The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel
aa r X i v : . [ m a t h . A P ] O c t The scaling hypothesis for Smoluchowski’s coagulation equationwith bounded perturbations of the constant kernel
Jos´e A. Ca˜nizo ∗ Sebastian Throm ∗ October 2019
Abstract
We consider Smoluchowski’s coagulation equation with a kernel of the form K = 2+ ǫW ,where W is a bounded kernel of homogeneity zero. For small ǫ , we prove that solutionsapproach a universal, unique self-similar profile for large times, at almost the same speedas the constant kernel case (the speed is exponential when self-similar variables are consid-ered). All the constants we use can be explicitly estimated. Our method is a constructiveperturbation analysis of the equation, based on spectral results on the linearisation of theconstant kernel case. To our knowledge, this is the first time the scaling hypothesis can befully proved for a family of kernels which are not explicitly solvable. Contents C K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Existence of time-dependent solutions and moment estimates . . . . . . . . . . 92.6 Stability of time-dependent solutions with respect to perturbations . . . . . . . 102.7 Asymptotic behaviour of solutions for the constant kernel . . . . . . . . . . . . 11 L estimate . . . . . . . . . . 153.4 Stability of profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ∗ Departamento de Matem´atica Aplicada, Universidad de Granada, 18071 Granada, Spain.
Email addresses: [email protected] , [email protected] Tools on the spectral gap of linear operators 23 L spaces 26 L spaces . . . . . . . . . . . . . . . 266.2 Extension of the spectral gap to weighted L spaces . . . . . . . . . . . . . . . 28 A Proof of Theorem 2.16 on L convergence for the constant kernel 40B An estimate on the Fourier transform 41C Proof of Lemma 4.6 on the transport semigroup 44 We study the long-time behaviour of solutions to Smoluchowski’s coagulation equation, whichreads ∂ τ φ ( τ, ξ ) = 12 Z ξ K ( ξ − η, η ) φ ( τ, ξ − η ) φ ( τ, η ) d η − φ ( τ, ξ ) Z ∞ K ( ξ, η ) φ ( τ, η ) d η . (1.1)This equation is a well-known model for coagulation processes in several contexts such asaerosol dynamics [14, 31], aggregation in planetary formation [3, 32] and biology [1, 2]. Theunknown φ = φ ( τ, ξ ) ≥ ξ > τ ≥
0, and K = K ( ξ, η ) = K ( η, ξ ) ≥ coagulation kernel giving the coagulation rateof clusters of size ξ with clusters of size η . We always consider the continuum version of thisequation, so the size ξ can take any positive value. A long-standing conjecture is that all(finite-mass, suitably decaying) solutions to (1.1) approach a universal self-similar shape astime τ → + ∞ , as long as K ( ξ, η ) is a homogeneous function of homogeneity degree γ ≤ K ( λξ, λη ) = λ γ K ( ξ, η ) for all λ, ξ, η ∈ (0 , ∞ )); this is known as the scaling hypothesis . Moreprecisely, one expects that there exists a self-similar profile G and a scaling function s ( τ ) → ∞ as τ → ∞ such that (cid:0) s ( τ ) (cid:1) φ (cid:0) τ, s ( τ ) ξ (cid:1) −→ G ( ξ ) as τ → ∞ , (1.2)in a suitable sense to be determined. This was established in the particular cases K ( ξ, η ) = 2(constant) and K ( ξ, η ) = ξ + η (linear) in [18,19] in the sense of weak convergence, with explicitrates given in [7, 33]. Convergence in stronger norms for the constant kernel was also foundin [7]. There is also a theory of fat-tailed profiles, which represent the asymptotic behaviourof solutions with slowly decaying tails. We do not consider them in this work, and we refer2he reader to [18] for explicitly solvable kernels, and to [26, 28, 34] for results on existence anduniqueness of fat-tailed self-similar profiles with infinite mass.In this paper we are able to prove the scaling hypothesis in the regime of finite mass, withan explicit rate, for small bounded perturbations of the constant kernel. That is, we considerkernels of the type K = K ε ( ξ, η ) = 2 + εW ( ξ, η ) , (1.3)where ε > W : (0 , + ∞ ) × (0 , + ∞ ) → R must be continuous, symmetric in ξ, η , satisfy the bound 0 ≤ W ( ξ, η ) ≤ ξ, η >
0, (1.4)and be homogeneous of degree zero: W ( λξ, λη ) = W ( ξ, η ) for all ξ, η, λ >
0. (1.5)Suitable examples of W include W ( ξ, η ) = Ψ( ξ α η − α + ξ − α η α ) , where α ∈ R and Ψ : (0 , + ∞ ) → [ − ,
1] is any continuous function. Averages of functions ofthis type for different α are also examples of coefficients satisfying (1.3)–(1.5). Remark . The choice of K = 2 is made for convenience, since by simple scaling argumentsone can consider perturbations of any constant kernel (see Section 2.2). Consequently, alsothe lower bound in (1.4) can be slightly weakened, i.e. our result also holds for perturbationswhich may change sign, satisfying | W ( x, y ) | ≤
1. In fact, replacing the constant kernel by2 − k W k L ∞ and the perturbation by f W = W + k W k L ∞ the assumption (1.4) is satisfied.We prove that for ε small enough, solutions to (1.1) approach a unique, universal self-similarprofile at an explicit algebraic rate (which becomes exponential when self-similar variables areconsidered; see below), in the sense of the k · k L k norm defined by k f k L k := Z ∞ | f ( x ) | (1 + x ) k d x . Our main result is summarised in the following theorem:
Theorem 1.2.
Let K = K ε be a bounded perturbation of the constant kernel satisfying (1.3) , (1.4) and (1.5) .1. There exists ε > such that for ≤ ε ≤ ε there exists a unique self-similar profile G ε with unit mass.2. Given R > and k > , there exists < ε ≤ ε (depending only on R and k ) and M (depending only on k ) such that for ≤ ε < ε any solution φ to the Smoluchowskiequation (1.1) with nonnegative initial condition φ such that Z ∞ ξφ ( ξ ) d ξ = 1 , Z ∞ (cid:12)(cid:12) φ ( ξ ) − G ε ( ξ ) (cid:12)(cid:12) (1 + ξ ) k d ξ ≤ R satisfies k f ( t, · ) − G ε k L k ≤ Ce − λ ε t k f − G ε k L k for all t ≥ , ith λ ε := − M ε , for some
C > depending on k and R , and where f ( t, x ) := e t φ ( e t − , xe t ) . Equivalently, Z ∞ (cid:12)(cid:12)(cid:12) ( τ + 1) φ ( τ, ( τ + 1) x ) − G ε ( x ) (cid:12)(cid:12)(cid:12) (1 + x ) k d x ≤ C (1 + τ ) − λ ε k φ − G ε k L k for all τ ≥ .All constants appearing in this theorem can be explicitly estimated. As far as we know, this is the first time the scaling hypothesis can be proved to hold forkernels which do not allow for an explicit solution of equation (1.1). Results for the so-called diagonal kernels were obtained by [17], and in this case the approach to self-similarity does nothappen for all initial conditions. The part of our result on uniqueness of the profiles is not new:it has been proved for more general perturbations by Laplace transform methods in [25, 27, 30]and via compactness arguments in L in [35]. An improvement here is that we are able to givean explicit estimate of ε in the case of bounded perturbations, and explicit estimates on thecloseness of G ε to G = e − x , the self-similar profile for the equation with K = 2. For kernels K with negative homogeneity degree it has been recently proved that self-similar profiles areunique [16].The strategy to prove our results is a perturbation argument, carried out in a constructiveway, using the fact that the case of constant coefficients is fairly well understood. This has beendone for kinetic equations involving the Boltzmann operator in [5, 20, 21, 23], but has not beendone for coagulation-type equations as far as we know. Results on convergence to equilibriumfor the Becker-D¨oring equation were developed in [4] using properties of the linearised operator,with techniques similar to those in Section 8. We need three main ingredients in order tocomplete our perturbation arguments:1. First, we need a good global exponential convergence result for the constant coefficientscase. Convergence without rate is known since [18], and in order to obtain rates onecan use results in [7, 33]. It turns out that a global convergence in an L or L space ismore convenient for us, so we use a refined version of the results in [7]. These results aregiven in Section 2.7, with precise estimates on the dependence of the constants given inAppendix A.2. One also needs results on the stability of self-similar profiles with respect to the pertur-bation; that is, we need to show that a profile G ε associated to the perturbed kernel K ε must be close to the unique profile G for the constant coefficient case. In the case ofperturbations by a bounded coagulation coefficient, it turns out that the perturbation ofthe operator is continuous in weighted L norms, so we are led to work in these spaces.This kind of stability results was studied in [35], and we are able to give a new proofwith explicit estimates in Section 3.3. Finally, we need to show that the linearised equation around the self-similar profile forthe constant case has a spectral gap. More importantly, we need to do this in a normwhich allows us to complete the perturbation argument, so again we are forced to workin weighted L norms. A spectral gap in these spaces is proved in Section 6, using resultsfrom Sections 4 and 5. 4he paper is organised as follows: Section 2 gathers some preliminary results which areknown or can be obtained almost directly from existing results. In Section 3 we give boundson self-similar profiles (some of which are new) and show a quantitative stability result inweighted L norms. The result itself is not new, but we give a new proof that makes it fullyquantitative. Sections 4–6 study the linearised operator and show it has a spectral gap in theweighted L spaces we need. Finally, Sections 7 and 8 use all of this to show uniqueness andexponential stability of self-similar profiles for small values of the perturbation parameter ε . By scaling arguments and mass conservation we obtain that in the case of kernels of homo-geneity zero the function s ( τ ) in (1.2) is given by s ( τ ) = (1 + τ ), up to a time shift. Thus,plugging the self-similar ansatz φ ( τ, ξ ) = (1 + τ ) − f (cid:0) log(1 + τ ) , (1 + τ ) − ξ (cid:1) into (1.1) we obtain ∂ t f ( t, x ) = 12 Z x K ( x − y, y ) f ( x − y, t ) f ( t, y ) d y − f ( t, x ) Z ∞ K ( x, y ) f ( t, y ) d y + 2 f ( t, x ) + x∂ x f ( t, x ) . (2.1)To simplify the notation, let us define the operator C K ( f, f ) := 12 Z x K ( x − y, y ) f ( x − y ) f ( y ) d y − f ( x ) Z ∞ K ( x, y ) f ( y ) d y , (2.2)which motivates the definition of the following symmetric bilinear form which will be usefullater: C K ( g, h ) := 12 Z x K ( x − y, y ) g ( x − y ) h ( y ) d y − g ( x ) Z ∞ K ( x, y ) h ( y ) d y − h ( x ) Z ∞ K ( x, y ) g ( y ) d y . (2.3)When the kernel K is K ε = 2 + εW , as it is almost always the case in this paper, we will write C K ε ≡ C ε . We can then write equation (2.1) in an abbreviated form as ∂ t f = C K ( f, f ) + 2 f + x∂ x f. We refer to this equation as the Smoluchowski equation in self-similarity variables, or simplythe self-similar Smoluchowski equation .To simplify the notation at some places (especially when the kernel K is constant), we mayalso use the following notation ( f ∗ g )( x ) = R x f ( x − y ) g ( y ) d y for the convolution.5 .2 Scale invariances We collect here some elementary properties about (2.1). It is well-known that (2.1) preservesthe total mass m ( t ) := R ∞ xf ( x, t ) d x , i.e. m ( t ) ≡ m (0), provided that the kernel K growsat most linearly at infinity. Yet, for kernels with superlinear growth, a loss of total mass infinite time occurs which is known as gelation (e.g. [10]).Furthermore, if f is a solution to (2.1) with kernel K one easily checks that for any α > g = α f solves the self-similar Smoluchowski equation with kernel αK , i.e. ∂ t g = C αK ( g, g ) + 2 g + x∂ x g. Moreover, one verifies that for each solution f of (2.1) also the rescaled function f a := af ( ax )is a solution to (2.1) with the same kernel. Note also that for both transformations themass changes. Summarising, we find that for f solving (2.1) with kernel K , the function h ( x ) = aα f ( ax ) is a solution to (2.1) with K replaced by αK . Moreover, if f has total mass m , i.e. R ∞ xf ( x ) d x = m we get for h that Z ∞ xh ( x ) d x = aα Z ∞ xf ( ax ) d x = 1 aα Z ∞ xf ( x ) d x = m aα . These computations allow to transform solutions to (2.1) for different constant kernels andmodify the total mass. As a consequence, we can assume without loss of generality in thefollowing that the constant kernel K is given by K = 2 and the total mass of the solutions andprofiles is given by one. Remark . Note also that in [7] the kernel was chosen to be K ≡
1. However, by theconsiderations above, all results can be easily rescaled to the case K = 2 which is what we willalways implicitly do during this work. We collect in this section the function spaces and corresponding notation which we use through-out this work. If nothing else is stated, all functions live on the set (0 , ∞ ). First, for a generalweight function w : (0 , ∞ ) → (0 , ∞ ) and p ∈ [0 , ∞ ) we define in the usual way the weighted L p space L p ( w ) := (cid:26) f ∈ L p (cid:12)(cid:12)(cid:12)(cid:12) Z ∞ | f ( x ) | p w ( x ) d x < ∞ (cid:27) with norm k f k L p ( w ) := (cid:0)R ∞ | f ( x ) | p w ( x ) d x (cid:1) /p . The most important case for this work will bethe choice p = 1 and w ( x ) = (1 + x ) k with k ≥ L ((1 + x ) k ) := (cid:26) f ∈ L (cid:12)(cid:12)(cid:12)(cid:12) Z ∞ | f ( x ) | (1 + x ) k d x < ∞ (cid:27) with corresponding norm k f k L ((1+ x ) k ) := R ∞ | f ( x ) | (1 + x ) k d x . To simplify the notation atsome places, we might also write L k = L ((1 + x ) k ) and we might use the abbreviations k·k k = k·k L k = k·k L ((1+ x ) k ) . For parts of this work, we also need several spaces with ratherweak norms defined via the primitive.In particular, we define the norm k h k W − , ∞ := sup x> (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ x h ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) . L space H − (e µx ) which arose naturally in [7].More precisely, for µ ≥ k h k H − (e µx ) = Z ∞ ( D − h ( y )) e µy d y . where D − h ( y ) = R ∞ y h ( x ) d x denotes the primitive. This norm originates from the followingscalar product h g, h i H − (e µx ) = Z ∞ ( D − g )( x )( D − h )( x )e µx d x . The corresponding (Hilbert) space W − , (e µx ) = H − (e µx )is then given as the completion of C ∞ c (0 , ∞ ) with respect to the norm k·k H − (e µx ) . For µ > H − (e − µx ) ∩ (cid:26)Z ∞ xh ( x ) d x = 0 (cid:27) is defined as the completion of ( C ∞ c (0 , ∞ ) ∩ { R ∞ xh ( x ) d x = 0 } ) with respect to k·k H − (e µx ) . Remark . The latter definition is justified by the following estimate which exploits integra-tion by parts as well as H¨older’s inequality: for h ∈ C ∞ c (0 , ∞ ) we have Z ∞ xh ( x ) d x = − Z ∞ x∂ x (cid:18)Z ∞ x h ( z ) d z (cid:19) d x = Z ∞ e − µ x e µ x (cid:18)Z ∞ x h ( z ) d z (cid:19) d x ≤ (cid:18)Z ∞ e − µx d x (cid:19) / k h k H − (e µx ) = µ − / k h k H − (e µx ) . Thus, by density the integral R ∞ xh ( x ) d x is meaningful for all h ∈ H − (e µx ).We also note the following continuous embeddings which will be especially important forthis work. Lemma 2.3.
