A One-dimensional kinetic model of plasma dynamics with a transport field

Motivated by the fundamental model of a collisionless plasma, the Vlasov-Maxwell (VM) system, we consider a related, nonlinear system of partial differential equations in one space and one momentum dimension. As little is known regarding the regularity properties of solutions to the non-relativistic version of the (VM) equations, we study a simplified system which also lacks relativistic velocity corrections and prove local-in-time existence and uniqueness of classical solutions to the Cauchy problem. For special choices of initial data, global-in-time existence of these solutions is also shown. Finally, we provide an estimate which, independent of the initial data, yields additional global-in-time regularity of the associated field.


Introduction.
A plasma is a partially or completely ionized gas. Such a form of matter occurs if the velocity of individual particles in a material achieves an enormous magnitude, perhaps a sizable fraction of the speed of light. Plasmas are widely used in solid state physics since they are great conductors of electricity due to their free-flowing abundance of ions and electrons. When a plasma is of low density or the time scales of interest are sufficiently small, it is deemed to be "collisionless", as collisions between particles become infrequent. Many examples of collisionless plasmas occur in nature, including the solar wind, galactic nebulae, the Van Allen radiations belts, and comet tails.
The fundamental equations which describe the time evolution of a collisionless plasma are given by the Vlasov-Maxwell system: in the first equation of (VM), called the Vlasov equation, and in the integrand of the current j. General references on the kinetic equations of plasma dynamics, such as (VM) and (RVM), include [7] and [13].
Over the past twenty-five years significant progress has been made in the analysis of (RVM), specifically, the global existence of weak solutions (which also holds for (VM); see [3]) and the determination of sufficient conditions which ensure global existence of classical solutions (originally discovered in [8], and later in [9], and [1]) for the Cauchy problem. Additionally, a wide array of information has been discovered regarding the electrostatic versions of both (VM) and (RVM) -the Vlasov-Poisson and relativistic Vlasov-Poisson systems, respectively. These models do not include magnetic effects within their formulation, and the electric field is given by an elliptic, rather than a hyperbolic equation. This simplification has led to a great deal of progress concerning the electrostatic systems, including theorems regarding global existence, stability, and long-time behavior of solutions; though a global existence theorem for classical solutions with arbitrary data in the relativistic case has remained elusive. Independent of these advances, many of the most basic existence and regularity questions remain unsolved for (VM). The main difficulty which arises is the loss of strict hyperbolicity of the kinetic system due to the possibility that particle velocities v may travel faster than the propagation of signals from the electric and magnetic fields, which do so at the speed of light c = 1. As one can see, this difficulty is remedied by the inclusion of relativistic velocity corrections which uniformly constrain velocities |v| < 1. In many physical systems one does not consider the effects of special (or general) relativity, but at the kinetic level such velocity corrections may play a fundamental role, even in the basic existence, uniqueness, and regularity properties of solutions. Hence, one of the primary goals of the current work is to understand how this difference in formulation affects such properties, and yield a partial answer to the question, "Are relativistic velocity corrections really necessary to ensure classical well-posedness?". It should be noted here that, whereas (RVM) is invariant under Lorentzian transformations, (VM) lacks invariance properties as it combines a Galilean-invariant equation for the particle distribution with a Lorentz-invariant field equation. To date, though, we are 1D PLASMA MODEL WITH TRANSPORT FIELD 3 unaware of any argument which truly utilizes the invariance properties of (RVM) or (VM) in order to arrive at an existence, uniqueness, or regularity theorem.
Often a remedy to the lack of progress on such a problem is to reduce the dimensionality of the system. Unfortunately, posing the problem in one-dimension (i.e., x, v ∈ R) eliminates the relevance of the magnetic field as the Maxwell system decouples, yielding the one-dimensional Vlasov-Poisson system: The lowest-dimensional reduction which includes magnetic effects is the so-called "one-and-one-half-dimensional" system which is constructed by taking x ∈ R but v ∈ R 2 : Surprisingly, the question of classical regularity remains open even in this simplified case. The noticeable difference between (1.5D VM) and (VP) is the introduction of electric and magnetic fields E 2 and B that are solutions to transport equations. Thus, in order to study the existence and regularity questions, but keep the problem posed in a one-dimensional setting, we consider the following nonlinear system of PDE which couples the Vlasov equation to a transport field equation: The system (1.1) is supplemented by given initial data Since the field equation in (1.1) is hyperbolic, we denote it with a magnetic field variable B, as opposed to the electric field E of (VP) which satisfies an elliptic equation. Notice that these equations retain the main difficulty of (VM), namely the interaction between characteristic particle velocities v and constant field velocities c = 1. Hence, we hope to analyze (1.1) and develop estimates or methods which can be generalized to deal with (1.5D VM). To our knowledge, this is the first analytic study of these kinetic equations, formed from a system of conservation laws coupled by a non-local field dependence on the particle densities. As such, the properties of solutions to (1.1) may also be of interest to mathematicians studying scalar, hyperbolic conservation laws in a two-dimensional phase space. A related model, similar to (1.1), was previously studied [5] in an attempt to understand the possible singularities generated by the intersection of Vlasov and transport characteristics. However, only minor results were derived in this work, all of which concerned a reduced system of ODEs rather than PDEs. Here, we present results for the full system and our results also generalize to their original system of PDEs. We also mention the work [2] as it contains a discussion of computational methods for (1.1).
This paper proceeds as follows. In the next section, we will derive a priori estimates in order to simplify the proof of the local-in-time existence and uniqueness of classical solutions to (1.1) with given smooth initial data (1.2), which follows in Section 3. In Section 4, we present two results concerning global existence, namely that certain particle distributions for which particle velocities travel at light speed do remain smooth globally in time. The unfortunate detail of Theorems 4.1 and 4.3, however, is that they do not necessarily extend to arbitrary initial data. Hence, it still remains an open problem to show that any solution launched from smooth initial data remains smooth for all time. As an intermediate step, we prove in Section 5 an additional regularity result for the associated field B using a decomposition of derivatives similar to that of [8]. More specifically, we show a priori that for any T > 0, B ∈ C 0,1/2 ([0, T ] × R). Throughout the paper the value C > 0 will denote a generic constant that may change from line to line. When necessary, we will specifically identify the quantities upon which C may depend (e.g., C T ). Since we are interested in classical solutions, we will also assume the initial data are smooth and bounded, i.e. f 0 ∈ C 1 c (R 2 ) and B 0 ∈ C 1 (R), for the entirety of the paper.

