ON LOCAL SMOOTH SOLUTIONS FOR THE VLASOV EQUATION WITH THE POTENTIAL OF INTERACTIONS ±r−2

For the initial value problem for the Vlasov equation with the potential of interactions ±r−2, we prove the existence and uniqueness of a local solution with values in the Schwartz space S of infinitely differentiable functions rapidly decaying at infinity.

1. Introduction, notation, and results.Vlasov equation and its various modifications are classical equations of physics.They appear in the mean-field approximations of the dynamics of a large number of interacting classical particles (molecules).Currently, there is a numerous literature devoted to its mathematical treatments.In particular, in [2,5,7,12] a well-posedness for this equation supplied with initial data and its derivation from molecular dynamics are considered in the case when the potential of interactions between particles is smooth and bounded.In [1,3,6,8,10,11,13,15], this equation is studied for the singular Coulomb potential U(r ) = ±r −1 (in [6,13], the Vlasov-Maxwell system and the two-dimensional Vlasov-Poisson system are considered, respectively).In [9], the local existence of smooth solutions in the case U(r ) = ±r −2 is studied.We also mention paper [14] where a well-posedness of this equation supplied with a joint distribution of particles at two moments of time is proved.
In the present paper, we consider the problem where all quantities are real, x, v ∈ R 3 , κ is a constant, ∇ x and ∇ v are the gradients in x and v, respectively, v • ∇ x f and ∇ v f • E(x, t) are the scalar products in R 3 , and f is an unknown function.For any fixed t, f (t,x,v) regarded as a function of (x, v) has the sense of a distribution function of particles in (x, v) ∈ R 3 × R 3 .Therefore, the following requirements are natural: Generally speaking, it is known that proving the existence of a solution for problem (1.1), (1.2), (1.3), and (1.4) is more difficult for a singular potential U than for a more regular one.Also, although the Vlasov equation appeared for the first time with the Coulomb potential U(r ) = ±r −1 for a description of plasma, it is well known that in statistical physics potentials with higher singularities occur, for example, the following one, the so-called Lennard-Jones potential, is known: U(r ) = Ar −12 − Br −6 .So, the author of the present paper believes that considerations of Vlasov equations for potentials with singularities of degrees higher than r −1 make sense.Here we consider the case U(r ) = ±r −2 proving the existence and uniqueness of a local smooth solution of the problem (1.1), (1.2), (1.3), and (1.4).This case is critical in a sense.A treatment of the problem in the case U(r ) = r −2−a with a > 0 is still left open.As for the case U(r ) = r −a with a ∈ (1, 2), here the problem becomes simpler, and our methods still hold for it.
The existence and uniqueness of a smooth solution of problem (1.1), (1.2), (1.3), and (1.4) is proved in [9].In fact, we prove a similar result by using another method that allows to treat the problem in a simpler and shorter way.We do not exploit the known method of characteristics but we use an approach known in the theory of nonlinear PDEs, too.Our results also hold for potentials of interaction of a more general kind U(r ) = κr −2 +U 1 (r ), where U 1 (r ) is a function continuously differentiable everywhere except maybe the point r = 0 where it may have a singularity of order |r | a−2 with 0 < a < 2 and U 1 (r ) must satisfy certain conditions of decay at infinity.Now, we introduce some notation.Let (1.5) The linear space S is equipped with a topology of open subsets becoming a complete topological space.This topology is generated by the system of seminorms where k = (k 1 ,k 2 ) with k i ,m = 0, 1, 2,....By C(I; S), where I ⊂ R is an interval, we denote the linear space of all continuous functions g : I → S such that each seminorm p k,m (g(t)) and q k,m (g(t)) is bounded uniformly in t ∈ I.
Our main result here is the following.

Lemma 2.2. There exists
Proof.As is well known, there exists C 1 > 0 such that (2.6) Also, and ).Hence, we have, for g ∈ S, Let [a] be the maximal integer not larger than a real a and let an integer m 0 > 0 be such that each Sobolev space H l+m 0 (R 3 ×R 3 ) with a positive integer l is embedded into In what follows, we exploit the following embeddings (g ∈ S): where 0 ≤ l = n + r ≤ s, i, j = 1, 2, 3, and m = m 0 ,m 0 + 1,m 0 + 2,... .To prove them, consider the partition of R 3 ×R 3 by cubes K α defined by where r i , p j run over all integers.Observe that for any integer for all α.Now, inequalities (2.9) follow by standard Sobolev embeddings applied to each cube K α with further summing in α.

Proof.
We derive our estimates only for t > 0 because the case t < 0 can be treated by analogy.Integrating by parts in v using the independence of

2, and the fact that
, and applying Hölder's inequality, we have, from (2.1), 1 2 (2.12) Consider 1 2 (2.13) To estimate the term with the (m 1 + 1)st derivative of f n , we integrate by parts in v and use the fact that E n (x, t) does not depend on v. Also, as above, we estimate the uniform norm of E n (x, t) by p (0,2),0 (f n (t, and apply Hölder's inequality.To estimate all other terms in the right-hand side of equation (2.13), observe that the order of one of the derivatives in )E n,j (x, t) is not larger than m 0 + 1 and therefore, either one has or the estimate in the fourth string of (2.9) takes place; in addition, we estimate the L 2 -norm of the other cofactor with the derivative of order larger than m 0 + 1 by the same quantity corresponding to it.So, applying Hölder's inequality, we arrive at the estimate 1 2 (2.15) Finally, we deduce by complete analogy 1 2 (2.16) Now, in view of (2.12), (2.13), (2.14), (2.15), and (2.16), denoting A = p 2 (0,2),0 (f n ) + p 2 (0,2),m 1 (f n ) and B = p 2 (0,2),0 (f n ) + q 2 (0,2),m 1 (f n ), where p (0,2),0 (f n ) ≡ q (0,2),0 (f n ), we obtain and our claim is proved.

Lemma 2.5. There exist
Proof.Again, we establish all our estimates only for t > 0. Let m ≥ 2m 0 + 3. It follows from (2.1) that 1 2 (2.24) The first term in the integrand in the right-hand side of this relation can be estimated as when proving Lemma 2.3: we integrate by parts in v and apply the estimate As for the other terms in the integrand in (2.24), again, the order of one of two derivatives in (1 x,v)/∂v j ∂x l i ) and (∂ m−l E n,j (x, t)/∂x m−l i ) is not larger than [m/2] + 1 and so, the uniform norm of the corresponding expression can be estimated by , respectively; the L 2 -norm of the other cofactor can be estimated in a similar way.So, as earlier, applying Hölder's inequality, we arrive at the estimate 1 2 (2.26) By complete analogy 1 2 (2.27) Now, the statement of our lemma follows from (2.26), (2.27), and Lemmas 2.3 and 2.4 by induction.
In view of Lemmas 2.2, 2.3, 2.4, and 2.5 and (2.1), the sequences {f n } and {df n /dt} are relatively compact in C([−T ,T ]; S).Without loss of generality, we can accept that these sequences converge and let f (t,x,v) and f 1 (t,x,v) be their limit points in this sense.Clearly, f t (t,x,v) ≡ f 1 (t,x,v) and f satisfies problem (1.1), (1.2), (1.3), and (1.4).Now, we prove the uniqueness of this solution.Assume the existence of two solutions f 1 and f 2 of the above class and set f = f 1 − f 2 .As earlier, we establish our estimates only for t > 0. One can obtain, as when proving Lemmas 2.3 and 2.4, therefore f (t,x,v) ≡ 0 for all t ∈ [0,T ], which completes our proof of the theorem.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation