WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD

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1 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD JENNIFER RAE ANDERSON 1. Introduction A plama i a partially or completely ionized ga. Nearly all (approximately 99.9%) of the matter in the univere exit in the tate of plama, a oppoed to a olid, fluid, or a gaeou tate. Such a form of matter occur if the velocity of individual particle in a material achieve an enormou magnitude, perhap a izable fraction of the peed of light. Hence all matter, if heated to a ignificantly great temperature, will enter a plama tate. In term of practical ue, plama are of great interet to the energy, aeronautical, and aeropace indutrie among other, a they are ued in the production of electronic, (plama) engine, and laer, a well a, in harneing the power of nuclear energy. Plama are widely ued in olid tate phyic ince they are great conductor of electricity due to their free-flowing abundance of ion and electron. When a plama i of low denity or the time cale of interet are ufficiently mall, it i deemed to be colliionle, a colliion between particle become infrequent. Many example of colliionle plama occur in nature, including the olar wind, galactic nebulae, the Van Allen radiation belt, and comet tail. The fundamental equation which decribe the time evolution of a colliionle plama are given by the Vlaov-Maxwell ytem: (VM) t f + v x f + (E + v B) v f = ρ(t, x) = f(t, x, v) dv, j(t, x) = vf(t, x, v) dv t E = B j, E = ρ t B = E, B =. Here, f repreent the denity of (poitively-charged) ion in the plama, while ρ and j are the charge and current denity, and E and B repreent electric and magnetic field generated by the charge and current. The independent variable, t and x, v R 3 repreent time, poition, and velocity, repectively, and phyical contant, uch a the charge and ma of particle, a well a, the peed of light, have been normalized to one. In the preence of large velocitie, relativitic correction become important and the correponding ytem to conider i the relativitic analogue of (VM), denoted by (RVM) and contructed by replacing v with v ˆv = 1 + v 2 in the firt equation of (VM), called the Vlaov equation, and in the integrand of the current j. General reference on the kinetic equation of plama dynamic, uch a (VM) Thi work i ubmitted in partial fulfillment of the requirement of the Mater of Science degree in Mathematic at the Univerity of Texa at Arlington, 21; Advior - S. Pankavich. 1

2 2 JENNIFER RAE ANDERSON and (RVM), include [4] and [8]. Over the pat twenty-five year ignificant progre ha been made in the analyi of (RVM), pecifically, the global exitence of weak olution (which alo hold for (VM); ee [2]) and the determination of condition which enure global exitence of claical olution (originally dicovered in [5], and later in [6], and [1]) for the Cauchy problem. Additionally, a wide array of information ha been dicovered regarding the electrotatic verion of both (VM) and (RVM) - the Vlaov-Poion and relativitic Vlaov-Poion ytem, repectively. Thee model do not include magnetic effect within their formulation, and the electric field i given by an elliptic, rather than a hyperbolic equation. Thi implification ha led to a great deal of progre concerning the electrotatic ytem, including theorem regarding global exitence, tability, and long-time behavior of olution; though a global exitence theorem for claical olution in the relativitic cae ha remained eluive. Independent of thee advance, many of the mot baic exitence and regularity quetion remain unolved for (VM). The main difficulty which arie i the lo of trict hyperbolicity of the kinetic ytem due to the poibility that particle velocitie v may travel fater than the propagation of ignal from the electric and magnetic field, which do o at the peed of light c = 1. Often a remedy to the lack of progre on uch a problem i to reduce the dimenionality of the ytem. Unfortunately, poing the problem in one-dimenion (i.e., x, v R) eliminate the relevance of the magnetic field a the Maxwell ytem decouple, yielding the onedimenional Vlaov-Poion ytem: t f + v x f + E v f = (VP) x E = fdv. The lowet-dimenional reduction which include magnetic effect i the o-called one-andone-half-dimenional ytem which i contructed by taking x R but v R 2. Surpriingly, the quetion of exitence remain open even in thi cae. Thu, in order to tudy thi quetion, but keep the problem poed in a one-dimenional etting, we conider the following nonlinear ytem of hyperbolic PDE: t f + v x f + B v f = (SVM) t B + x B = f(t, x, v)dv. Since the field equation in (SVM) i hyperbolic, we denote it with a magnetic field variable B, a oppoed to the electric field E of (VP) which atifie an elliptic equation. Notice that thee equation retain the main difficulty of (VM), namely the interaction between characteritic particle velocitie v and contant field velocitie c = 1. The ytem (SVM) i upplemented by given initial data (IC) f(, x, v) = f (x, v), B(, x) = B (x) where f C 1 c (R 2 ) and B Lip(R). Here, t i again time, x R i pace, v R i momentum (and velocity ince the ma ha been normalized), f = f(t, x, v) repreent the denity of ion in phae pace (x, v) over time, and B = B(t, x) i effectively a magnetic field. We wih to prove the local-in-time exitence of a unique olution in the pace of Lipchitz function. Theorem 1.1. There exit T > and unique function f Lip([, T ] R 2 ), B Lip([, T ] R R) uch that f and B atify (SVM) and (IC).