2.
A priori estimates. To begin, we will first prove a lemma that will allow us to represent and bound the particle density, its derivatives, and the associated field. (a). Let T > 0 be given and f be the solution of the Vlasov equation with given initial data ) and for any t ∈ [0, T ] we have the estimates where C depends only upon the initial data.
(b). Let T > 0 be given and B be the solution to the field equation with a given initial condition, namely for some given f ∈ C 1 ([0, T ]; C 1 c (R 2 )) and define Then, B ∈ C 1 ([0, T ] × R) and for any t ∈ [0, T ], we have the estimate where C depends only upon the initial data. Proof. To prove the first result, we begin by introducing characteristics for the Vlasov equation. Define the curves X(s, t, x, v) and V (s, t, x, v) as solutions to the system of ODEs Often, the (t, x, v) dependence of these curves will be suppressed so, for example, X(s, t, x, v) will be denoted by X(s) for brevity. Then, the Vlasov equation can be expressed as a derivative along the characteristic curves by Thus, we find Finally, taking the supremum of this equality over x, v ∈ R yields the first estimate in (a). The remaining estimates involve derivatives, so differentiating the Vlasov equation in v yields and upon integrating along characteristics we find Taking supremums in x and v gives the last estimate for (a). The second estimate is derived similarly. Differentiating with respect to x in the Vlasov equation, we find and integrating as before along characteristics yields