For each k ≥ and µ > the space L (e µx ) embeds continuously into L ((1 + x ) k ) .Proof. Using the splitting 1 = e − µx/ e µx/ and H¨older’s inequality we find k h k L k = Z ∞ | h ( x ) | (1 + x ) k d x = Z ∞ | h ( x ) | e µ x e − µ x (1 + x ) k d x ≤ (cid:18)Z ∞ | h ( x ) | e µx d x (cid:19) / (cid:18)Z ∞ e − µx (1 + x ) k d x (cid:19) / ≤ C ( µ, k ) k h k L (e µx ) . Lemma 2.4.
For each µ > the space L (e µx ) embeds continuously into H − (e µx ) . roof. It suffices to verify the embedding for the dense subset C ∞ c (0 , ∞ ). In this case, inte-gration by parts yields k h k H − (e µx ) = Z ∞ e µx (cid:18)Z ∞ x h ( z ) d z (cid:19) d x = 1 µ Z ∞ ∂ x (e µx ) (cid:18)Z ∞ x h ( z ) d z (cid:19) d x = − µ (cid:18)Z ∞ h ( z ) d z (cid:19) + 2 µ Z ∞ e µx (cid:18)Z ∞ x h ( z ) d z (cid:19) h ( x ) d x . Note, that we additionally exploit here that the boundary term at infinity vanishes since h ∈ C ∞ c . To proceed, we use that the first expression on the right-hand side is non-positivewhile the second one can be estimated by means of H¨older’s inequality together with thesplitting e µx = e µ x e µ x which results in k h k H − (e µx ) ≤ µ k h k H − (e µx ) k h k L (e µx ) . Cauchy’s inequality with parameter 2 /µ finally yields k h k H − (e µx ) ≤ k h k H − (e µx ) + 2 µ k h k L (e µx ) which finishes the proof by providing k h k H − (e µx ) ≤ /µ k h k L (e µx ) . C K The spaces L ((1+ x ) k ) are convenient to work with because, for a bounded kernel K , the coag-ulation operator C K is a continuous bilinear form on them, as we show in the next proposition.Notice that this is not true for weighted L spaces, for example. Proposition 2.5.
For K : (0 , ∞ ) → (0 , ∞ ) bounded, the bilinear form C K given by (2.3) iscontinuous from L ((1 + x ) k ) to itself for each k ≥ and we have kC K ( g, h ) k L k ≤ k K k L ∞ k g k L k k h k L k . Proof.
From (2.3) together with Fubini’s theorem we find kC K ( g, h ) k L k ≤ Z ∞ Z ∞ | K ( x, y ) || g ( x ) | + | h ( y ) | (cid:0) x + y ) (cid:1) k d x d y + 12 Z ∞ Z ∞ | K ( x, y ) || g ( x ) || h ( y ) | (1+ x ) k d x d y + 12 Z ∞ Z ∞ | K ( x, y ) || g ( y ) || h ( x ) | (1+ x ) k d x d y . Exploiting that (1 + ( x + y )) k ≤ (1 + x ) k (1 + y ) k and 1 ≤ (1 + x ) k if x, y ∈ (0 , ∞ ) and k ≥ kC K ( g, h ) k L k ≤ k K k L ∞ (cid:18) | g ( x ) | (1 + x ) k d x (cid:19)(cid:18)Z ∞ | h ( y ) | (1 + y ) k d y (cid:19) = 32 k K k L ∞ k g k L k k h k L k . .5 Existence of time-dependent solutions and moment estimates Following [11] we will use the following concept of (mild) solutions to (2.1).
Definition 2.6.
For k ≤ f ∈ L k . A function f ∈ C ([0 , ∞ ) , L k ) is denoted a (mild)solution to (2.1) if it satisfies f = S t f + Z t S t − s C K ( f, f )( s ) d s (2.4)where ( S t ) t ≥ is the semigroup generated by the operator h h + xh ′ , i.e. ( S t h )( x ) =e t h (e t x ). Remark . According to [11], the following two solution concepts are equivalent to mildsolutions if f ∈ C ([0 , ∞ ) , L k ):1. f is a weak or distributional solution, i.e. it satisfies Z T Z ∞ (cid:16) f ∂ t ϕ + C K ( f, f ) ϕ − f (2 ϕ − ∂ x ( xϕ ) (cid:17) d x d t = Z ∞ f ϕ (0 , · ) d x for all T > ϕ ∈ C c ([0 , T ) × (0 , ∞ )).2. f is a renormalised solution, i.e. it satisfiesdd t Z ∞ β ( f ) ϕ d x = Z ∞ C K ( f, f ) β ′ ( f ) ϕ − f (2 ϕ − ∂ x ( xϕ ) d x in the sense of distributions in [0 , ∞ ) for all β ∈ C ( R ) ∩ W , ∞ ( R ) and ϕ ∈ C (0 , ∞ ) ∩ L ∞ (0 , ∞ ).The self-similar profiles are then seen to be the stationary solutions to (2.1). Definition 2.8.
A function G ∈ L is a self-similar profile of (1.1) if it is a stationary solutionto (2.1), i.e. it is a fixed-point for (2.4). Remark . Due to Remark 2.7 we see that G is a self-similar if it is a weak stationary solutionto (2.1), i.e. with left-hand side zero. Remark . Note that when dealing with self-similar profiles frequently an integrated ver-sion of (the stationary) equation (2.1) is used, i.e. x G ( x ) = R x R ∞ x − y yK ( y, z ) G ( y ) G ( z ) d z d y (e.g. [25, 27, 29]). However, in view of [9, Lemma 2.11] together with Proposition 3.3 bothformulations are equivalent.The next proposition is a classical result which provides the existence of solutions to (2.1)under the assumption of a bounded coagulation coefficient. A proof can be found for examplein [11, Lemma 2.8] Proposition 2.11.
Let K be a bounded, symmetric and homogeneous of degree zero. For k ≥ let f ∈ L ((1 + x ) k ) . Then there exists a unique solution f ∈ L ((1 + x ) k ) to (2.1) on (0 , ∞ ) with f (0 , · ) = f . Moreover, there exists C k , which only depends on k f k L k increasingly,such that sup t ≥ k f ( t, · ) k L k ≤ C k . emark . Note that the bound on the norm provided by [11, Lemma 2.8] is local in time.However, arguing analogously as in the proof of Proposition 3.3 one can easily verify that theestimate holds globally as stated above.The next statement provides a more explicit estimate on the integral of solutions to (2.1)for bounded perturbations of the constant kernel.
Proposition 2.13.
Let W satisfy (1.3) – (1.5) and let f ε be a solution to (2.1) with kernel K ε = 2 + εW and denote m ( t ) := R ∞ f ε ( x, t ) d x . Then the estimate m ( t ) ≤ e t m (0) + e t − ≤ max { m (0) , } holds for all t ≥ .Proof. It is well-known that C K satisfies the relation Z ∞ C K ( f, f )( x ) d x = − Z ∞ Z ∞ K ( x, y ) f ( x ) f ( y ) d x d y which follows from Fubini’s Theorem. Thus, integrating (2.1) and noting that integration byparts yields R ∞ xf ′ ε ( x, t ) d x = − m ( t ) we obtain ∂ t m ( t ) = 2 m ( t ) − m ( t ) − Z ∞ Z ∞ K ε ( x, y ) f ε ( x ) f ε ( y ) d x d y = m ( t ) − m ( t ) − ε Z ∞ Z ∞ W ( x, y ) f ε ( x ) f ε ( y ) d x d y ≤ m ( t ) − m ( t ) . In the last step, we exploited that
W, f ε ≥
0. Solving this differential inequality explicitly, theclaim directly follows.
Remark . The result of the previous proposition holds in general for any kernel K ≥
2, aslong as time-dependent solutions can be proved to exist for that kernel. We have stated it for K = 2 + εW since it is the only case used in the rest of this paper. Proposition 2.11 together with the continuity results in Proposition 2.5 easily implies that ona fixed time interval solutions to (2.1) for the perturbed and unperturbed kernel stay close atorder O ( ε ): Lemma 2.15.
Let K ε satisfy (1.3) – (1.5) and let f ε and f be the solutions to (2.1) withkernels K ε and K , respectively, and with the same initial condition f ε (0 , · ) = f (0 , · ) . Thenthere exists a constant C > depending only on k f k L k such that k f ε ( t, · ) − f ( t, · ) k L k ≤ εC (e Ct − . Also, the constant
C > depends increasingly on k f k L k . roof. We take the difference of (2.1) for ε and ε = 0 and multiply by sgn( f ε − f ) whichallows to rewrite ∂ t | f ε ( t, · ) − f ( t, · ) | = (cid:0) C εW ( f ε , f ε ) − C ( f , f ) (cid:1) sgn( f ε − f )+ 2 | f ε ( t, · ) − f ( t, · ) | + x∂ x | f ε ( t, · ) − f ( t, · ) | . We multiply by (1 + x ) k , integrate over (0 , ∞ ) and use that integration by parts allows torewrite and estimate Z ∞ x (1 + x ) k ∂ x | f ε ( t, x ) − f ( t, x ) | d x = −k f ε ( t, · ) − f ( t, · ) k L ((1+ x ) k − k Z ∞ x (1 + x ) k − | f ε ( t, x ) − f ( t, x ) | d x ≤ −k f ε ( t, · ) − f ( t, · ) k L k . Thus, we obtainddt k f ε − f k L k ≤ kC εW ( f ε , f ε ) − C ( f , f ) k L k + k f ε − f k L k . Notice that we have omitted the variables ( t, · ) for brevity. Using that C εW ( f ε , f ε ) −C ( f , f ) = C ( f ε − f , f ε + f ) + ε C W ( f ε , f ε ) we can estimate this further, using Propo-sition 2.5:ddt k f ε − f k L k ≤ kC ( f ε − f , f ε + f ) k L k + ε kC W ( f ε , f ε ) k L k ≤ k f ε − f k L k k f ε + f k L k + ε k f ε k L k . From Proposition 2.11 we know that there exists C > k f (0 , · ) k L k (andincreasingly) such that k f ε k L k + k f k L k ≤ C for all t ≥ k f ε − f k L k ≤ C k f ε − f k L k + εC . From Gronwall’s inequality we get k f ε − f k L k ≤ εC C t − , which implies the statement for C := 3 C . First, we recall from [7, Lemma 6.1] the following statement which provides exponential con-vergence in L to the unique profile for the constant coagulation kernel K = 2.11 heorem 2.16. Let f be a solution to (2.1) for the constant kernel K = 2 with total massone and initial condition f such that f ∈ L (d x ) ∩ L ( x d x ) . Let G ( x ) = e − x be theunique stationary solution to (2.1) with total mass . There exists an explicit constant C > depending only on k f k L and k f k L such that k f ( t, · ) − G k L ≤ C e − t for all t ≥ . Also, the constant
C > depends increasingly on k f k L and k f k L . The statement we give here is slightly different to that in [7, Lemma 6.1] in that we say onecan find an explicit constant C depending only on k f k L and k f k L . Although this constantwas not specified in [7], this can be seen from the proof of the lemma, which consists on explicitestimates on the Fourier transform. We give a justification of this in Appendix A.The aim of the following two lemmas is to transfer the convergence in Lemma 2.16 to L ((1 + x ) k ). For this, we first prove an elementary interpolation inequality (Lemma 2.17)which then allows to extend the convergence in Lemma 2.16 to L ((1 + x ) k ) (Lemma 2.18). Lemma 2.17.
Let k ∗ > k and α ∈ (0 , min { k ∗ − k )1+2 k ∗ , } ) be given and assume that f ∈ L ((1 + x ) k ∗ ) ∩ L (0 , ∞ ) . There exists a constant C which depends on α , k and k ∗ such that k f k L k ≤ C k f k αL k f k − αL k ∗ . Proof.
The claim follows from a straightforward application of H¨older’s inequality with thethree exponents p = p = 2 /α and p = 1 / (1 − α ). In fact, we have k f k L k = Z ∞ | f ( x ) | (1 + x ) k d x = Z ∞ | f ( x ) | α | f ( x ) | − α (1 + x ) (1 − α ) k ∗ (1 + x ) k − (1 − α ) k ∗ d x ≤ k f k αL (0 , ∞ ) k f k − αL ((1+ x ) k ∗ ) (cid:18)Z ∞ (1 + x ) k − (1 − α ) k ∗ ) α d x (cid:19) α . Since α < − α ) k ∗ − k ), the remaining integral on the right-hand side can easily be computedas (cid:18)Z ∞ (1 + x ) k − (1 − α ) k ∗ ) α d x (cid:19) α = (cid:16) α − α ) k ∗ − k ) − α (cid:17) α , which finishes the proof. Lemma 2.18.