3 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD Notation. In order to prove the theorem, we begin by defining the following norm. For f C(R 2 ), we define Similarly for B C(R), we ue f = up f(x, v). x,v B = up B(x) x where the norm on the domain R 2 and R are ditinguihed by context. Furthermore, we will ue f Lip = infc : f(x, v) f(y, u) c x y 2 + v u 2, x, y, v, u R} for any f Lip(R 2 ) and the analogou definition for function which have domain R. Finally, for any T > and f C([, T ]; Lip(R 2 )) we define the norm f L = up f(t) Lip. t [,T ] Each will be utilized in the local exitence proof to follow. Additionally, we define the upremum of the velocity upport of a given ion ditribution by P f (t) := up v : x uch that f(t, x, v) }. Before beginning the proof of Theorem 1.1, we firt derive ome a priori etimate on function which atify (SVM) with (IC). 2. A priori bound We begin by etimating the magnetic field, auming the ion ditribution and initial data are known in advance. Lemma 2.1. Let T >, B Lip(R), and f Lip([, T ] R 2 ) be given. Conider B Lip([, T ] R) to be the unique olution of the linear Cauchy problem t B + x B = fdv B(, x) = B (x). then, for every t [, T ] (1) B(t) B + 2 and P f () f() d T (2) B L B Lip + 2 f L P f ()d. Proof. We will ue the method of characteritic. Let χ(, t, x) : [, T ] [, T ] R R atify χ(, t, x) = 1 χ(t, t, x) = x.

4 4 JENNIFER RAE ANDERSON Then χ i linear in and o it mut atify χ(, t, x) = t + x. Uing thi we ee d d B(, t + x) = tb(, t + x) + x B(, t + x) = f(, t + x, v)dv Integrating with repect to from to t yield (3) B(t, x) = B(, x t) Pf () = f(, t + x, v)dv P f () Pf () P f () Taking the upremum over all x R, we conclude or up x B(t, x) up B(, x) + 2 x B(t) B + 2 which prove (1). Further, from (3) B(t, x) B(t, y) = B (x t) B (y t) Pf () for every x, y R. So, taking x y, we have B(t, x) B(t, y) x y P f () = B (x t) B (y t) + x y f(, t + x, v)dvd. P f () up f(, x, v)d x,v P f () f() d (f(, t + x, v) f(, t + y, v)) dvd Pf () P f () f(, t + x, v) f(, t + y, v) dvd. x y Since thi yield x y = (x t) (y t) = ( t + x) ( t + y), B(t, x) B(t, y) x y B Lip + B Lip + 2 Pf () P f () f() Lip dvd P f () f L d T B Lip + 2 f L P f ()d. Taking the infemum over uch value of x and y how (2). With thi reult, we can now etimate the Lipchitz norm of the reulting characteritic curve induced by the magnetic field.