CHARLES NGUYEN, JENNIFER ANDERSON, AND STEPHEN PANKAVICH
Taking the supremum on the right side, inserting the estimate on ∂ v f (s) ∞ above, and using the bounded data gives us Finally, taking the supremum gives the estimate for For the claim in (b), we may use the method of characteristics to solve for B in terms of f . First, we write the differential equation for B as a derivative along the curves (s, Hence, integrating along these curves, we arrive at Using the bounded data, compact velocity support, and uniform bound on the particle density yields Finally, taking the supremum in x concludes the proof.
3. Local-in-time existence of classical solutions. With these a priori estimates and classical convergence theorems from analysis, we now have the necessary tools to prove the local-in-time existence theorem.
and B 0 ∈ C 1 (R) be given. Then, there exists T > 0 and a unique classical solution . Moreover, if we denote the maximal lifespan of the solution by T * then for T * < ∞ we must have lim sup Proof. The outline of our proof generally follows the structure of [10], [12], and [11]. We begin with the existence argument, which utilizes the method of successive approximations. Hence, we define an iterative sequence of solutions to linear PDEs 1D PLASMA MODEL WITH TRANSPORT FIELD 7 and show that it must converge to a solution of the nonlinear system (1.1). We take Additionally, for every n ∈ N, define f n ∈ C 1 ([0, ∞); C 1 c (R 2 )) and B n ∈ C 1 ([0, ∞)× R) by solving the linear initial-value problems respectively. Notice that if f n → f and B n → B in the appropriate sense as n → ∞ then f and B will satisfy (1.1). Now as in Lemma 2.1, we further define the sequence of velocity support functions for every t ≥ 0, n ∈ N, 3.1. Uniform boundedness. For the first portion of the proof, we consider T > 0 given and estimate on the time interval [0, T ]. In order to uniformly bound the sequence B n , we utilize the estimates on the velocity support of f n . First, by Lemma 2.1, we have the bound To bound the velocity support in terms of the field, we express the solution of the Vlasov equation in terms of the associated characteristics. For every n ∈ N define the characteristic curves X n (s, t, x, v) and V n (s, t, x, v) by As before, the (t, x, v) dependence of these curves will be suppressed for brevity. Then, using the argument in the proof of the first result of Lemma 2.1, we find for every n ∈ N. Inverting the characteristics using the identities

CHARLES NGUYEN, JENNIFER ANDERSON, AND STEPHEN PANKAVICH
within this last equality, then utilizing the compact support of f 0 and the definition of P n it follows that sup for every t ≥ 0. From the velocity characteristic equation (3.4), we can integrate to find Taking the supremum over characteristics along which f n = 0, we find We can now use (3.3) to arrive at a recursive bound for P n , namely Since f 0 has compact support, we know P n (0) is finite and constant in n, thus for every n ∈ N and on every bounded time interval [0, T ], where C depends upon f 0 and T . Using this recursive relation, we immediately deduce Thus, on any bounded time interval [0, T ] the function P n (t) is uniformly bounded and from (3.3) so is B n (t) ∞ .

3.2.
Uniform boundedness of derivatives. Now we focus on obtaining uniform bounds on derivatives, sketching the proof for x derivatives, with t and v derivatives following similarly. From the definition of the iterates we can differentiate the representation for B n (2.4) after integrating along characteristics with speed one, so that Using the bound on the velocity support above, we arrive at for every n ∈ N. Using Lemma 2.1(a) we also have the estimate 1D PLASMA MODEL WITH TRANSPORT FIELD 9 We combine these inequalities to find With this definition, the previous inequality yields is the maximal solution of the integral equation corresponding to (3.7): This solution exists on some time interval [0, T * ) with T * > 0 determined by f 0 and B 0 . This yields a uniform bound on ∂ x f n (t) ∞ on [0, T ] for every n ∈ N and T < T * . Additionally, ∂ x B n (t) ∞ is bounded on the same interval by (3.6). The argument can be repeated in the same manner to bound 3.3. Uniform Cauchy property. For the remainder of the proof, we will estimate on the bounded time interval [0, T ], where T ∈ (0, T * ). To show that the sequences f n and B n are Cauchy, we directly estimate differences of terms of the sequences. Let n, m ∈ N be given and define the functions Since (3.1) holds for any n ∈ N, we subtract the f m equation from that for f n to find so that by rearranging terms this becomes As Now, integrating both sides with respect to s, we find and since both f n and f m satisfy the same initial condition (3.1), it follows that f n,m (0, x, v) = 0. Therefore, the equality simplifies to We know ∂ v f m (s) ∞ is uniformly bounded (from Section 3.2) so we can bound the right side to find Now, since (3.2) must also hold for all n ∈ N, we subtract the B m equation from that for B n and arrive at which we can write as a derivative along curves with slope one as Integrating in s and using the initial conditions (3.2) to conclude that B n,m (0, x) = 0 for every x ∈ R, this becomes Since the velocity support of f , denoted by P n , is uniformly bounded from Section 3.11, we can bound B n,m by for any n ∈ N with n ≥ 2. Using this recursive relation for successive differences, we deduce as n, m → ∞ for every t ∈ [0, T ]. Therefore, B n (t, x) is uniformly Cauchy for all t ∈ [0, T ] and x ∈ R. Similarly, using (3.9) we see that f n,m (t) ∞ → 0 as n → ∞ for every t ∈ [0, T ] and it follows that f n (t, x, v) is uniformly Cauchy for every t ∈ [0, T ] and x, v ∈ R.
To conclude this section, we use the newly discovered boundedness of derivatives and Cauchy property of the field to show that the characteristics given by (3.4) are uniformly Cauchy. First, we let m, n ∈ N be given and define with the analogous definition for V n,m (s). Upon integrating the ODEs of (3.4) and subtracting the equations for X m from those of X n , we find Doing the same for the V n (s) equations, we simply use the Mean Value Theorem to find Since field derivatives are bounded from Section 3.2, this implies Finally, we let Z n,m (s) = X n,m (s) ∞ + V n,m (s) ∞ and add (3.3) and (3.12) together to find We then use Gronwall's Inequality (cf. [4]) to deduce the bound Since we know B n,m (τ ) ∞ → 0 uniformly for τ ∈ [0, T ], this implies that Z n,m (s) does so as well, and finally that X n and V n are uniformly Cauchy.