Let k ∗ > k ≥ , and let f be a solution to (2.1) for the constant kernel K = 2 with total mass one and initial condition f ∈ L ∩ L k ∗ , and let G ( x ) = e − x be the uniquestationary solution to (2.1) with total mass one. There exist constants C, β > such that k f ( t, · ) − G k L k ≤ C e − βt for all t ≥ .The constant C > depends only on k , k ∗ , k f k L , and k f k L k ∗ (and depends increasingly onthe latter two). The constant β > depends only on k and k ∗ . roof. Due to [11, Lemma 2.8] we have that f ∈ L ∞ (0 , T, L ) for all T >
0. Thus, fromLemma 2.17 we know that, with an appropriate choice of α ∈ (0 ,
1) (depending on k and k ∗ ), k f ( t, · ) − G k L k ≤ C k f ( t, · ) − G k αL k f ( t, · ) − G k − αL k ∗ , for some C > k and k ∗ . Thus, Young’s inequality (with a parameter δ )for p = 1 /α and p = 1 / (1 − α ) yields k f ( t, · ) − G k L k ≤ C δ α − α k f ( t, · ) − G k L + C δ k f ( t, · ) − G k L k ∗ , again for C > k , k ∗ . Using Proposition 2.11 and Theorem 2.16 we obtain k f ( t, · ) − G k L k ≤ C δ α − α e − t + δC , for some C > k , k ∗ , k f k L and k f k L , and C > k ∗ and k f k L k ∗ . Taking δ = e − αt/ we deduce that k f ( t, · ) − G k L k ≤ C e − α t + C e − α t = ( C + C )e − α t . In sum, the constants C and C depend on k , k ∗ , k f k L , k f k L and k f k L k ∗ . Since k f k L ≤k f k L k ∗ (and C , C are increasing in k f k L ), one can always modify the constants to havethem depend only on k , k ∗ , k f k L , and k f k L k ∗ . Taking β := α/ C := C + C , thisshows the result. We gather in this section some basic results on existence and bounds which apply in particularto the self-similar profiles for the perturbed equation. More importantly, we give some stability results showing that any self-similar profile G ε with mass one for the kernel K ε must be closeto G ( x ) = e − x , in distances given by suitable norms. In general, these stability results cannotbe obtained from the linearisation methods in this paper, so we borrow them from elsewhereor prove them using different methods. (However, linearisation methods do give some resultson local stability of profiles, assuming we are in a certain neighbourhood of the profile G ; seeLemma 7.2.) Existence of self-similar profiles for large classes of non-solvable kernels with power-law struc-ture was shown in [9, 11, 12], and precise results in the case of homogeneity zero are givenin [29]. Except for the works mentioned in the introduction, uniqueness of scaling profiles innot known for most coagulation kernels. However, there is a number of works providing apriori regularity and asymptotics of self-similar solutions for small and large cluster sizes (e.g.[6, 13, 29]).In our particular setting of homogeneity zero, we cite the following result from [29, Prop.1.1] which provides existence of self-similar profiles with finite mass for the kernels we consider(see also Remark 2.10):
Proposition 3.1 (Existence of profiles) . Let K be homogeneous of degree zero and let k , K > and κ ∈ (0 , be constants such that K ( x, y ) ≤ K (( x/y ) α + ( y/x ) α ) for all x, y ∈ (0 , ∞ ) with α ∈ [0 , and min | x − y |≤ κ ( x + y ) K ( x, y ) ≥ k . Then there exists a self-similar profile G ∈ C (0 , ∞ ) ∩ L ( x d x ) of (1.1) . .2 Bounds on self-similar profiles In this subsection, we provide several a-priori estimates for self-similar profiles. More precisely,Proposition 3.2 provides precise upper and lower bounds for the integral of perturbed self-similar profiles while Proposition 3.3 states that self-similar profiles are uniformly bounded inthe L k norm. Proposition 3.2.
Let K ε satisfy (1.3) – (1.5) and let G ε be a stationary solution of (2.1) , i.e.a self-similar profile. Then we have
11 + ε ≤ Z ∞ G ε ( x ) d x ≤ for all ε ≥ .Proof. By assumption G ε satisfies (2.1) with left-hand side zero. Integrating this equation, wefind2 Z ∞ G ε ( x ) d x + Z ∞ xG ′ ε ( x ) d x + 12 Z ∞ Z x (2 + εW ( x − y, y )) G ε ( x − y ) G ε ( y ) d y d x − Z ∞ Z ∞ (2 + εW ( x, y ) G ε ( x ) G ε ( y ) d x d y = 0 . Denoting m = R ∞ G ε ( x ) d x , integrating by parts in the second term on the left-hand side andusing Fubini’s Theorem for the two double integrals, this reduces to0 = 2 m − m + m − m + ε Z ∞ Z ∞ W ( x, y ) G ε ( x ) G ε ( y ) d x d y − ε Z ∞ Z ∞ W ( x, y ) G ε ( x ) G ε ( y ) d x d y . Combining terms, we end up with m = m + ε Z ∞ Z ∞ W ( x, y ) G ε ( x ) G ε ( y ) d x d y . Due to (1.4) and the non-negativity of G ε we have 0 ≤ W ( x, y ) G ε ( x ) G ε ( y ) d x d y ≤ m whichleads to m ≤ m ≤ (cid:16) ε (cid:17) m from which the claim directly follows.Based on the previous proposition, we can also show the following statement which givesuniform boundedness of all non-negative moments for self-similar profiles. We also note thatthis result also follows from estimates in [29] but to be self-contained, we include the completeproof. Proposition 3.3.
Let K ε satisfy (1.3) – (1.5) and k ≥ . Then there exists a uniform constant C k (depending only on k ) such that k G ε k L k = Z ∞ G ε ( x )(1 + x ) k d x ≤ C k for all self-similar profiles G ε and all ε ∈ [0 , . roof. We note that (1+ x ) k ≤ C (1+ x k ) for some C = C ( k ) >
0. According to Proposition 3.2it thus suffices to show that R ∞ x k G ε ( x ) d x ≤ C k for all k ∈ N with k ≥ k ∈ N with k ≥ k − R ∞ x ℓ G ε ( x ) d x ≤ C ℓ for all ℓ ≤ k −
1. For
R > ϕ R be the linear continuation of x x k , i.e. ϕ R ( x ) = ( x k if x ≤ RkR k − x − ( k − R k if x > R. Since by assumption G ε has finite moments of order zero and one, we can take ϕ R as testfunction in the weak formulation of self-similar profiles which yields Z ∞ G ε ( x ) (cid:0) ( xϕ R ( x )) ′ − ϕ R ( x ) (cid:1) = (cid:18) Z ∞ K ( x, y ) G ε ( x ) G ε ( y ) (cid:2) ϕ R ( x + y ) − ϕ R ( x ) − ϕ R ( y ) (cid:3) d y (cid:19) d x . (3.1)It is easy to check that ϕ R ( x + y ) − ϕ R ( x ) − ϕ R ( y ) ≤ b C k (cid:0) x k − y + xy k − (cid:1) for all x, y > R . Moreover, a direct computation yields( xϕ R ( x )) ′ − ϕ R ( x ) = ( ( k − x k if x ≤ R ( k − R k if x > R. . Thus, we deduce from (3.1) together with (1.3) and (1.4) that( k − Z R x k G ε ( x ) d x + ( k − R k − Z ∞ R G ε ( x ) d x ≤ b C k Z ∞ Z ∞ G ε ( x ) G ε ( y ) (cid:2) x k − y + xy k − (cid:3) d x d y . Due to the non-negativity of G ε we get in particular the estimate( k − Z R x k G ε ( x ) d x ≤ b C k (cid:18)Z ∞ x k − G ε ( x ) d x (cid:19)(cid:18)Z ∞ yG ε ( y ) d y (cid:19) from which the claim follows since the right-hand side is uniformly bounded by the inductionassumption. L estimate The main goal of this subsection is to provide a uniform bound in L for self-similar profiles(Proposition 3.8). The main task for this consists in deriving the behaviour of the self-similarprofiles for small values of x which will be done in the following sequence of lemmata. As abyproduct, we also obtain an a-priori estimate for certain negative moments, depending on theperturbation parameter ε (Lemma 3.7).The first lemma provides a lower integral bound on the profiles which shows that theself-similar solutions can not concentrate at zero.15 emma 3.4. Let K ε satisfy (1.3) – (1.5) . For each a ∗ ∈ (0 , there exists a constant c ∗ > such that Z ∞ a ∗ G ε ( x ) d x ≥ c ∗ for all self-similar profiles G ε with R ∞ xG ε ( x ) d x = 1 .Proof. Splitting the integral we find together with Cauchy’s inequality and Proposition 3.2that1 = Z ∞ xG ε ( x ) d x = Z a ∗ xG ε ( x ) d x + Z ∞ a ∗ xG ε ( x ) d x ≤ a ∗ Z a ∗ G ε ( x ) d x + Z ∞ a ∗ xG ε ( x ) d x ≤ a ∗ + (cid:18)Z ∞ a ∗ x G ε ( x ) d x (cid:19) / (cid:18)Z ∞ a ∗ G ε ( x ) d x (cid:19) / . Together with Proposition 3.3 we find1 − a ∗ ≤ C / (cid:18)Z ∞ a ∗ G ε ( x ) d x (cid:19) / where C is a uniform bound on the second moment which is provided by Proposition 3.3.Thus, since a ∗ ∈ (0 ,
1) we conclude Z ∞ a ∗ G ε ( x ) d x ≥ (1 − a ∗ ) C and the claim follows with c ∗ = (1 − a ∗ ) /C .The next statement gives an estimate on the primitive for self-similar profiles close to zero. Lemma 3.5.
Let K ε satisfy (1.3) – (1.5) . There exist constants C ∗ > and ε ∗ ∈ (0 , suchthat for ε ∈ (0 , ε ∗ ) each self-similar profile G ε satisfies Z x G ε ( y ) d y ≤ C ∗ x − ε ε for all x ≤ . In particular, this implies R x G ε ( y ) d y ≤ C ∗ x − ε .Proof. To simplify the notation, we denote P ( x ) := R x G ε ( y ) d y . We first note that it sufficesto prove the claim for x ≤ a ∗ with a ∗ ∈ (0 ,
1) fixed. In fact, for x > a ∗ we obtain by means ofProposition 2.13 that P ( x ) ≤ ≤ x/a ∗ and thus the claimed estimate holds with C ∗ = 1 /a ∗ .To prove the statement for x ≤ a ∗ , we integrate the stationary version of (2.1) for K = K ε over [0 , x ] to obtain2 P ( x ) + Z x zG ′ ε ( z ) d z + 12 Z x Z z K ε ( z − y, y ) G ε ( z − y ) G ε ( y ) d y d z − Z x Z ∞ K ε ( z, y ) G ε ( y ) G ε ( z ) d y d z = 0 . z z + y we find2 P ( x ) + xP ′ ( x ) − P ( x ) + 12 Z x Z x − y K ε ( z, y ) G ε ( y ) G ε ( z ) d z d y − Z x Z ∞ K ε ( z, y ) G ε ( y ) G ε ( z ) d y d z = 0 . Inserting K ε = 2 + εW and summarising, this simplifies to(1 − P ( ∞ )) P ( x ) + xP ′ ( x ) + ( P ∗ G ε )( x )+ ε (cid:18) Z x Z x − y W ( y, z ) G ε ( y ) G ε ( z ) d z d y − Z x Z ∞ W ( y, z ) G ε ( y ) G ε ( z ) d y d z (cid:19) = 0 . We estimate the left-hand side from above by noting that the expression in parentheses isnon-positive since the domain of integration for the negative term is larger than the one forthe positive (which in addition has a factor of 1 / P ∗ G ε )( x ) ≤ P ( x ). Together this implies0 ≤ (1 − P ( ∞ )) P ( x ) + xP ′ ( x ) + P ( x ) . Denoting α := 2 P ( ∞ ) − R ∞ G ε ( x ) d x − − αP ( x ) + xP ′ ( x ) ≥ − P ( x ) . This differential inequality can be solved explicitly. In fact, using the integration factor x − α − we get dd x (cid:0) x − α P ( x ) (cid:1) = x − α − ( − αP ( x ) + xP ′ ( x )) ≥ − x − α − P ( x ) = − x α − (cid:0) x − α P ( x ) (cid:1) . We note that P is monotonically non-decreasing. Thus, if P ( a ∗ ) = 0 the claim is trivial. Wetherefore assume P ( a ∗ ) > a, a ∗ ) (note that we onlyhave to consider the region where P is non-zero): − dd x (cid:0) x − α P ( x ) (cid:1) − = dd x (cid:0) x − α P ( x ) (cid:1)(cid:0) x − α P ( x ) (cid:1) − ≥ − x α − . Integrating this inequality over ( x, a ∗ ) we obtain − P ( a ∗ ) + x α P ( x ) ≥ − α ( a α ∗ − x α )or equivalently x α P ( x ) ≥ P ( a ∗ ) − a α ∗ α + x α α . (3.2)The definitions of α and P imply1 P ( a ∗ ) − a α ∗ α = 1 αP ( a ∗ ) (cid:0) α − a α ∗ P ( a ∗ ) (cid:1) = 1 αP ( a ∗ ) (cid:18) Z ∞ G ε ( x ) d x − − a α ∗ Z a ∗ G ε ( x ) d x (cid:19) . α > a ∗ ∈ (0 ,
1) we have a α ∗ ≤ ε sufficiently small, we have1 P ( a ∗ ) − a α ∗ α ≥ αP ( a ∗ ) (cid:18) Z ∞ G ε ( x ) d x − − Z a ∗ G ε ( x ) d x (cid:19) = 1 αP ( a ∗ ) (cid:18)Z ∞ a ∗ G ε ( x ) d x + Z ∞ G ε ( x ) d x − (cid:19) ≥ αP ( a ∗ ) (cid:16) c ∗ + 11 + ε − (cid:17) = 1 αP ( a ∗ ) (cid:16) c ∗ − ε ε (cid:17) . (3.3)Thus, if ε is small enough, the right-hand side is strictly positive (note that α is strictly positivedue to Proposition 3.2). With this, we deduce from (3.2) and (3.3) that P ( x ) ≤ x α P ( a ∗ ) − a α ∗ α + x α α ≤ αP ( a ∗ ) c ∗ − ε ε x α = 1 c ∗ − ε ε (cid:18) Z ∞ G ε ( y ) d y − (cid:19) Z a ∗ G ε ( y ) d y x α . Together with Proposition 3.2 and the non-negativity of G ε the right-hand side can be furtherestimated to get P ( x ) ≤ c ∗ − ε ε x α . Finally, we recall again Proposition 3.2 to deduce α = 2 R ∞ G ε ( x ) d x − ≥ / (1 + ε/ − ≥ (2 − ε ) / (2 + ε ) from which the claimed estimate follows.Based on the preparation above, we can now provide a pointwise estimate on the behaviourof self-similar profiles close to zero. Lemma 3.6.
Let K ε satisfy (1.3) – (1.5) . There exist constants C ∗ > and ε ∗ > such thateach self-similar profiles G ε satisfies G ε ( x ) ≤ C ∗ x − ε ε ≤ C ∗ x − ε for almost all x ≤ if ε ≤ ε ∗ .Proof. We recall from Remark 2.10 that G ε satisfies the equation G ε ( x ) = 1 x Z x Z ∞ x − y yK ε ( y, z ) G ε ( y ) G ε ( z ) d z d y . The assumptions (1.3) and (1.4) together with ε ≤ K ε ≤ G ε yields G ε ( x ) ≤ x Z x Z ∞ x − y yG ε ( y ) G ε ( z ) d z d y ≤ x Z x Z ∞ G ε ( y ) G ε ( z ) d z d y . Together with Lemma 3.4 and Proposition 3.2 we thus conclude G ε ( x ) ≤ x Z x G ε ( y ) d y ≤ C ∗ x − ε ε . Lemma 3.7.
Let ε ∗ ∈ (0 , and let K ε satisfy (1.3) – (1.5) with ≤ ε ≤ ε ∗ . For each α ∈ ( ε ∗ − , ∞ ) there exists a constant C α such that each self-similar profile G ε satisfies Z ∞ x α G ε ( x ) d x ≤ C α . Proof.