5 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD 5 Lemma 2.2. Let T > and B Lip([, T ] R) be given and define X, V Lip([, T ] 2 R 2 ) a the unique olution to the ytem of ODE with Cauchy data then X(, t, x, v) = V (, t, x, v) V (, t, x, v) = B(, X(, t, x, v)) X(t, t, x, v) = x V (t, t, x, v) = v (4) X L + V L 2 [1 + T max1, B L }( X L + V L )]. Proof. Firt oberve from the characteritic ODE and Thu, we have and X(, t, x, v) = x V (, t, x, v) = v X(, t, x, v) X(, t, y, v) = x y X(, t, x, v) X(, t, x, u) = V (, t, x, v) V (, t, y, v) = V (, t, x, v) V (, t, x, u) = v u Taking abolute value yield, V (τ, t, x, v)dτ B(τ, X(τ, t, x, v))dτ. (V (τ, t, x, v) V (τ, t, y, v)) dτ, (V (τ, t, x, v) V (τ, t, x, u)) dτ, (B(τ, X(τ, t, x, v)) B(τ, X(τ, t, y, v))) dτ, X(, t, x, v) X(, t, y, v) = x y + and imilarly for the difference in v X(, t, x, v) X(, t, x, u) = (B(τ, X(τ, t, x, v)) B(τ, X(τ, t, x, u))) dτ. x y (1 + V (τ, t, x, v) V (τ, t, y, v) dτ V L dτ) x y (1 + T V L ) v u V (τ, t, x, v) V (τ, t, x, u) dτ V L dτ v u T V L.

6 6 JENNIFER RAE ANDERSON We proceed analogouly for the velocity characteritic, o that and V (, t, x, v) V (, t, y, v) = = x y V (, t, x, v) V (, t, x, u) v u + B(τ, X(τ, t, x, v)) B(τ, X(τ, t, y, v)) dτ B(τ, X(τ, t, x, v)) B(τ, X(τ, t, y, v)) X(τ, t, x, v) X(τ, t, y, v) X(τ, t, x, v) X(τ, t, y, v) dτ B L X L dτ T x y B L X L, v u (1 + Adding the etimate on X together we find and taking the upremum over and t, We do the ame for V, which give u and thu B(τ, X(τ, t, x, v)) B(τ, X(τ, t, x, u)) dτ B L X L dτ) v u (1 + T B L X L ). X(, t) Lip 1 + 2T V L, X L 1 + 2T V L. V (, t) Lip 1 + 2T B L X L, V L 1 + 2T B L X L. Moreover, we combine the etimate on characteritic o that and thi complete the proof. X L + V L 2 [1 + T max1, B L }( X L + V L )] Here, we note that characteritic will be abbreviated in the future a X() and V () repectively. Thu, we will uppre their dependence on t, x, and v for brevity. Next, we etimate that particle ditribution auming that the field i known. Lemma 2.3. Let T > and B Lip([, T ] R) be given. Define the characteritic X and V a in Lemma 2.2 and let f Lip([, T ] R 2 ) be the unique olution to t f + v x f + B v f = f(, x, v) = f (x, v) then for all t [, T ] (5) f(t) = f and (6) f L f Lip ( X L + V L ).

7 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD 7 Proof. The former reult follow from the conervation of ion along characteritic. Uing X and V, we may write the Vlaov equation a d d f(, X(), V ()) = tf(, X(), V ())+V () x f(, X(), V ())+B(, X()) v f(, X(), V ()) =. Integrating with repect to from to t we obtain, f(t, x, v) = f(, X(), V ()) = f (X(), V ()). Hence it follow, f(t) = f Further, let u denote for fixed x, y, v, u R the quantity R(t) = X(, t, x, v) X(, t, y, u) 2 + V (, t, x, v) V (, t, y, u) 2. Then, etimating the Lipchitz norm of f, we find f(t, x, v) f(t, y, u) x y 2 + v u 2 f(, X(, t, x, v), V (, t, x, v)) f(, X(, t, y, u), V (, t, y, u)) R(t) = R(t) R() X(, t, x, v) X(, t, y, u) + V (, t, x, v) + V (, t, y, u) f Lip R() f Lip X(, t, x, v) X(, t, y, u) x y 2 + v u 2 + f Lip V (, t, x, v) + V (, t, y, u) x y 2 + v u 2 f Lip X(, t) Lip + f Lip V (, t) Lip whence f(t) Lip f Lip X L + f Lip V L. Taking the upremum over t [, T ] give (6). 3. Local exitence and proof of Theorem 1.1 Uing the a priori etimate of the previou ection, we can now begin to prove the exitence theorem. Since, f i given and ha compact upport, let R, R 1 > be uch that upp(f ) [ R, R ] [ R 1, R 1 ]. For f Lip([, T ] R 2 ) we will refer to the following propertie: (P1) f(, x, v) = f (x, v) (P2) upp(f(t,, )) [ R 1, R + 1] [ R 1 1, R 1 + 1], t [, T ] (P3) f L 4 f Lip. For any T >, we define the function pace G T := f Lip([, T ] R 2 ) : f atifie (P 1) (P 3) }. Notice that G T i nonempty a the function f(t, x, v) = f (x, v) for all t [, T ] i certainly an element of G T. We claim that when equipped with the norm, f GT = up t [,T ] f(t), G T i a complete, cloed ubet of the Banach pace Lip([, T ] R 2 ) for any T >. To how thi, all that mut be demontrated i that G T i complete. If we conider a Cauchy equence of function f n in G T, becaue we are uing the C ([, T ] R 2 ) norm and that pace i complete, there exit a limit f C([, T ] R 2 ) uch that f n f uniformly (i.e.,