Uniform Cauchy property of derivatives.
In order to prove that the resulting limits of f n and B n are differentiable, we will show that the sequence of derivatives ∂ (t,x,v) f n and ∂ (t,x) B n are Cauchy as well. We begin by bounding ∂ x B n,m (t) ∞ . Using the representation from (3.5) and subtracting the equation for ∂ x B m from that for ∂ x B n we obtain Using the boundedness of the velocity support from Section 3.1, this relationship implies Hence, we must estimate derivatives of f n,m also. We first use the representation for ∂ v f n from (2.2) and subtract this equation for ∂ v f m from that of ∂ v f n to find Here, the first and second terms tend to zero uniformly as n, m → ∞ using the Cauchy property of characteristics and the uniform boundedness of derivatives of the iterates. Thus, taking supremums, we find where α n,m → 0 uniformly on [0, T ] as n, m → ∞.

1D PLASMA MODEL WITH TRANSPORT FIELD 13
Similarly, we use the representation for ∂ x f n from (2.3) and take the difference of this equation for ∂ x f n and ∂ x f m to find From Section 3.2 we have uniform bounds on ∂ v f n (s) ∞ and ∂ x B m−1 (s) ∞ . Thus, the first and second terms tend to zero uniformly as n, m → ∞ and the last line simplifies. With this, we have where β n,m → 0 uniformly on [0, T ] as n, m → ∞. Using (3.14), this inequality becomes which simplifies to (with γ n,m = α n,m + β n,m ) Using Gronwall's Inequality we find (3.15) Combining (3.13) and (3.15) yields for any n, m ∈ N. We can iterate (3.16) to arrive at x) is uniformly Cauchy for all t ∈ [0, T ] and x ∈ R. Similarly, using (3.14) and (3.15) we see that ∂ (x,v) f n,m (t) ∞ → 0 as n, m → ∞ for every t ∈ [0, T ] and it follows that The same argument can then be used to show that ∂ t f n (t, x, v) and ∂ t B(t, x) are also uniformly Cauchy.