The statement is a direct consequence of Proposition 3.3 and Lemma 3.6.The preparation above now allows us to obtain uniform estimates on the L norm of self-similar profiles. Proposition 3.8.
Let K ε satisfy (1.3) – (1.5) . There exist constants C ∗ > and ε ∗ > suchthat each self-similar profiles G ε satisfies k G ε k L (1) = (cid:18)Z ∞ | G ε ( x ) | d x (cid:19) / ≤ C ∗ if ε ≤ ε ∗ .Proof. We recall from the proof of Lemma 3.6 that G ε ( x ) = 1 x Z x Z ∞ x − y yK ε ( y, z ) G ε ( y ) G ε ( z ) d z d y . Thus, multiplying by G ε and integrating, together with Fubini’s Theorem, we deduce k G ε k L (1) = Z ∞ | G ε ( x ) | d x = Z ∞ Z ∞ yK ε ( y, z ) G ε ( y ) G ε ( z ) Z y + zy G ε ( x ) x d x d z d y . Due to (1.3) and (1.4) and ε ≤ k G ε k L (1) ≤ Z ∞ Z ∞ yG ε ( y ) G ε ( z ) Z ∞ y G ε ( x ) x d x d z d y . Next, we note that Proposition 3.3 and Lemma 3.6 directly imply that R ∞ y G ε ( x ) /x d x ≤ Cy − − ε for all y > y > k G ε k L (1) ≤ Z ∞ y − ε G ε ( y ) d y Z ∞ G ε ( z ) d z . The claim then follows from Proposition 3.3 and Lemma 3.7 if ε < / Regarding stability, the following statement is a particular case of [35, Thm. 2.4] for boundedperturbations W . Proposition 3.9 (Stability of profiles) . Let W be a bounded kernel satisfying (1.4) and (1.5) .For ε ≥ , denote K ε := 2 + εW . For any k ≥ there exists a function δ = δ ( ε ) dependingonly k and ε , with δ ( ε ) → as ε → such that any self-similar profile G ε with mass ofSmoluchowski’s equation with kernel K ε satisfies k G ε − G k L k ≤ δ ( ε ) . δ = δ ( ε ) was obtained in [35]via compactness arguments, and hence one cannot give any constructive estimate on it. As aconsequence, using Proposition 3.9 as given, the constants in our main result in Theorem 1.2would become non-constructive: we would not be able to estimate the size of ε or ε , even ifwe know there must be one satisfying the statement.In order to improve this situation we give an alternative way to show Proposition 3.9, whichyields an explicit estimate on the rate δ ( ε ).The main idea to obtain constructive estimates is to use available quantitative informationon the asymptotic behaviour of solutions to the self-similar equation ∂ t f = C ( f, f )+2 f + x∂ x f with constant coefficients. In a broad sketch, if we know that1. the equation with constant coefficients relaxes to equilibrium, with explicit rates, in acertain norm,2. the dynamics of solutions depends continuously on the perturbation ε , in the same norm,3. and the norm of any profile G ε is bounded by a uniform constant,then we can conclude that any profile G ε must be close to G for small ε , in the same normwe are considering. Point 2 seems to be the least problematic of the three, and we will useour Lemma 2.15. Let us see what is available regarding point 1. Since [18] it is known thatsolutions in the constant coefficients case converge to equilibrium, and a quantitative estimateof the rate at which this happens was given in [7], in several norms including L and weighted L norms. A clean statement with explicit constants was then given in [33, Theorem 1.1], for k·k W − , ∞ . Since we want to show that k G − G ε k L k is small, we are forced to use the L normconvergence result in [7] to fulfill point 1 since a simple interpolation then allows us to controlthe L k norm. We have stated this result in our Lemma 2.18. Notice that we are also restrictedby point 3, since uniform estimates of profiles are available in L k , but not for example in L ∞ or W , . For point 3, we use uniform estimates of profiles in L given in Proposition 3.8, whichas far as we know were not available elsewhere.These ideas give us a proof of Proposition 3.9 with an explicit δ ( ε ): Proof Proposition 3.9.
Take any solution f ε to (2.1) with kernel K = 2 + εW , and any solution f to (2.1) with constant kernel K = 2. We have k f ε ( t, · ) − G k L k ≤ k f ε ( t, · ) − f ( t, · ) k L k + k f ( t, · ) − G k L k . (3.4)Let G ε be any self-similar profile with mass one for the kernel K ε , and choose the initialcondition for both f ε and f to be equal to G ε . In particular, f ε is then equal to the constant-in-time profile G ε . From Lemma 2.15, k f ε ( t, · ) − f ( t, · ) k L k = k G ε − f ( t, · ) k L k ≤ εC e C t . for some C > k G ε k L k , in an increasing way. Since we know from Lemma3.3 that k G ε k L k is uniformly bounded by a constant C k depending only on k , we conclude thatthe constant C can be chosen to depend only on k as well. On the other hand, Lemma 2.18shows that k f ( t, · ) − G k L k ≤ C e − βt C > k , k G ε k L and k G ε k L k +1 (for example; any momentlarger than k will do, not necessarily k + 1). The constant β > k . In asimilar way as before, since we know from Propositions 3.3 and 3.8 that these norms of G ε areuniformly bounded by constants that depend only on k we conclude that the constant C canbe chosen to depend only on k . Using our last two estimates in (3.4), k G ε − G k L k = k f ε ( · , t ) − G k L k ≤ εC e C t + C e − βt , t ≥ . for each solution f ε to (2.1) with K = 2 + εW . Choosing t = log( C β/ ( εC )) / ( C + β ) theclaim follows with δ ( ε ) = ε β/ ( C + β ) . Remark . In Section 7.1 we will give a further improvement of Proposition 3.9, i.e. we willshow that actually δ ( ε ) = O ( ε ) as ε → In this section we will introduce the linearised coagulation operator L in self-similar variables.Precisely, if we linearise the stationary version of (2.1) around the profile G ( x ) = e − x we get L [ h ] = x∂ x h + 2( G ∗ h ) − G Z ∞ h ( y ) d y . (4.1)Since this expression contains a derivative with respect to x which is not defined in the spaceswe consider, L has to be understood as an unbounded operator. However, L is obviouslywell-defined on C ∞ c (0 , ∞ ) by the following equivalent representation formulas: Lemma 4.1.
On the space C ∞ c (0 , ∞ ) the operator L as given in (4.1) can be equivalentlywritten as L [ h ]( x ) = xh ′ ( x ) + 2 e − x Z x h ( y )( e y −
1) d y − e − x Z ∞ x h ( y ) d y (4.2) and L [ h ]( x ) = xh ′ ( x ) − H ( x ) + 2 Z x H ( y )e − ( x − y ) d y (4.3) where H ( x ) = R ∞ x h ( z ) d z .Proof. The expression (4.2) is obvious by splitting and re-combining the two integrals. For (4.3)we rewrite h ( y ) = − ∂ y ( R ∞ y h ( z ) d z ) in the convolution expression of (4.1) and integrate by partswhich yields L [ h ]( x ) = xh ′ ( x ) + 2 Z x − ∂ y Z ∞ y h ( z ) d z e − ( x − y ) d y − − x Z ∞ h ( y ) d y = xh ′ ( x ) − Z ∞ x h ( z ) d z + 2 Z ∞ h ( z ) d z e − x + 2 Z x Z ∞ y h ( z ) d z e − ( x − y ) d y − − x Z ∞ h ( y ) d y = − H ( x ) + 2 Z x H ( y )e − ( x − y ) d y . L , defined on a suitable domain, generatesa strongly continuous semigroup in the spaces in which we will be working later: Theorem 4.2.
There are semigroups defined on each of the spaces L ((1 + x ) k ) for all k ≥ , L (e µx ) and H − (e µx ) for all µ ≥ , such that1. their generators are all defined on C ∞ c , and they are equal to L on C ∞ c ,2. and C ∞ c is a core for their generators (see Definition 4.3 below).These semigroups can all be restricted to the corresponding spaces with mass zero, i.e. theirintersections with { R ∞ xf ( x ) d x = 0 } When talking about L on any of these spaces, it is implicit that we mean the genera-tor of the corresponding semigroup (equivalently, the closure of L , defined on C ∞ c , in thecorresponding norm).The rest of this section is devoted to the proof of Theorem 4.2. To simplify working withunbounded operators and in particular with the corresponding domains in the following weintroduce the notion of a core which usually allows to restrict to dense subsets instead of thefull domain of the operator. The following definition is taken from [8, Ch. I, Definition 1.6]: Definition 4.3 (Core) . For an unbounded operator U : D ( U ) ⊂ X → X a subspace S ⊂ D ( U )is denoted core of U if S is dense in D ( U ) for the graph norm k h k U = k h k + k U h k .The next lemma is an extension of the Bounded Perturbation Theorem stating that thelatter also preserves the core of an unbounded operator. Since this result seems not to beproved in [8], we present the short proof for completeness. Lemma 4.4.
Let U : D ( U ) ⊂ X → X be the generator of a strongly continuous semigroup e Ut with core S ⊂ D ( U ) . Furthermore, let V : X → X be bounded. Then U + V is the generatorof a strongly continuous semigroup e ( U + V ) t on X with domain D ( U ) and core S .Proof. According to the Bounded Perturbation Theorem, the operator U + V is the generatorof a strongly continuous semigroup on X with domain D ( U ). To see that S is still a core, itsuffices to prove that the graph norms of U and U + V , i.e. k·k U and k·k U + V are equivalent.For this, we fix κ > k V k which yields k h k U + V = k h k + k ( U + V ) h k ≥ k h k + 1 κ k ( U + V ) h k ≥ k h k + 1 κ k U h k − κ k V h k≥ (cid:16) − k V k κ (cid:17) k h k + 1 κ k U h k ≥ κ (cid:0) k h k + k U h k (cid:1) = 1 κ k h k U . Conversely, we find k h k U + V = k h k + k ( U + V ) h k ≤ k h k + k U h k + k V h k ≤ (cid:0) k V k (cid:1) k h k + k U h k≤ (cid:0) k V k (cid:1)(cid:0) k h k + k U h k (cid:1) ≤ (cid:0) k V k (cid:1) k h k U . This finishes the proof.The following remark states more precisely how the action of linear operators on the space H − (e µx ) will be understood in the following.22 emark H − ) . Since H − (e µx ) is a rather weak space, itseems most appropriate to define linear operators on it via a density argument. Precisely, dueto the definition of H − (e µx ), the elements of this space are represented by equivalence classesof Cauchy sequences with respect to k·k H − (e µx ) . In particular, for each class, we can alwaysfind a representative sequence which is contained in C ∞ c (0 , ∞ ). The approach then consists indefining a given linear operator on this space and extend it again by completion with respectto the norm on H − (e µx ). This procedure of course requires that the operator defined thisway in fact maps to H − (e µx ) and that the definition is independent of the specific choice of asequence. However, both properties are obviously satisfied if the operator U to be defined thisway satisfies k U h k H − (e µx ) ≤ C k h k H − (e µx ) for all h ∈ C ∞ c (0 , ∞ ) , i.e. the restriction of U to C ∞ c (0 , ∞ ) is bounded. For the operators considered in this work, thelatter property will typically be satisfied and in this case, we implicitly use the constructiondescribed before.The following lemma provides that h xh ′ ( x ), which appears in the linearised coagulationoperator, is the generator of a strongly continuous semigroup in the spaces L k , L (e µx ) and H − (e µx ). Lemma 4.6.
The family of operators ( T t ) t ≥ given by the formula ( T t h )( x ) = h ( x e t ) definesa strongly continuous semigroup on L ((1 + x ) k ) for all k ≥ as well as on L (e µx ) and H − (e µx ) for all µ ≥ . Moreover, this defines also a semigroup on the corresponding spacesrestricted to total mass equal to zero, i.e. L ((1 + x ) k ) ∩ { R ∞ xf ( x ) d x = 0 } for all k ≥ as well as L (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } and H − (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } for µ > . Inall cases the generator B is given by B h = x dd x : h xh ′ ( x ) (while, by abuse of notation,we use the same notation for the generator on different spaces) and the space C ∞ c (0 , ∞ ) or C ∞ c (0 , ∞ ) ∩ { R ∞ xf ( x ) d x = 0 } respectively is a core.Moreover, we have k T t h k L k ≤ k h k L k e − t for all h ∈ L k k T t h k L (e µx ) ≤ k h k L (e µx ) e − t for all h ∈ L (e µx ) k T t h k H − (e µx ) ≤ k h k H − (e µx ) e − t for all h ∈ H − (e µx ) and all t ≥ . Since the semigroup is explicitly given, the proof is straightforward but for the sake ofcompleteness, we include it in Appendix C.
Proof of Theorem 4.2.
We just apply the bounded perturbation theorem to the semigroupsgiven by Lemma 4.6, since the reader can check that the remaining terms in the definition of L are bounded operators (in the L and L cases one can choose expression (4.2) for this; inthe H − case one can choose expression (4.3)). The following technique allows us to obtain spectral gaps in different spaces, once a spectralgap in some space has been proved. These ideas stem from classical perturbation theory of23inear operators, with constructive estimates given in [24] and a general theory developed in[15]. The simple approach described here was already used in [4], and we describe it below.
We often refer to our estimates on the decay of several semigroups as “spectral gap estimates”.The main decay property that we are interested in is more precisely called hypodissipativity : Definition 5.1 (Hypodissipative semigroup) . Let X be a Banach space and A : D ( A ) ⊂ X → X the generator of a strongly continuous semigroup e A t . We say that A is hypodissipative (orthat the semigroup ( e A t ) t ≥ is hypodissipative) if there exist constants C ≥ λ > k e A t h k X ≤ C k h k X e − λt for all h ∈ X .An operator A is usually called dissipative if the above definition holds with C = 1. If A isthe generator of a strongly continuous semigroup on a Banach space X , we say A has a spectralgap if its kernel is nonzero (i.e., there are equilibria of the evolution), and the correspondingsemigroup is hypodissipative when restricted to a suitable subspace of X which does notcontain the kernel of A (usually the subspace perpendicular to the kernel in a suitable scalarproduct). Since we do not define this “suitable subspace” in general, every time we mention aspectral gap result it should be clear that we always refer to a specific decay property of thecorresponding semigroup.Of course, this is intimately related to the property that the spectrum of A consists of 0,plus an additional set contained in { z ∈ C | ℜ ( z ) ≤ − λ } , but there is not a simple equivalencewithout further decay properties of the resolvent (by the Hille-Yosida theorem). This is whywe prefer to work only with estimates on the decay of the associated semigroups. We state a result similar to [4, Theorem 3.1] or [22, Theorem 1.1], dealing with restriction ofthe spectral gap of a linear operator instead of extension. Strictly speaking, the results belowallow us to transfer the hypodissipativity property between semigroups. In order to use themto transfer a spectral gap property, we will later apply them to the subspaces perpendicular tothe equilibrium in a suitable sense.
