 − e − sx ) µ ( ds ) , x ∈ (0 , ∞ ) . Acknowledgement
TSV thanks Bob Pego and Philippe Laurençot for the introduction to the C-Fas well as many helpful discussions. Proposition 4.3 and Remark 4.4 are basedon an observation of Laurençot through private communications. TSV alsothanks Hausdorff Research Institute for Mathematics (Bonn), through theJunior Trimester Program on Kinetic Theory, for their welcoming environmentduring the period that he learned most about the questions about C-Faddressed in this paper. HT thanks Tuoc Phan for a discussion on regularity45f degenerate parabolic equations, and Hiroyoshi Mitake for some usefulsuggestions. We also would like to thank the anonymous referee very much forcarefully reading our manuscript and giving very helpful comments to improveits presentation. HT is supported in part by NSF grant DMS-1664424 andNSF CAREER grant DMS-1843320.
References [AB79] Michael Aizenman and Thor A. Bak. “Convergence to equilibriumin a system of reacting polymers”. In:
Comm. Math. Phys. issn : 0010-3616. url : http://projecteuclid.org/euclid.cmp/1103904874 .[Ald99] David J. Aldous. “Deterministic and stochastic models for coales-cence (aggregation and coagulation): a review of the mean-fieldtheory for probabilists”. In: Bernoulli issn :1350-7265. doi : .[BC90] J. M. Ball and J. Carr. “The discrete coagulation-fragmentationequations: existence, uniqueness, and density conservation”. In: J. Statist. Phys. issn : 0022-4715. doi : . url : https://doi.org/10.1007/BF01013961 .[Ber06] Jean Bertoin. Random fragmentation and coagulation processes .Vol. 102. Cambridge Studies in Advanced Mathematics. CambridgeUniversity Press, Cambridge, 2006, pp. viii+280. isbn : 978-0-521-86728-3; 0-521-86728-2. doi : .[BLL19] Jacek Banasiak, Wilson Lamb, and Philippe Laurencot. Analyticmethods for coagulation-fragmentation models . Vol. 1&2. CRCPress, 2019.[BT45] P. J. Blatz and A. V. Tobolsky. “Note on the Kinetics of Sys-tems Manifesting Simultaneous Polymerization-DepolymerizationPhenomena”. In:
The Journal of Physical Chemistry doi :
10 . 1021 / j150440a004 . eprint: https : / / doi .org/10.1021/j150440a004 .46Cañ07] José A. Cañizo. “Convergence to equilibrium for the discretecoagulation-fragmentation equations with detailed balance”. In:
J.Stat. Phys. issn : 0022-4715. doi : . url : https://doi.org/10.1007/s10955-007-9373-2 .[Car92] J. Carr. “Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case”. In: Proc. Roy. Soc. Edinburgh Sect. A issn : 0308-2105. doi : . url : https://doi.org/10.1017/S0308210500027888 .[CEL84] M. G. Crandall, L. C. Evans, and P.-L. Lions. “Some propertiesof viscosity solutions of Hamilton-Jacobi equations”. In: Trans.Amer. Math. Soc. issn : 0002-9947. doi : . url : https://doi.org/10.2307/1999247 .[CL83] Michael G. Crandall and Pierre-Louis Lions. “Viscosity solutionsof Hamilton-Jacobi equations”. In: Trans. Amer. Math. Soc. issn : 0002-9947. doi : . url : https://doi.org/10.2307/1999343 .[Cos15] F. P. da Costa. “Mathematical aspects of coagulation-fragmentationequations”. In: Mathematics of energy and climate change . Vol. 2.CIM Ser. Math. Sci. Springer, Cham, 2015, pp. 83–162.[DH98] P. Daskalopoulos and R. Hamilton. “Regularity of the free bound-ary for the porous medium equation”. In:
J. Amer. Math. Soc. issn : 0894-0347. doi : . url : https://doi.org/10.1090/S0894-0347-98-00277-X .[DLP17] Pierre Degond, Jian-Guo Liu, and Robert L Pego. “Coagulation–Fragmentation Model for Animal Group-Size Statistics”. In: Jour-nal of Nonlinear Science
Comm. Math. Phys. issn : 0010-3616. doi : . url : https://doi.org/10.1007/s00220-002-0680-9 .47Esc+03] M. Escobedo, Ph. Laurençot, S. Mischler, and B. Perthame. “Gela-tion and mass conservation in coagulation-fragmentation models”.In: J. Differential Equations issn : 0022-0396. doi : .[Eva10] Lawrence C. Evans. Partial differential equations . Second. Vol. 19.Graduate Studies in Mathematics. American Mathematical Soci-ety, Providence, RI, 2010, pp. xxii+749. isbn : 978-0-8218-4974-3. doi : . url : https://doi.org/10.1090/gsm/019 .