Properties of limiting functions.
Assembling the previous steps, we may prove that our sequences and their derivatives converge to solutions of (1.1). Let T * again denote the maximal existence time of the solution to (3.8). Using the Cauchy property of f n , X n , V n , B n , and their derivatives, we may conclude that each sequence of functions converges uniformly on the time interval [0, T ] for any T < T * and uniformly for x, v ∈ R, n ∈ N. Since the space of continuous functions is complete with respect to the norm of uniform convergence, we may conclude that these sequences converge to continuous functions. Therefore, let us define Then, we similarly define the field Thus, using the uniform convergence of f n (t, x, v), we can pass the limit inside these integrals to find for every t ∈ [0, T ], x ∈ R Further, we define It follows from (3.4) and the uniform field bound that ∂V ∂s = B(s, X(s, t, x, v)), and by the continuity of f 0 , we see that whence for every s ∈ [0, t], t, x, v)).
Since the approximating sequence of derivatives (e.g., ∂ x f n ) of these functions converge uniformly, this implies that f and B are C 1 and their derivatives are necessarily the limits of the respective sequences, meaning Furthermore, the uniform bound on P n (t) implies the compact x and v support of f (t, x, v) for every t ∈ [0, T ]. Thus, for every T < T * , f ∈ C 1 ([0, T ]; C 1 c (R 2 )). Using (3.17) and taking derivatives, we see that the field equation for B of (1.1) holds. Upon taking the derivative with respect to s in (3.19) and using (3.18), we see that the Vlasov equation of (1.1) holds. Additionally, the solutions (3.19) and (3.17) satisfy the initial conditions (1.2). Therefore, the continuously differentiable functions f and B satisfy (1.1) with (1.2). Hence, we have shown the existence of such a solution (f, B). Notice that this argument can be continued to a time interval of arbitrary size so long as ∂ x f (t) ∞ and ∂ v f (t) ∞ remain bounded.
3.6. Uniqueness of solutions. Finally, we turn to uniqueness. Let us first suppose that the functions (f (1) , B (1) ) and (f (2) , B (2) ) are two solutions to the system (1.1) on some time interval [0, T ] which satisfy (1.2). Also, for every t ∈ [0, T ] and x, v ∈ R define the difference of these solutions Then, we subtract the first equation of (1.1) for f (2) from that for f (1) to find (2) so that by rearranging terms this becomes The left side of this equation can be expressed as a derivative along characteristic curves as where the curves X (1) (s) and V (1) (s) are defined by the, now well-known, system of characteristic ordinary differential equations (3.20) s, t, x, v)), Here, we have abbreviated X (1) (s, t, x, v) by X (1) (s) and similarly for V (1) (s). Now, integrating both sides of the above equation with respect to s, we find and since both solutions satisfy the same initial condition (1.2), as before this implies f (0, x, v) ≡ 0. Therefore, the equality simplifies to and we can bound the right side to find Integrating in s and using (1.2) to conclude that B(0, x) ≡ 0, this becomes Since both f (1) and f (2) are solutions, the velocity support of f is controlled and we can bound B by
4. Global Existence. Now that we know smooth solutions exist on some time interval, the next logical question is whether this can be extended for all times. Unfortunately, a complete answer remains unknown. The fundamental issue is that the Vlasov characteristics in the density equation propagate at an uncontrollable speed v and hence, are able to intersect the field characteristics which propagate with speed 1. Though we cannot current prove that all initial data launch a globalin-time solution, we can provide an answer for certain classes of initial data. In what follows we will use the unidirectional nature of the transport operator in (1.1) to answer the question of global existence in the affirmative for a class of initial data (f 0 , B 0 ). Then, we shall utilize a new scaling invariance of the system to apply the global existence result to additional solutions. We begin with the former result: 2) and f (t, ·, ·) compactly supported for every t ∈ [0, T ].
Proof. Let f and B be the local solution guaranteed by Theorem 3.1 and T > 0 be given. First, we represent B and use the sign of the data to find Thus, if we now let (X(t), V (t)) be characteristics along which f is nonzero. Then, a lower bound on velocity characteristics follows Since the velocity support of f 0 is strictly bounded below by v = 1, it follows that the velocity support of f satisfies this same property.
With this, we may utilize the field representation of [8] to control derivatives of the density and field (as in the proof of (5.1)). Thus, we write the field derivative in terms of the derivative of the density, as in (3.5), so that Now, we would like to eliminate the x-derivative of the density in this equation, so similar to [6] we transform ∂ x into derivatives along the characteristic curves of the system. Define the operators Then, for v = 1 we may write the inverse transformation For t ∈ [0, T ], let and recall P (t) ≤ C T for all t ∈ [0, T ]. Then, differentiation of the representation for the field (3.17) yields Here we have used the Vlasov equation of (1.1) to write the integrand as a pure v-derivative so that Since the velocity support of f is bounded away from v = 1, we see that the integrands are non-singular and we integrate by parts to find Using the previously-derived L ∞ bounds on f , P , and B, which follow from the iterates, we find ∂ x B(t) ∞ ≤ C T Hence, using Lemma 2.1(a) we find Taking the supremum in t and using Gronwall's Inequailty implies and using 2.1(a) again we find for any T > 0 and t ∈ [0, T ]. Thus, the local-in-time solution can be extended to arbitrarily large time.
Next, we utilize a new invariance of (1.1) to extend this result to additional solutions.
Proof. Clearly, if (f u , B u ) satisfies these properties then choose u = 0 and (f, B) will satisfy the same equations. So, assume that (f, B) solve (1.1) and define (f u , B u ) by (4.2) and (4.3) respectively. Then, a brief calculation shows that (f u , B u ) satisfy (1.1) and the corresponding initial conditions. Denoting  Proof. We assume f 0 (x, v) ≥ 0 for x, v ∈ R with supp(f (x, ·)) ⊂ (−∞, 1) for every x ∈ R and B 0 (x) ≤ 0 for x ∈ R. Let f and B be the local solution guaranteed by Theorem 3.1 and T > 0 be given. By Lemma 4.2, there is a solution of (1.1) denoted (f u , B u ) with u = −2 and initial data