Theorem 5.2.
Consider two Banach spaces
Y ⊆ Z with corresponding norms Y and Z such that k h k Z ≤ C Y k h k Y for all h ∈ Y for some C Y > . Let L Y : D ( L Y ) → Y be anunbounded operator which extends to an unbounded operator L Z : D ( L Z ) → Z be unbounded,i.e. D ( L Y ) ⊆ D ( L Z ) and L Z | D ( L Y ) = L Y . Given that:1. L Z is the generator of a strongly continuous semigroup e L Z t on Z .2. which satisfies k e L Z t h k Z ≤ C e − λ t k h k Z , h ∈ Z , : : t ≥ with C > and λ ∈ R .3. L Z = A + B with linear operators A , B on Z which satisfy a) A : Z → Y is continuous, i.e. k Ah k Y ≤ C A k h k Z for all h ∈ Z and C A > ,(b) B is the generator of a strongly continuous semigroup e Bt on Y satisfying k e Bt h k Y ≤ C e − λ t k h k Y , h ∈ Y , : : t ≥ with C > , and λ = λ .the operator L Z (and thus L Y ) has also a spectral gap on Y , i.e. it satisfies k e L Z t h k Y ≤ C e − min { λ ,λ } t k h k Y , h ∈ Y , : : t ≥ for C = C (cid:16) C A C C Y | λ − λ | (cid:17) .Proof. We can use Duhamel’s formula to writee L Z t h = e Bt h + Z t e B ( t − s ) (cid:0) A e L Z s h (cid:1) d s ∀ h ∈ Y , : t ≥ . Thus, for fixed h ∈ Y and t ≥
0, we have k e L Z t h k Y ≤ k e Bt h k Y + Z t k e B ( t − s ) (cid:0) A e L Z s h (cid:1) k Y d s ≤ C e − λ t k h k Y + C Z t e − λ ( t − s ) k A e L Z s h k Y d s ≤ C e − λ t k h k Y + C C A Z t e − λ ( t − s ) k e L Z s h k Z d s ≤ C e − λ t k h k Y + C C A C k h k Z Z t e − λ ( t − s ) e − λ s d s = C e − λ t k h k Y + C C A C k h k Z λ − λ ( e − λ t − e − λ t ) ≤ C (cid:18) C A C C Y | λ − λ | (cid:19) e − min { λ ,λ } t k h k Y . (5.3)This shows the result. For convenience we recall [4, Theorem 3.1] which allows to extend the spectral gap from oneBanach space to a larger one.
Theorem 5.3.
Consider two Banach spaces
Y ⊂ Z with corresponding norms k·k Y and k·k Z and such that k h k Z ≤ C Y k h k Y for all h ∈ Y . Let L Y : D ( L Y ) → Y be an unbounded op-erator which extends to an unbounded operator L Z : D ( L Z ) → Z , i.e. D ( L Y ) ⊂ D ( L Z ) and L Z | D ( L Y ) = L Y . Given that1. L Y is the generator of a strongly continuous semigroup e L Y t on Y
2. which satisfies k e L Y t h k Y ≤ C e − λ t k h k Y for h ∈ Y and t ≥ with C > and λ ∈ R . L Z = A + B with linear operators A , B on Z which satisfy(a) A : Z → Y is continuous, i.e. kA h k Y ≤ C A k h k Z for all h ∈ Z and C A > ,(b) B is the generator of a strongly continuous semigroup e B t on Z satisfying k e B t h k Z ≤ C e − λ t k h k Z for all h ∈ Z and t ≥ with C > and λ > λ the operator L Z is the generator of a strongly continuous semigroup e L Z t on Z which extends e L Y t and satisfies k e L Z t h k Z ≤ C e − λ t k h k Z for all h ∈ Z and t ≥ with C = C + C Y C C C A ( λ − λ ) − . L spaces Our overall plan for the linearised coagulation operator consists in showing that we can restrictthe known H − ( e µx ) spectral gap to the smaller Hilbert space L (e µx ) and from there extendto the space L ((1 + x ) k ) with sufficiently large k >
0, using the techniques from the previoussection. In this section, we will transfer the spectral gap for L which has been obtained in [7]for the class of spaces H − (e µx ) to the spaces of the form L ((1 + x ) k ). For this, we will relyon the restriction/extension methods recapitulated in Section 5.In [7] a spectral gap for L was obtained in H − (e µx ). Precisely, we recall from [7, Prop.3.11 & Lem. 3.12] the following result: Theorem 6.1.
For any µ ∈ (0 , , the operator L as given by (4.3) has a spectral gap ofsize in H − (e µx ) , that is: on this space it generates a strongly continuous semigroup e L t satisfying k e L t h k H − (e µx ) ≤ k h k H − (e µx ) e − t for t ≥ and all h ∈ H − (e µx ) ∩ { R ∞ xh ( x ) d x = 0 } .Remark . In [7] the constant kernel was chosen to be K ≡
1, but the above result is adaptedto our choice K ≡ L spaces In this subsection, we will prove the following proposition which states that the spectral gapfor L in H − (e µx ) can be restricted to the subspace L (e µx ). Proposition 6.3.
The operator L as given by (4.3) generates a strongly continuous semigroup e L t on L (e µx ) and for each λ ∈ ( −∞ , / there exists C λ > such that k e L t h k L (e µx ) ≤ C λ e − λt k h k L (e µx ) for all h ∈ L (e µx ) with Z ∞ xh ( x ) d x = 0 and all t ≥ . roof. The proof follows from an application of Theorem 5.2. For this, we choose Z = H − (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } and Y = L (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } . Moreover, L Y and L Z are both given as unbounded operators by the expression (4.3) on the respective spaces(see also Remark 6.4 below).1. According to [7, Proposition 3.11] (see Theorem 6.1 above) the operator L Z generates astrongly continuous semigroup e L Z t on the space H − (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } .2. From [7, Lemma 3.12] (see also Theorem 6.1 above) we have that k e L Z t h k H − (e µx ) ≤ C k h k H − (e µx ) e − t for h ∈ H − (e µx ) ∩ (cid:26)Z ∞ xh ( x ) d x = 0 (cid:27) and all t ≥ L Z as follows: L Z = A + B with A [ h ]( x ) = − H ( x ) + 2 Z x H ( y )e − ( x − y ) d y = A [ h ]( x ) + A [ h ]( x ) B [ h ]( x ) = x∂ x h ( x ) . Note that we use here the notation H ( x ) = R ∞ x h ( y ) d y for the primitive of h .(a) We have to show that A : Z → Y is bounded. Since both L Z and B preserve theconstraint, the same is true for A and thus it suffices to show that A : H − (e µx ) → L (e µx ) is bounded. Obviously, we have k A [ h ] k L (e µx ) ≤ k h k H − (e µx ) . Thus, itsuffices to consider A : k A [ h ] k L (e µx ) = 4 Z ∞ (cid:18)Z x H ( y )e − ( x − y ) d y (cid:19) e µx d x = 4 Z ∞ (cid:18)Z x H ( y )e µ y e( µ − ) ( x − y ) d y (cid:19) d x = 4 (cid:13)(cid:13)(cid:13)(cid:0) H ( · )e µ · (cid:1) ∗ e( µ − ) · (cid:13)(cid:13)(cid:13) L (d x ) . Here, ∗ denotes the convolution given by ( f ∗ g )( x ) = R x f ( x − y ) g ( y ) d y for x ∈ (0 , ∞ ). From Young’s inequality for convolutions, we thus deduces k A [ h ] k L (e µx ) ≤ k h k H − (e µx ) k e( µ − ) · k L (0 , ∞ ) = 82 − µ k h k H − (e µx ) . (b) According to Lemma 4.6 the operator B generates a strongly continuous semigroupe Bt on L (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } which satisfies k e Bt h k L (e µx ) ≤ k h k L (e µx ) e − t for all h ∈ L (e µx ) ∩ (cid:26)Z ∞ xf ( x ) d x = 0 (cid:27) and t ≥ Remark . To be able to use Theorem 5.2 in the previous proof, we have to make surethat L Z is an extension of L Y . This follows from the following consideration: The proof ofProposition 6.3 (i.e. 3a) shows that A is bounded from Z into Y . According to Lemma 2.4it is thus in particular bounded from Z into itself as well as from Y to itself. Lemmas 4.4and 4.6 thus ensure that both L Z and L Y are generators of strongly continuous semigroupswith common core C ∞ c (0 , ∞ ) on which the two operators coincide.27 .2 Extension of the spectral gap to weighted L spaces From Proposition 6.3 we know that L has a spectral gap in L (e µx ) subject to the constraintthat R ∞ xf ( x ) d x = 0. In this subsection we will now prove, using Theorem 5.3, that the lattercan be extended to L ((1 + x ) k ). This is the main spectral gap result that will be used in therest of this paper: Theorem 6.5.
Take k > . The semigroup e t L defined on the space L ((1 + x ) k ) (see 4.2)has a spectral gap, in the sense that there is C = C ( k ) > such that k e L t h k k ≤ C e − t k h k k for all t ≥ ,for all h ∈ L ((1 + x ) k ) with R ∞ xh ( x ) d x = 0 . In particular, by the Hille-Yosida theoremwe see that kL ( h ) k L k ≥ C k h k L k := 1 M k h k L k for all h in the domain of L with R ∞ xh ( x ) d x = 0The proof of this statement will rely on an application of Theorem 5.3. Precisely, we choose Y = L (e µx ) and Z = L ((1 + x ) k ) and we will verify the following steps:1. L given by (4.2) generates a strongly continuous semigroup e L t on Y .2. For each λ ≤ /
2, this semigroup satisfies k e L t h k L (e µx ) ≤ C e − λ t k h k L (e µx ) for all h ∈ L (e µx ) ∩ { R ∞ xh ( x ) d x = 0 } and t ≥ L = A + B such that:(a) A : L k → L (e µx ) is continuous, i.e. kA h k L (e µx ) . k h k L k for all h ∈ L k and R ∞ x A [ h ]( x ) d x = 0 for all h ∈ L k with R ∞ xh ( x ) d x = 0.(b) B generates a strongly continuous -semigroup e B t on L k satisfying k e B t h k L k ≤ C e − t k h k L k for all h ∈ L k with Z ∞ xh ( x ) d x = 0 . In order to simplify the structure of the actual proof, we collect first several preparatoryresults while the proof of Theorem 6.5 will then be given at the end of this section.We will choose the following splitting L = A + B of the operator L with A [ h ]( x ) = 2e − x Z x h ( y ) χ { y ≤ R } (e y −
1) d y − − x Z ∞ x h ( y ) d y + Z ∞ z e − z Z z h ( y ) χ { y>R } (e y −
1) d y d z e − x = A + A + A B [ h ]( x ) = xh ′ ( x ) + 2e − x Z x h ( y ) χ { y>R } (e y −
1) d y − Z ∞ z e − z Z z h ( y ) χ { y>R } (e y −
1) d y d z e − x = B + B + B (6.1)where R is a sufficiently large constant which has to be fixed in the proof of Theorem 6.5below. The last expression on the right-hand side of A and B ensures that R ∞ x A [ h ]( x ) d x =28 = R ∞ x B [ h ]( x ) d x . At this point, we also exploit that L preserves this constraint, i.e. weconstruct B such that the first moment is zero which implies that the same is automaticallytrue for A since L = A + B .The following three lemmata provide estimates on auxiliary integrals which will turn outto be useful during subsequent computations. Lemma 6.6.
Let k ∈ R . For each β > there exists R β > such that Z ∞ y e − x (1 + x ) k d x ≤ β e − y (1 + y ) k if y ≥ R β .Proof. An application of l’Hˆopital’s rule yields R ∞ y e − x (1 + x ) k d x ∼ e − y (1 + y ) k as y → ∞ from which the claim directly follows. Lemma 6.7.
For each k ≥ we have the estimate Z ∞ ( x + 1) k e − x d x ≤ Γ( k + 1)e . Proof.
The definition of Γ( · ) together with the change of variables x x − Z ∞ ( x + 1) k e − x d x = Z ∞ x k e − x − d x = 1e Z ∞ x ( k +1) − e − x d x ≤ Γ( k + 1)e . Lemma 6.8.
For each k ≥ there exists a constant C k > such that Z ∞ y e − x (1 + x ) k d x ≤ C k e − y (1 + y ) k for all y > .Proof. Due to Lemma 6.6 there exists R > R ∞ y e − x (1 + x ) k d x ≤ − y (1 + y ) k if y ≥ R . Moreover, if y ≤ R we deduce together with Lemma 6.7 that Z ∞ y e − x (1 + x ) k d x ≤ Γ( k + 1)e ≤ Γ( k + 1)e e R (1 + y ) k e − y . Combining both estimates, the claim follows with C k = max { , Γ( k + 1)e R / e } .The next lemma shows that the operators B and B are bounded. Lemma 6.9.
For any β > there exists R β > such that the operators B , B : L ((1+ x ) k ) → L ((1 + x ) k ) as defined in (6.1) are bounded with kB h k L k ≤ β k h k L k and kB h k L k ≤ k + 1)e( R + 1) k − k h k L k if R > R β . Moreover, there exists C k > such that kB h k L k ≤ C k k h k L k for all R ≥ . roof. We first consider the correction term A = −B given in (6.1) which we rewrite bymeans of Fubini’s theorem and the relation R ∞ y z e − z d z = ( y + 1)e − y which yields Z ∞ z e − z Z z h ( y ) χ { y>R } (e y −
1) d y d z = 2 Z ∞ h ( y ) χ { y>R } (e y − Z ∞ y z e − z d z d y = 2 Z ∞ R h ( y )( y + 1)(1 − e − y ) d y = 2 Z ∞ R h ( y )( y + 1)(1 − e − y ) d y . From this, we deduce in particular the estimate (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ z e − z Z z h ( y ) χ { y>R } (e y −
1) d y d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞ R | h ( y ) | ( y + 1) k ( y + 1) − k (1 − e − y ) d y ≤ R + 1) − k Z ∞ R | h ( y ) | ( y + 1) k d y ≤ R + 1) − k k h k L k . (6.2)Thus, together with (6.1) and Lemma 6.7 we immediately get kB h k L k ≤ k + 1)e( R + 1) k − k h k L k . To bound B we note that by means of Fubini’s theorem we have kB h k L k ≤ Z ∞ e − x Z x | h ( y ) | χ { y>R } (e y −
1) d y (1 + x ) k d x = 2 Z ∞ R | h ( y ) | (e y − Z ∞ y e − x (1 + x ) k d x d y . Thus, on the one hand fixing β > kB h k L k ≤ β Z ∞ R | h ( y ) | (1 + y ) k (1 − e − y ) d y ≤ β k h k L k . if R > R β . On the other hand, Lemma 6.8 yields kB h k L k ≤ C k Z ∞ | h ( y ) | (1 + y ) k (1 − e − y ) d y ≤ C k k h k L k . We next prove that the operator B generates a strongly continuous semigroup (which isthe first part of Item 3b above). Lemma 6.10.