[FP13] Paul M. N. Feehan and Camelia A. Pop. “A Schauder approach todegenerate-parabolic partial differential equations with unboundedcoefficients”. In: J. Differential Equations issn : 0022-0396. doi : . url : https://doi.org/10.1016/j.jde.2013.03.006 .[GMT16] Yoshikazu Giga, Hiroyoshi Mitake, and Hung V. Tran. “On Asymp-totic Speed of Solutions to Level-Set Mean Curvature Flow Equa-tions with Driving and Source Terms”. In: SIAM J. Math. Anal. doi : .[Koc98] H. Koch. Non-Euclidean Singular Integrals and the Porous MediumEquation . Verlag nicht ermittelbar, 1998. url : https://books.google.com/books?id=nNjnPgAACAAJ .[Kry96] N. V. Krylov. Lectures on elliptic and parabolic equations in Hölderspaces . Vol. 12. Graduate Studies in Mathematics. American Math-ematical Society, Providence, RI, 1996, pp. xii+164. isbn : 0-8218-0569-X. doi :
10 . 1090 / gsm / 012 . url : https : / / doi . org / 10 .1090/gsm/012 .[Lau19a] Philippe Laurençot. “Mass-conserving solutions to coagulation-fragmentation equations with balanced growth”. In: arXiv:1901.08313[math.AP] (2019). url : https://arxiv.org/abs/1901.08313 .[Lau19b] Philippe Laurençot. “Stationary solutions to coagulation-fragmentationequations”. In: Annales de l’Institut Henri Poincaré C, Anal-yse non linéaire (2019). issn : 0294-1449. doi : https : / / doi .org / 10 . 1016 / j . anihpc . 2019 . 06 . 003 . url : .48Lie96] Gary M. Lieberman. Second order parabolic differential equa-tions . World Scientific Publishing Co., Inc., River Edge, NJ, 1996,pp. xii+439. isbn : 981-02-2883-X. doi :
10 . 1142 / 3302 . url : https://doi.org/10.1142/3302 .[LNP19] Jian-Guo Liu, B. Niethammer, and Robert L. Pego. “Self-similarspreading in a merging-splitting model of animal group size”. In: J. Stat. Phys. issn : 0022-4715. doi :
10 . 1007 / s10955 - 019 - 02280 - w . url : https : / / doi. org / 10 .1007/s10955-019-02280-w .[LSU68] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva. Linearand quasilinear equations of parabolic type . Translated from theRussian by S. Smith. Translations of Mathematical Monographs,Vol. 23. American Mathematical Society, Providence, R.I., 1968,pp. xi+648.[McL62] J. B. McLeod. “On an infinite set of non-linear differential equa-tions”. In:
Quart. J. Math. Oxford Ser. (2)
13 (1962), pp. 119–128. issn : 0033-5606. doi : .[Mel57] Z. A. Melzak. “A scalar transport equation”. In: Trans. Amer.Math. Soc.
85 (1957), pp. 547–560. issn : 0002-9947. doi : .[MP04] Govind Menon and Robert L. Pego. “Approach to self-similarityin Smoluchowski’s coagulation equations”. In: Comm. Pure Appl.Math. issn : 0010-3640. doi : .[MP06] Govind Menon and Robert L. Pego. “Dynamical scaling in Smolu-chowski’s coagulation equations: uniform convergence”. In: SIAMRev. issn : 0036-1445. doi : .[MP08] Govind Menon and Robert L. Pego. “The scaling attractor andultimate dynamics for Smoluchowski’s coagulation equations”. In: J. Nonlinear Sci. issn : 0938-8974. doi : .49Nor99] James R. Norris. “Smoluchowski’s coagulation equation: unique-ness, nonuniqueness and a hydrodynamic limit for the stochasticcoalescent”. In: Ann. Appl. Probab. issn :1050-5164. doi : .[NV13] B. Niethammer and J. J. L. Velázquez. “Self-similar solutionswith fat tails for Smoluchowski’s coagulation equation with locallybounded kernels”. In: Comm. Math. Phys. issn : 0010-3616. doi : . url : https://doi.org/10.1007/s00220-012-1553-5 .[Smo16] M. V. Smoluchowski. “Drei Vortrage uber Diffusion, BrownscheBewegung und Koagulation von Kolloidteilchen”. In: Zeitschriftfur Physik
17 (1916), pp. 557–585.[SSV12] René L. Schilling, Renming Song, and Zoran Vondraček.
Bernsteinfunctions . Second. Vol. 37. De Gruyter Studies in Mathematics.Theory and applications. Walter de Gruyter & Co., Berlin, 2012,pp. xiv+410. isbn : 978-3-11-025229-3; 978-3-11-026933-8. doi :
10 . 1515 / 9783110269338 . url : https : / / doi . org / 10 . 1515 /9783110269338 .[Tra19] Hung V. Tran. Hamilton–Jacobi equations: viscosity solutionsand applications . 2019. url : http://math.wisc.edu/~hung/HJ%20equations-viscosity%20solutions%20and%20applications-v2.pdf .[VZ89] R. D. Vigil and R. M. Ziff. “On the stability of coagulation–fragmentation population balances”. In: Journal of Colloid andInterface Science
133 (Nov. 1989), pp. 257–264. doi :10.1016/0021-9797(89)90300-7