5.
Additional field regularity. Though we cannot currently show global existence of solutions to (1.1) for arbitrary initial data, we now provide an estimate which yields additional regularity of the field B. If one could show a priori for any T > 0 that B ∈ C 1 ([0, T ] × R) then global existence would follow as this bound would imply smoothness of characteristics and f ∈ C 1 ([0, T ] × R 2 ). Hence, the local solution could be continued up to the arbitrary existence time T . Instead of the result that B gains a full derivative, however, we are able to show that it gains half of a derivative in space-time.
Proof. Let T > 0 be given. We will prove the result for the gain of regularity in x, while a similar argument leads to the additional 1/2 derivative in t. Let h > 0 be given. As in Theorem 4.1 we wish to use the operators As before, for v = 1 we may write the inverse transformation For t ∈ [0, T ], let P (t) = sup{|v| : ∃x ∈ R with f (t, x, v) = 0} ≤ C T .
Since the (∂ t , ∂ x ) → (S, T ) transformation is only valid for v bounded away from one, we decompose the v-integral in (4.1) over [−P (t), P (t)] into integrals over the disjoint sets A ǫ = {v : |1 − v| < ǫ} ∩ [−P (t), P (t)] and B ǫ = {v : |1 − v| > ǫ} ∩ [−P (t), P (t)] where ǫ > 0 is to be chosen. Then, beginning with the representation for the field (3.17) we have Using the bound on f we estimate II and find where µ denotes the Lebesgue measure on R. As all velocities in B ǫ are bounded away from 1, we can then use the transformation of derivatives for the estimate of III. We include the x-derivative of the density and transform ∂ x into terms involving S and T so that where the quantities (∂ t f + v∂ x f )(τ, y, v) dy dv dτ separate the T and S terms. To estimate III T we change variables in the y integral with z = y − (x − t + τ ), switch the order of integration, and integrate by parts in τ to find Notice that on the set B ǫ , we have |1 − v| > ǫ and thus |1 − v| −1 < ǫ −1 . In addition, µ(B ǫ ) ≤ 2P (t) ≤ C T . Therefore, Estimating III S , we again use the Vlasov equation of (1.1) to write the integrand as a pure v-derivative so that Sf (τ, y, v) = (∂ t f + v∂ x f )(τ, y, v) = −∂ v (Bf )(τ, y, v).

CHARLES NGUYEN, JENNIFER ANDERSON, AND STEPHEN PANKAVICH
Then, we integrate by parts in v and use the bounds on f and B, yielding Finally, we combine the estimates to find We choose ǫ = √ h optimally so that ǫ = h/ǫ, and finally Thus, B(t, ·) ∈ C 0,1/2 (R) for every t ∈ [0, T ] and a similar argument can be used to establish the Hölder continuity in t for fixed x ∈ R.
6. Acknowledgements. The third author would like to thank the Isaac Newton Institute at Cambridge University for hosting him during the program "Partial Differential Equations in Kinetic Theories" and providing an amazing environment to promote collaboration and the exchange of mathematical ideas, including many helpful discussions with S. Calogero.