Under the assumptions of Theorem 6.5 the operator B as defined in (6.1) generates a strongly continuous semigroup both on L k and L k ∩ { R ∞ xf ( x ) d x = 0 } . Moreover,the space C ∞ c (0 , ∞ ) or C ∞ c (0 , ∞ ) ∩ { R ∞ xh ( x ) d x = 0 } respectively is a core.Proof. The statement is a consequence of the Bounded Perturbation Theorem (e.g. [8, Ch. III,Sec. 1.3]) since B = x∂ x generates a strongly continuous semigroup according to Lemma 4.6while B and B are bounded linear operators as shown in Lemma 6.9. Moreover, the operator B has been constructed explicitly to preserve the constraint R ∞ xf ( x ) d x = 0. The statementon the core is a direct consequence of Lemma 4.4.30ith the preparations above, we can now give the proof of Theorem 6.5. Proof of Theorem 6.5.
As already indicated above, the proof relies on an application of The-orem 5.3, i.e. we have to verify Items 1–3. We recall from Proposition 6.3 that L as givenby (4.3) generates a C -semigroup e L t on L (e µx ) which satisfies for each λ ≤ / k e L t h k L (e µx ) ≤ C λ e − λ t k h k L (e µx ) for all h ∈ L (e µx ) ∩ (cid:26)Z ∞ xh ( x ) d x = 0 (cid:27) (6.3)and all t ≥
0. Moreover, C ∞ c (0 , ∞ ) is a core for L given by (4.3) (see Remark 6.4) and thus,according to Lemma 4.1 the generator L is equivalently represented by (4.2), i.e. it generatesthe same semigroup satisfying (6.3).It thus remains to verify Item 3 above while we consider first 3a. As already noted, theoperator A has been explicitly constructed such that the first moment is zero. Thus, it onlyremains to prove the continuity, i.e. that A is regularising. To see this, we consider A , A and A separately. To begin with A , we find kA [ h ] k L (e µx ) = 4 Z ∞ e − x (cid:18)Z x h ( y ) χ { y ≤ R } (e y −
1) d y (cid:19) e µx d x ≤ (cid:18)Z R | h ( y ) | (e y −
1) d y (cid:19) Z ∞ e − (2 − µ ) x d x ≤ R − µ (cid:18)Z R | h ( y ) | d y (cid:19) ≤ R − µ k h k L k . (6.4)For A we get similarly kA [ h ] k L (e µx ) = 4 Z ∞ e − x (cid:18)Z ∞ x h ( y ) d y (cid:19) e µx d x ≤ (cid:18)Z ∞ | h ( y ) | d y (cid:19) Z ∞ e − (2 − µ ) x d x = 42 − µ (cid:18)Z ∞ | h ( y ) | d y (cid:19) ≤ − µ k h k L k . (6.5)Finally, recalling (6.2) from the proof of Lemma 6.9 we have kA [ h ] k L (e µx ) ≤ R + 1) − k ) k h k L k Z ∞ e − (2 − µ ) x d x = 4(2 − µ )( R + 1) k − k h k L k . (6.6)Summarising (6.4)–(6.6) we obtain kA [ h ] k L (e µx ) ≤ − µ (cid:16) e R + 1 + ( R + 1) − k ) (cid:17) k h k L k which shows that A is continuous from L k to L (e µx ).Finally, we prove Item 3b, i.e. that B generates a strongly continuous -semigroup e B t on L k which satisfies k e B t h k L k ≤ C e − λ t k h k L k for all h ∈ L k with R ∞ xh ( x ) d x = 0 and λ ≤
1. According to Lemma 6.10 the operator B generates a strongly continuous semigroupon L k which preserves the constraint on the first moment. Thus, it only remains to prove theindicated estimate on the semigroup and according to Lemma 6.10 we can restrict to the core C ∞ c (0 , ∞ ). Thus, for h ∈ C ∞ c (0 , ∞ ) let h = h ( x, t ) = e B t h such that ∂ t h = B h . We thushave the relation ∂ t k h k L k = Z ∞ (cid:0) ∂ t h ( x, t ) (cid:1) sgn( h ( x, t ))(1 + x ) k d x = Z ∞ B [ h ]( x ) sgn( h ( x, t ))(1 + x ) k d x . (6.7)31e require estimates on the right-hand side of this equation. Again, we treat the expressions B , B and B separately and to simplify the notation, we only write h ( x ), i.e. neglecting thetime-dependence, in the following. With | h | ′ = h ′ sgn( h ) integration by parts yields Z ∞ B [ h ]( x ) sgn( h ( x ))(1 + x ) k d x = Z ∞ | h ( x ) | ′ x (1 + x ) k d x = − Z ∞ | h ( x ) | (cid:0) (1 + x ) k + kx (1 + x ) k − (cid:1) d x = −k h k L k − k Z ∞ | h ( x ) | x (1 + x ) k − d x . (6.8)Next we consider B for which we obtain together with Fubini’s theorem and Lemma 6.6 that Z ∞ B [ h ]( x ) sgn( h ( x ))(1+ x ) k d x = 2 Z ∞ e − x Z x h ( y ) χ { y>R } (e y −
1) d y sgn( h ( x ))(1+ x ) k d x ≤ Z ∞ R e − x Z xR | h ( y ) | (e y −
1) d y (1 + x ) k d x = 2 Z ∞ R | h ( y ) | (e y − Z ∞ y e − x (1 + x ) k d x d y ≤ β Z ∞ R β | h ( y ) | (1 − e − y )(1 + y ) k d y . (6.9)Finally, recalling (6.2) from the proof of Lemma 6.9 we estimate B together with Lemma 6.7as Z ∞ B [ h ]( x ) sgn( h ( x ))(1 + x ) k d x ≤ R + 1) − k Z ∞ R | h ( y ) | (1 + y ) k d y Z ∞ e − x sgn( h ( x ))(1 + x ) k d x ≤ R + 1) − k Z ∞ R | h ( y ) | (1 + y ) k d y Z ∞ e − x (1 + x ) k d x ≤ k + 1)e( R + 1) k − Z ∞ R | h ( y ) | (1 + y ) k d y . (6.10)Summarising (6.8)–(6.10) we obtain Z ∞ B [ h ]( x ) sgn( h ( x ))(1 + x ) k d x ≤ −k h k L k − k Z ∞ | h ( x ) | x (1 + x ) k − d x + 2 β Z ∞ R β | h ( y ) | (1 − e − y )(1 + y ) k d y + 2Γ( k + 1)e( R β + 1) k − Z ∞ R β | h ( y ) | (1 + y ) k d y = −k h k L k − k Z R β | h ( x ) | x (1 + x ) k − d x + Z ∞ R β | h ( x ) | (1 + x ) k − (cid:16) β (1 + x )(1 − e − x ) + 2Γ( k + 1)(1 + x )e( R β + 1) k − − kx (cid:17) d x . (6.11)We fix β > k > β (notice this is where the restriction on the values of k comesinto play) and choose then R β sufficiently large such that (cid:16) β (1 + x )(1 − e − x ) + 2Γ( k + 1)(1 + x )e( R β + 1) k − − kx (cid:17) < x ≥ R β . R β is large enough to satisfy for example R β > (cid:16) k + 1)e( k − β ) (cid:17) k − − (cid:16) β + Γ( k + 1)e(1 + R β ) k − − k (cid:17) ( R β + 1) < − k. For this choice of R β , we deduce from (6.11) that Z ∞ B [ h ]( x ) sgn( h ( x ))(1 + x ) k d x ≤ −k h k L k . Recalling (6.7), Gr¨onwall’s inequality yields the desired estimate on the semigroup generatedby B . Remark . The fact that the operator L defined by (4.2) as an unbounded operator on L k is an extension of the same expression defined on L (e µx ) follows by an argument analogousto that in Remark 6.4. Precisely, from the proof of Proposition 6.3 we know that C ∞ c (0 , ∞ )is a core for L on L (e µx ). Since Lemma 6.9 and the proof of Theorem 5.3 provide that A and B (for R = 0) are bounded from L k to itself we deduce from Lemmas 4.4 and 4.6 that C ∞ c (0 , ∞ ) is also a core for L on L k and on this common core, both operators coincide. The spectral gap estimates proved in the previous section allow us to show that small pertur-bations of the equation for K = 2 have essentially the same behaviour, at least when solutionsare not far from the self-similar profile e − x for K = 2. We gather all local results of this typein this section. We call stability of the self-similar profiles with respect to the perturbation we are consideringthe property that for small ε the unit-mass profiles are close to the unique unit-mass profile G for ε = 0. We prove now a local version of this result, which states that this is true providedthe profiles are contained in a ball of a specific radius around G . Global versions are given inSection 3.4, but the advantage of the local statements we give now is that they give an optimalrate of stability, and they use only the properties of the linearised operator L .Our first observation is that the nonlinear operators defining the equation for ε = 0 andits perturbation are not far from each other in the k · k L k norm: Lemma 7.1.
Denote by N ε the self-similar Smoluchowski operator with kernel K ε ; that is, N ε ( f ) := 2 f + x∂ x f + C ε ( f, f ) . For any k ≥ and any f ∈ L k we have kN ε ( f ) − N ( f ) k L k ≤ ε k f k L k . Proof.
We have N ε ( f ) := 2 f + x∂ x f + C ε ( f, f ) , so kN ε ( f ) − N ( f ) k L k = kC ε ( f, f ) − C ( f, f ) k L k = ε kC W ( f, f ) k L k ≤ ε k f k L k , where the last inequality is due to Lemma 2.5.33e now give our local result on the stability of profiles: Lemma 7.2 (Local stability of profiles) . Take ≤ ε < , let G ε be a self similar profile forthe kernel K ε , and assume that k G ε − G k ≤ M , where M > is the one in Theorem 6.5.Take k ≥ . There exists an (explicit) constant M = M ( k ) > such that k G ε − G k L k ≤ εM . Proof.
Denote N ε the same operator as in Lemma 7.1, and let G ε be any self-similar profilefor the kernel K ε . Since N ε ( G ε ) = 0 we have kN ( G ε ) k L k = kN ( G ε ) − N ε ( G ε ) k L k ≤ ε k G ε k L k , (7.1)where we have used Lemma 7.1. Now N ( G ε ) = L ( G ε − G ) + C ( G ε − G , G ε − G ) , so kN ( G ε ) k L k = kL ( G ε − G ) + C ( G ε − G , G ε − G ) k L k ≥ kL ( G ε − G ) k L k − kC ( G ε − G , G ε − G ) k L k ≥ M k G ε − G k L k − k G ε − G k L k . Together with (7.1) this gives1 M k G ε − G k L k ≤ ε k G ε k L k + 3 k G ε − G k L k . If we assume that k G ε − G k L k ≤ M then this implies12 M k G ε − G k L k ≤ ε k G ε k L k . Proposition 3.3 then shows that the right hand side is finite and depends only on k .As an immediate consequence of Lemma 7.2 and Proposition 3.9, we then also obtain thefollowing global stability result (notice that in Section 3.4 the constant in Proposition 3.9 wasexplicitly estimated, so the D k in the following result is constructive). Also, we remark thata global result like Proposition 3.9 is essential here, since the stability of all possible solutionsto the self-similar equation cannot be proved by studying only its linearisation. Corollary 7.3 (Global stability of profiles) . For each k ≥ there exists a constant D k suchthat each self-similar profile G ε for the kernel K ε with ≤ ε ≤ satisfies k G ε − G k L k ≤ D k ε. .2 Uniqueness We first show a local uniqueness result which states that self-similar profiles are unique forsmall epsilon, provided they are within a certain distace of G . As in the previous resultsin this section, we prefer to state this local result because it depends only on linearisationarguments involving the operator L . Using the stability results in Section 3.4, it immediatelygives a full uniqueness result.In order to state our local uniqueness result we first show that the perturbed linear operators L ε around a self-similar profile G ε which is in a certain ball around G , also have a spectralgap in the L k spaces for small ε . This is not strictly needed, but it makes the later proof a biteasier. Notice that the operators L ε are just bounded perturbations of the operator L , so it isunderstood that they are defined in the same way as in Theorem 4.2, with the same domain. Lemma 7.4.
Take k ≥ and < ε < , and call L ε the linearised self-similar Smoluchowskioperator in the space L k , with kernel K ε , around a given self-similar profile G ε with mass .There is an explicit constant M = M ( k ) such that kL ε − L k ≤ εM Proof.
In terms of C K as given in (2.3) the operators L ε and L read L [ h ]( x ) = 2 h + xh ′ + 2 C ( G , h ) L ε [ h ]( x ) = 2 h + xh ′ + 2 C ε ( G ε , h ) . This yields L ε [ h ]( x ) − L [ h ]( x ) = 2 C ε ( G ε , h ) − C ( G , h ) = 2 C ( G ε − G , h ) + 2 ε C W ( G ε , h ) . With Propositions 2.5 and 3.3 and Corollary 7.3 we thus deduce kL ε [ h ] − L [ h ] k L k ≤ k G ε − G k L k k h k L k + 3 ε k G ε k L k k h k L k ≤ (6 D k ε + 3 C k ε ) k h k L k . According to Proposition 3.3 and Corollary 7.3 the coefficient M ( k ) := (6 D k + 3 C k ) dependsonly on k , which proves the result. Lemma 7.5 (Spectral gap of L ε ) . Let L ε be the linearised self-similar Smoluchowski operatorwith kernel K ε , around a given self-similar profile G ε with mass and k > . Then for ε < CM =: ε , where M = M ( k ) is from Lemma 7.4 and C = C ( k ) is from Theorem 6.5,the operator L ε has a spectral gap in L k of size / − CM ε . That is: with the same C = C ( k ) from Theorem 6.5 we have k e t L ε h k L k ≤ C k h k L k e − ( − CM ε ) t , t ≥ for all h ∈ L k with R xh = 0 . We sometimes state the result above by saying that, under these conditions, the linearisedoperator L ε has a spectral gap in L k of size 1 / − CM ε . Remark . In particular, under the assumptions of the previous result, by the Hille-Yosidatheorem we have kL ε h k L k ≥ − CM ε C k h k L k (7.2)for all h ∈ L k with R xh ( x ) d x = 0. 35 roof of Lemma 7.5. All norms used in this proof are k · k L k , and we omit the subscript tosimplify the notation. From Lemma 7.4 we have kL ε − L k ≤ εM =: δ. (7.3)We consider the equation ∂ t h = L ε h = L h + ( L ε − L ) h and write, using Duhamel’s formula and setting h t := e t L ε h , h t = e t L h + Z t e ( t − s ) L ( L ε − L ) h s d s . Hence, using Theorem 6.5 for L k h t k ≤ Ce − t k h k + C Z t e − ( t − s ) k ( L ε − L ) h s k d s . Hence, from (7.3), k h t k ≤ Ce − t k h k + Cδ Z t e − ( t − s ) k h s k d s . Calling u ( t ) := k h t k e t we see that u ( t ) ≤ C k h k + Cδ Z t u ( s ) d s , so by Gronwall’s Lemma we have u ( t ) ≤ C k h k e Cδt , that is k h t k ≤ C k h k e ( Cδ − ) t , t ≥ . Recalling (7.3) this shows the claim.We can finally give the proof of local uniqueness of the profiles:
Theorem 7.7 (Local uniqueness of self-similar profiles) . Take any k > . For all < ε < ε (with ε from Lemma 7.5), Smoluchowski’s coagulation equation with kernel K ε has at mostone self-similar profile G ε with mass satisfying k G ε − G k L k ≤ − ε/ε C (1 + ε ) with C = C ( k ) from Theorem 6.5.Proof. Let N ε be the the self-similar Smoluchowski operator with kernel K ε . Assume we havetwo different self-similar profiles G , G with mass 1 for the kernel K ε : N ε ( G ) = N ε ( G ) = 0 , and that they both satisfy k G − G k L k ≤ − ε/ε C (1 + ε ) , k G − G k L k ≤ − ε/ε C (1 + ε ) . (7.4)36all L ε the linearised self-similar Smoluchowski operator with kernel K ε , around the profile G . Since N ε ( f ) = L ε ( f − G ) + C εW ( f − G , f − G ),0 = N ε ( G ) − N ε ( G ) = L ε ( G − G ) − C εW ( G − G , G − G ) . Using Lemma 7.5 (see equation (7.2) in particular) and Proposition 2.5,1 − ε/ε C k G − G k ≤ kL ε ( G − G ) k ≤
32 (1 + ε ) k G − G k , so, since G = G , and assuming always ε < k G − G k ≥ − ε/ε C (1 + ε )This contradicts (7.4), since k G − G k ≤ k G − G k + k G − G k ≤ − ε/ε C (1 + ε ) . We can then use this local result for any fixed k >
2, together with the stability results inSection 3.4, to obtain that there is a unique unit-mass self-similar profile in L k . Since we knowfrom Section 3.1 that all profiles must be in L k , we immediately obtain a uniqueness result: Corollary 7.8 (Uniqueness of profiles for small perturbations) . There exists ε > suchthat for all ≤ ε ≤ ε Smoluchowski’s coagulation equation with kernel K ε has exactly oneself-similar profile G ε with mass . Proposition 8.1 (Local exponential convergence to equilibrium) . Take k > , and consider ε from Corollary 7.8. For any ≤ ε < ε , let G ε be the unique self-similar profile with mass for the kernel K ε . There exist constants C ∗ , M, ε , r > depending on k only such that forany ≤ ε ≤ ε , any solution f to the self-similar Smoluchowski equation (2.1) with kernel K ε with initial condition f ∈ L k such that Z ∞ xf ( x ) d x = 1 , k f − G ε k L k ≤ r satisfies k f ( t, · ) − G ε k L k ≤ C ∗ e − ( − Mε ) t k f − G ε k L k for all t ≥ .Proof. Since we have information on the spectral properties of the linearised operator L ε aroundthe profile G ε , the proof becomes a standard perturbation argument: we write the self-similarSmoluchowski equation as ∂ t f = C εW ( f, f ) + 2 f + x∂ x f = L ε ( f − G ε ) + C εW ( f − G ε , f − G ε ) . By Duhamel’s formula, calling h := f − G ε ,h t = e t L ε h + Z t e ( t − s ) L ε (cid:0) C εW ( h s , h s ) (cid:1) d s , λ ε := 1 / − CM ε (with C and M from Lemma 7.5), k h t k L k ≤ Ce − λ ε t k h k L k + C Z t e − λ ε ( t − s ) kC εW ( h s , h s ) k L k d s ≤ Ce − λ ε t k h k L k + 2 C k K ε k ∞ Z t e − λ ε ( t − s ) k h s k L k d s . If we define u ( t ) := k h ( t, · ) k L k e λ ε t we have u ( t ) ≤ Cu (0) + 2 C k K ε k ∞ Z t u ( s ) e − λ ε s d s . Gronwall’s lemma applied to this integral inequality then shows that u ( t ) ≤ (cid:18) Cu (0) − C k K ε k ∞ λ ε (1 − e − λ ε t ) (cid:19) − , which remains bounded for all t ≥ u (0) < λ ε C k K ε k ∞ . For example, if we assume u (0) < λ ε C k K ε k ∞ then u ( t ) ≤ Cu (0) for all t ≥ k h ( t, · ) k ≤ Ce − λ ε t k h k , which is what we wanted to show. If we additionally use our knowledge that solutions to the unperturbed problem with kernel K = 2 converge to equilibrium globally we can obtain a slight improvement of the above result.Namely, that the size R of the region in which we have convergence can be taken as large asone wants, provided ε is close enough to zero: Theorem 8.2 (Exponential convergence to equilibrium in large regions for small ε ) . Let k > and W be a bounded kernel of homogeneity , take R > , and take ≤ ε ≤ ε (with ε theone from Corollary 7.8 ensuring uniqueness of profiles). Denote K ε := 2 + εW , and call G ε the unique self-similar profile with mass for the kernel K ε . There exist constants C and M (depending only on k ) and ε > (depending on W , R and k ) such that any solution f to theself-similar Smoluchowski equation (2.1) with kernel K ε with ≤ ε ≤ ε and initial condition f ∈ L k with f ≥ almost everywhere and Z ∞ xf ( x ) d x = 1 , k f − G ε k ≤ R satisfies k f ( t, · ) − G ε k ≤ Ce − ( − Mε ) t k f − G ε k for all t ≥ . roof. The idea that we want to exploit is that for small ε , solutions to our perturbed equationare not too far from solutions to the equation for the constant kernel. Since we know that theequation for the constant kernel converges to equilibrium exponentially fast, we can show thatsolutions to the perturbed equation will eventually fall inside the local region where we canapply Proposition 8.1.For R > ε >
0, take k > f ∈ L k with mass 1 and k f − G ε k ≤ R . Call f ε the solution to the self-similar Smoluchowski equationwith kernel K ε , and f the solution to the self-similar Smoluchowski equation with constantkernel K = 2, both with initial condition f . From Lemma 2.15 we know that these twosolutions remain close for some time: for some C , C > k f ε ( t, · ) − f ( t, · ) k L k ≤ C εe C t . Also, according to Theorem 2.16 the solution f converges exponentially fast to G ( x ) = e − x : k f ( t, · ) − G k L k ≤ C e − t . Hence together with Corollary 7.3 k f ε ( t, · ) − G ε k L k ≤ k f ε ( t, · ) − f ( t, · ) k L k + k f ( t, · ) − G k L k + k G − G ε k L k ≤ C εe C t + C e − t + D k ε. We can choose large enough t (which we call t ), and then small enough ε , so that this quantityis less than the r in Proposition 8.1. Then, from Proposition 8.1, k f ε ( t, · ) − G ε k L k ≤ Ce − (1 / − Mε )( t − t ) k f ε ( t , · ) − G ε k L k for all t ≥ t .It is also easy to see that, for some C > k f ε ( t , · ) − G ε k L k ≤ e C t k f − G ε k L k , which then gives k f ε ( t, · ) − G ε k L k ≤ Ce − (1 / − Mε )( t − t ) e C t k f − G ε k L k for all t ≥ t .This shows the result for t ≥ t , and for t ≤ t we can easily obtain k f ε ( t, · ) − G ε k L k ≤ e C t k f − G ε k L k by similar calculations as in Proposition 8.1, using that we already know from 2.11 that k f ε ( t, · ) k L k is uniformly bounded for all times. This is enough to obtain the result. Acknowledgements
JAC and ST were supported by project MTM2017-85067-P, funded by the Spanish govern-ment and the European Regional Development Fund. ST has been funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 396845724.The authors would like to acknowledge the support of the Hausdorff Institute for Mathematics,since part of this work was completed during their stay at the Trimester Program on kinetictheory. 39
Proof of Theorem 2.16 on L convergence for the constantkernel We gather here the proof of Theorem 2.16, which is a small modification of [7, Lemma 6.1].Since the proof is independent of the rest of the paper and is a small improvement of theaforementioned one, we prefer to give it in an appendix.Our starting point is [7, Lemma 6.1]:
Lemma A.1.
Let f be a solution to (2.1) for the constant kernel K = 2 with total mass and initial condition f such that f ∈ L (d x ) ∩ L ( x d x ) . Let G ( x ) = e − x be the uniquestationary solution to (2.1) with total mass . There exist constants C, T > depending onlyon f such that k f ( t, · ) − G k L ≤ C e − t for all t ≥ T. We want to make two modifications to this statement, namely: 1. that a bound can begiven for all t ≥
0, and 2. that the constants can be explicitly calculated and depend only on k f k L and k f k L . The first modification is very simple, and we give it first: Lemma A.2.
In the conditions of Lemma A.1, there exists a constant
C > depending onlyon f such that k f ( t, · ) − G k L ≤ C e − t for all t ≥ . Proof.
For t ≥ T it is clearly true from Lemma A.1. For 0 ≤ t ≤ T we can use any availablebound on the growth of the L norm of a solution. For example, one can easily calculate that12 ddt k f k L = 32 k f k L + Z ∞ Z x f ( x ) f ( x − y ) f ( y ) d y d x − k f k L Z ∞ f ( x ) d x ≤ k f k L + k f k L Z ∞ f ( x ) d x , where we have used Cauchy-Schwarz’s inequality on the integral term and disregarded thenegative one. Since R f can be calculated explicitly, we see R f ≤ max { , R f } =: C , so k f k L ≤ k f k L exp ((3 + 2 C ) t ) for all t ≥ . In particular, for all 0 ≤ t ≤ T , k f k L ≤ k f k L exp (cid:18)(cid:0)
32 + C (cid:1) T (cid:19) e T e − t = k f k L exp ((2 + C ) T ) e − t , so k f − G k L ≤ k f k L + k G k L ≤ k f k L exp ((2 + C ) T ) e − t + 12 . We obtain then k f ( t, · ) − G k L ≤ C e − t for all t ≥ , with C := max { C,
12 + k f k L exp ((2 + C ) T ) } , where C and T are those from Lemma A.1. 40he final version we want to give is Theorem 2.16, which is the same as Lemma A.2 withthe addition that the constant C depends only on k f k L and k f k L . Let us give the proof ofthis: Proof of Theorem 2.16.
We notice that for t ∈ [0 , T ], the constant we obtain in the proof ofLemma A.2 depends only on k f k L and R ∞ f (increasingly), so it can be made to depend onlyon k f k L and k f k L , as we want. Hence we just need to check that the constants obtained inthe proof of [7, Lemma 6.1] depend only on the specified norms of f . One may assume that R f = 1, since one can always reduce the proof to that case by a change of variables. One cansee from the proof in [7] that all constants are explicit, except for the one called ε , defined by ε := inf | ξ | >ε | − φ ( ξ ) | , (A.1)where ε is a quantity that depends only on k f k L , and φ is the Fourier transform of f : φ ( ξ ) := Z ∞ e − iξx f ( x ) d x , ξ ∈ R . (Notice that we have adapted the definition of ε to our current choice of constant kernel K = 2instead of K = 1, as used in [7]; this is not essential). We need then to find an explicit lowerbound of ε that depends only on ε , k f k L and k f k L . This is given by Lemma B.4, whichwe prove in the remaining part of this appendix. B An estimate on the Fourier transform
In order to estimate the ε in (A.1) we need to understand the following. The Fourier transformof f is always less than or equal to 1 in absolute value, since f ≥ ξ = 0, and we need to find a quantitativeestimate of this phenomenon. Our final result is given in Lemma B.4, but we will need a fewlemmas to arrive there. The next one contains the central part of the argument: Lemma B.1.
Given
R > , take a nonnegative function f ∈ L ( − R, R ) ∩ L ( − R, R ) . Then Z R − R f ( x ) sin( x ) d x ≤ (1 − α ) Z R − R f ( x ) d x , where α := M πn k f k L , n := 1 + R π , M := Z R − R f ( x ) d x . Proof.
It is clearly enough to prove it in the case R R − R f ( x ) d x = 1, so we make this assumptionthroughout. For 0 < ε < π to be fixed later, we call A ε the ε -neighbourhood of the points in[ − R, R ] where sin x = 1: A ε := n x ∈ [ − R, R ] | (cid:12)(cid:12)(cid:12) x − (4 k + 1) π (cid:12)(cid:12)(cid:12) < ε for some odd integer k o , and B ε its complement in [ − R, R ]: B ε := [ − R, R ] \ A ε . B ε does not contain the points where sin x = 1, there is a positive function δ = δ ( ε ) suchthat Z B ε f ( x ) sin x d x ≤ (1 − δ ( ε )) Z B ε f ( x ) d x . For example, Lemma B.2 gives a simple explicit bound of δ ( ε ). For convenience, we call m A := Z A ε f ( x ) d x , m A := Z B ε f ( x ) d x = 1 − m A . Hence, Z R − R f ( x ) sin x d x = Z A ε f ( x ) sin x d x + Z B ε f ( x ) sin x d x ≤ m A + (1 − δ ( ε )) m B = m A + (1 − δ ( ε ))(1 − m A ) = 1 − δ ( ε )(1 − m A ) . Now, by Cauchy-Schwarz’s inequality we notice that m A = Z A ε f ( x ) d x ≤ k f k L p | A ε | ≤ k f k L r (cid:0) R π (cid:1) ε, since the Lebesgue measure of A ε is at most 2 (cid:0) R π (cid:1) ε . For convenience, call n := 1 + R π . Wethen choose ε such that k f k L √ nε = 12 , that is, ε := (8 n k f k L ) − and we obtain Z R − R f ( x ) sin x d x ≤ − δ ( ε ) . Using our bound of δ ( ε ) from Lemma B.2 we finally obtain the result. Lemma B.2.
For every < ε < π/ , δ ( ε ) := 1 − sup It is easy to check thatsin x ≤ − (cid:0) x − π (cid:1) π for all 0 ≤ x ≤ π , which easily implies the statement. Lemma B.3. Take a nonnegative function f ∈ L ( R ) ∩ L ( R ) with R ∞−∞ | x | f ( x ) d x < + ∞ .Then Z + ∞−∞ f ( x ) sin( x ) d x ≤ (1 − α ) Z + ∞−∞ f ( x ) d x , where α := M R k f k L , M := Z + ∞−∞ f ( x ) d x , R := max n M, Z + ∞−∞ | x | f ( x ) d x o . (In the trivial case that f = 0 , it is understood that the right hand side is also .) roof. Again, it is clearly enough to prove it when M = R ∞−∞ f = 1, so let us assume this. Call K := Z + ∞−∞ | x | f ( x ) d x . If we take any R ≥ K , then Z | x | >R f ( x ) d x ≤ K Z R | x | f ( x ) d x = K K = 12 . Now, call m R := Z | x | Take a nonnegative function f ∈ L ( R ) ∩ L ( R ) with R ∞−∞ | x | f ( x ) d x < + ∞ .Then for all ξ ∈ R we have (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞−∞ f ( x ) e − ixξ d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − α ) Z + ∞−∞ f ( x ) d x , where α = α ( f, ξ ) := ξ M R k f k L , M := Z + ∞−∞ f ( x ) d x , R := 2 ξ Z + ∞−∞ | x | f ( x ) d x + 2 πM. (In the trivial case that f = 0 , it is understood that the right hand side is also .) roof. By the change of variables y xξ , it is enough to prove it when ξ = 1. By scaling f as before, we may also assume that M = R ∞−∞ f = 1. We use the following trick to rewrite themodulus as an integral similar to that in Lemma B.3: if we call a := Z ∞−∞ f ( x ) cos x d x , b := Z ∞−∞ f ( x ) sin x d x , then a + b = 1 and there exists some θ ∈ [0 , π ) such that a = sin θ , b = cos θ . (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞−∞ f ( x ) e − ix d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18)Z + ∞−∞ f ( x ) cos x d x (cid:19) + (cid:18)Z + ∞−∞ f ( x ) sin x d x (cid:19) = sin θ Z + ∞−∞ f ( x ) cos x d x + cos θ Z + ∞−∞ f ( x ) sin x d x = Z + ∞−∞ f ( x ) sin( x + θ ) d x = Z + ∞−∞ f ( x − θ ) sin x d x . We then apply Lemma B.3 to ˜ f ( x ) := f ( x − θ ) to obtain the result. Notice that k ˜ f k = k f k , R ∞−∞ ˜ f = R ∞−∞ f , and Z ∞−∞ | x | ˜ f ( x ) d x = Z ∞−∞ | x + θ | f ( x ) d x ≤ Z ∞−∞ | x | f ( x ) d x + 2 πM. C Proof of Lemma 4.6 on the transport semigroup Proof of Lemma 4.6. We first show that for each t ≥ T t is well-defined on therespective spaces, while for H − (e µx ) we also recall Remark 4.5. In fact, using the change ofvariables x x e − t we find for L k that k T t h k L k = Z ∞ | h ( x e t ) | (1 + x ) k d x = e − t Z ∞ | h ( x ) | (1 + x e − t ) k d x ≤ e − t Z ∞ | h ( x ) | (1 + x ) k d x = e − t k h k L k . Similarly, we get k T t h k L (e µx ) = Z ∞ | h ( x e t ) | e µx d x = e − t Z ∞ | h ( x ) | e µx e − t d x ≤ e − t Z ∞ | h ( x ) | e µx d x = e − t k h k L (e µx ) . Finally, for H − (e µx ) we obtain k T t h k k H − (e µx ) = Z ∞ e µx (cid:18)Z ∞ x h k ( z e t ) d z (cid:19) d x = e − t Z ∞ e µx (cid:18)Z ∞ x e t h k ( z ) d z (cid:19) d x = e − t Z ∞ e µx e − t (cid:18)Z ∞ x h k ( z ) d z (cid:19) d x ≤ e − t Z ∞ e µx (cid:18)Z ∞ x h k ( z ) d z (cid:19) d x = e − t k h k k H − (e µx ) . 44n particular, this yields the estimates k T t k L k → L k ≤ e − t k T t k L (e µx ) → L (e µx ) ≤ e − t k T t k H − (e µx ) → H − (e µx ) ≤ e − t (C.1)for all t ≥ t → T t h = h for all h in a dense subset D ⊂ L ((1+ x ) k ), D ⊂ L (e µx ) or D ⊂ H − (e µx ) respectively. Thus, taking for example D = C ∞ c ((0 , ∞ )) wefind for h ∈ D that k T t h − h k L k = Z ∞ | h ( x e t ) − h ( x ) | (1 + x ) k d x = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z t ∂ τ h ( x e τ ) d τ (cid:12)(cid:12)(cid:12)(cid:12) (1 + x ) k d x = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z t x e τ h ′ ( x e τ ) d τ (cid:12)(cid:12)(cid:12)(cid:12) (1 + x ) k d x ≤ Z t e τ Z ∞ | h ′ ( x e τ ) | x (1 + x ) k d x d τ = Z ∞ e − τ Z ∞ | h ′ ( x ) | x (1 + x e − τ ) k d x d τ ≤ k xh ′ ( x ) k L k (1 − e − t ) . For t → L ((1 + x ) k ). For L (e µx ) we argue similarly and get k T t h − h k L (e µx ) = Z ∞ | h ( x e t ) − h ( x ) | e µx d x = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z t ∂ τ h ( x e τ ) d τ (cid:12)(cid:12)(cid:12)(cid:12) e µx d x = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z t x e τ h ′ ( x e τ ) d τ (cid:12)(cid:12)(cid:12)(cid:12) e µx d x . Together with Cauchy-Schwartz and the change of variables x x e − τ , we find k T t h − h k L (e µx ) ≤ Z ∞ t Z t x e τ | h ′ ( x e τ ) | d τ e µx d x = t Z t e τ Z ∞ x | h ′ ( x e τ ) | e µx d x d τ ≤ t Z t e − τ Z ∞ x | h ′ ( x ) | e µx d x d τ = k xh ′ ( x ) k L (e µx ) t (1 − e − t ) . Again, the right-hand side converges to zero as t → L (e µx ). Finally, for H − (e µx ) we get analogously k T t h − h k H − (e µx ) = Z ∞ e µx (cid:18)Z ∞ x h ( z e t ) − h ( z ) d z (cid:19) d x = Z ∞ e µx (cid:18)Z ∞ x Z t ∂ τ h ( z e τ ) d τ d z (cid:19) d x = Z ∞ e µx (cid:18)Z ∞ x Z t z e τ h ′ ( z e τ ) d τ d z (cid:19) d x = Z ∞ e µx (cid:18)Z t e − τ Z ∞ x e τ zh ′ ( z ) d z d τ (cid:19) d x . k T t h − h k H − (e µx ) ≤ Z ∞ e µx (cid:18)Z t e − τ d τ (cid:19)(cid:18)Z t (cid:18)Z ∞ x e τ zh ′ ( z ) (cid:19) d τ (cid:19) d x = 12 (1 − e − t ) Z ∞ e µx Z t (cid:18)Z ∞ x e τ zh ′ ( z ) d z (cid:19) d τ d x . Since we are interested in the limit t → 0, we can assume that t ≤ h ∈ C ∞ c (0 , ∞ ) the integral on the right-hand side is bounded. Therefore, for t → H − (e − µx ).To determine the generator, we take h ∈ C ∞ c (0 , ∞ ) and computelim t → t ( T t h − h )( x ) = lim t → t (cid:0) h ( x e t ) − h ( x ) (cid:1) = xh ′ ( x ) . This shows that C ∞ c (0 , ∞ ) ⊂ D ( B ) and B | C ∞ c (0 , ∞ ) = x∂ x . Thus, to conclude the proof itsuffices to prove that C ∞ c is a core for B .According to [8, Ch. I, 1.7 Proposition], it suffices to verify that C ∞ c (0 , ∞ ) is invariantunder the action of ( T t ) t ≥ and that C ∞ c (0 , ∞ ) is dense in L ((1 + x ) k ), L (e µx ) and H − (e µx ),respectively. Due to the explicit formula ( T t h )( x ) = h ( x e t ) the invariance is clear while densityis also well-known or clear by construction.The claim for the spaces L ((1 + x ) k ) ∩ { R ∞ xf ( x ) d x = 0 } , L (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } and H − (e µx ) ∩ { R ∞ xf ( x ) d x = 0 } directly follows by restricting the semigroup once we noticethat ( T t ) t ≥ preserves the constraint. References [1] Azmy S. Ackleh. Parameter estimation in a structured algal coagulation-fragmentationmodel. Nonlinear Anal. , 28(5):837–854, 1997.[2] Azmy S. Ackleh and Ben G. Fitzpatrick. Modeling aggregation and growth processes inan algal population model: analysis and computations. J. Math. Biol. , 35(4):480–502,1997.[3] Eric J. Allen and Pierre Bastien. On coagulation and the stellar mass spectrum. TheAstrophysical Journal , 452:652, Oct 1995.[4] Jos´e A. Ca˜nizo and Bertrand Lods. Exponential convergence to equilibrium for subcriticalsolutions of the Becker–D¨oring equations. Journal of Differential Equations , 255(5):905–950, September 2013.[5] Jos´e A. Ca˜nizo and Bertrand Lods. Exponential trend to equilibrium for the inelasticBoltzmann equation driven by a particle bath, July 2015.[6] Jos´e A Ca˜nizo and St´ephane Mischler. Regularity, local behavior and partial uniquenessfor Smoluchowski’s coagulation equation. Revista Matem´atica Iberoamericana , 27(3):803–839, 2011.[7] Jos´e A. Ca˜nizo, St´ephane Mischler, and Cl´ement Mouhot. Rate of convergence to self-similarity for Smoluchowski’s coagulation equation with constant coefficients. SIAM Jour-nal on Mathematical Analysis , 41(6):2283–2314, 2010.468] Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolutionequations , volume 194 of Graduate Texts in Mathematics . Springer-Verlag, New York,2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel,D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.[9] M. Escobedo and S. Mischler. Dust and self-similarity for the Smoluchowski coagulationequation. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire , 23(3):331–362, 2006.[10] M. Escobedo, S. Mischler, and B. Perthame. Gelation in coagulation and fragmentationmodels. Comm. Math. Phys. , 231(1):157–188, 2002.[11] M. Escobedo, S. Mischler, and M. Rodriguez Ricard. On self-similarity and stationaryproblem for fragmentation and coagulation models. Ann. Inst. H. Poincar´e Anal. NonLin´eaire , 22(1):99–125, 2005.[12] Nicolas Fournier and Philippe Lauren¸cot. Existence of self-similar solutions to Smolu-chowski’s coagulation equation. Comm. Math. Phys. , 256(3):589–609, 2005.[13] Nicolas Fournier and Philippe Lauren¸cot. Local properties of self-similar solutions toSmoluchowski’s coagulation equation with sum kernels. Proc. Roy. Soc. Edinburgh Sect.A , 136(3):485–508, 2006.[14] Sheldon K. Friedlander. Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics .Topics in Chemical Engineering. Oxford University Press, 2000.[15] Maria P. Gualdani, St´ephane Mischler, and Cl´ement Mouhot. Factorization for non-symmetric operators and exponential H-theorem. M´emoires de la Soci´et´e Math´ematiquede France , 2018. To appear.[16] Philippe Lauren¸cot. Uniqueness of Mass-Conserving Self-similar Solutions to Smo-luchowski’s Coagulation Equation with Inverse Power Law Kernels. J. Stat. Phys. ,171(3):484–492, 2018.[17] Philippe Lauren¸cot, Barbara Niethammer, and Juan J. L. Vel´azquez. Oscillatory dynam-ics in Smoluchowski’s coagulation equation with diagonal kernel. Kinet. Relat. Models ,11(4):933–952, 2018.[18] Govind Menon and Robert L. Pego. Approach to self-similarity in Smoluchowski’s coag-ulation equations. Communications on Pure and Applied Mathematics , 57(9):1197–1232,2004.[19] Govind Menon and Robert L. Pego. Dynamical scaling in Smoluchowski’s coagulationequations: uniform convergence. SIAM J. Math. Anal. , 36(5):1629–1651, 2005.[20] S. Mischler and C. Mouhot. Cooling process for inelastic Boltzmann equations for hardspheres. II. Self-similar solutions and tail behavior. J. Stat. Phys. , 124(2-4):703–746, 2006.[21] S. Mischler and C. Mouhot. Stability, convergence to self-similarity and elastic limit for theBoltzmann equation for inelastic hard spheres. Communications in Mathematical Physics ,288(2):431–502, June 2009. 4722] S. Mischler and C. Mouhot. Exponential stability of slowly decaying solutions tothe kinetic-Fokker-Planck equation. Archive for Rational Mechanics and Analysis ,221(2):677–723, Aug 2016.[23] S. Mischler, C. Mouhot, and M. Rodr´ıguez Ricard. Cooling Process for Inelastic Boltz-mann Equations for Hard Spheres, Part I: The Cauchy Problem. Journal of StatisticalPhysics , V124(2):655–702, 2006.[24] Cl´ement Mouhot. Rate of convergence to equilibrium for the spatially homogeneousBoltzmann equation with hard potentials. Communications in Mathematical Physics ,261(3):629–672, February 2006.[25] B. Niethammer, S. Throm, and J. J. L. Vel´azquez. A revised proof of uniqueness ofself-similar profiles to Smoluchowski’s coagulation equation for kernels close to constant. Preprint arXiv:1510.03361v3 , October 2015.[26] B. Niethammer, S. Throm, and J. J. L. Vel´azquez. Self-similar solutions with fat tails forSmoluchowski’s coagulation equation with singular kernels. Ann. Inst. H. Poincar´e Anal.Non Lin´eaire , 33(5):1223–1257, 2016.[27] B. Niethammer, S. Throm, and J. J. L. Vel´azquez. A uniqueness result for self-similarprofiles to Smoluchowski’s coagulation equation revisited. J Stat Phys , 164(2):399–409,Jun 2016.[28] B. Niethammer and J. J. L. Vel´azquez. Self-similar solutions with fat tails for Smo-luchowski’s coagulation equation with locally bounded kernels. Comm. Math. Phys. ,318(2):505–532, 2013.[29] B. Niethammer and J. J. L. Vel´azquez. Exponential tail behavior of self-similar solutions toSmoluchowski’s coagulation equation. Comm. Partial Differential Equations , 39(12):2314–2350, 2014.[30] B. Niethammer and J. J. L. Vel´azquez. Uniqueness of self-similar solutions to Smolu-chowski’s coagulation equations for kernels that are close to constant. Journal of StatisticalPhysics , 157(1):158–181, 2014.[31] H.R. Pruppacher and J.D. Klett. Microphysics of Clouds and Precipitation . SpringerNetherlands, 2010.[32] J. Silk and T. Takahashi. A statistical model for the initial stellar mass function. TheAstrophysical Journal , 229:242, Apr 1979.[33] R. Srinivasan. Rates of convergence for Smoluchowski’s coagulation equations. SIAMJournal on Mathematical Analysis , 43(4):1835–1854, 2011.[34] Sebastian Throm. Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagu-lation equation for a perturbation of the constant kernel.