8 8 JENNIFER RAE ANDERSON in the upremum norm). Certainly, f atifie condition (P1) - (P3). In addition, it follow from Arzelá-Acoli (cf. [7]) that the uniform limit of Lipchitz function i alo Lipchitz and will hare the ame uniform Lipchitz contant a thoe in the equence. Thu, f G T and the ubet i complete. Next, we will contruct a map φ : G T Lip([, T ] R 2 ) a follow. Given f G T define B Lip([, T ] R) to be the unique olution of t B + x B = fdv B(, x) = B (x). From B, define g Lip([, T ] R 2 ) a the unique olution to t g + v x g + B v g = g(, x, v) = f (x, v) and let the mapping φ be defined by φ(f) = g. In order to ue the Contraction Mapping Principle, we mut prove that for every ufficiently mall T >, we have φ : G T G T and φ i a contraction on thi pace. The following two lemma do jut thi. Lemma 3.1. There exit T 1 > uch that for any T (, T 1 ) we have φ : G T G T. Proof. To how thi we mut how, g L 4 f Lip and upp(f(t, )) [ R 1, R + 1] [ R 1 1, R 1 + 1]. Notice that condition (P1) hold by contruction of φ. From Lemma 2.3 we know g L f Lip ( X L + V L ) where X and V are the characteritic defined from B in Lemma 2.2. Uing Lemma 2.2 we can reduce the right ide to knowledge of the Lipchitz norm of B ince X L + V L 2 [1 + T max1, B L }( X L + V L )]. Now, utilizing Lemma 2.1 and the fact that f G T for ome T >, we find Let T B L B Lip + 2 f L P f ()d B Lip + 2 f L T (R 1 + 1) B Lip + 8 f Lip T (R 1 + 1) 1 T 1 = min 4, 1 } 4 [ B Lip + 2 f Lip (R 1 + 1)] 1. Then, it follow from the bound on B L that X L + V L ( X L + V L ) on the interval [, T 1 ]. Thi implie X L + V L 4 and we have, g L 4 f Lip where the L norm i taken over the interval [, T 1 ]. To how our upport contraint i alo atified, we compute for characteritic along which g,

9 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD 9 P g (t) : = up v : x R with g(t, x, v) } = up v : x R with g(, X(, t, x, v), V (, t, x, v)) } = up v : x R with f (X(, t, x, v), V (, t, x, v)) } If we invert our characteritic (ee [4] for more detail) and allow y = X(, t, x, v) w = V (, t, x, v) then it follow that x = X(t,, y, w) v = V (t,, y, w). Thu we conclude, P g (t) = up V (t,, y, w) : y with f (y, w) }. Uing the definition of characteritic and integrating over [, t] give u V (t, t, x, v) V (, t, x, v) = Taking t = we have V (,, x, v) V (,, x, v) = If we then let = t thi become V (,, x, v) V (t,, x, v) = or V (t,, x, v) V (,, x, v) = B(τ, X(τ, t, x, v))dτ B(τ, X(τ,, x, v))dτ. B(τ, X(τ,, x, v))dτ B(τ, X(τ,, x, v))dτ B(τ) dτ Hence, for all x, v R and any characteritic along which f, we have V (t,, x, v) V (,, x, v) + B(τ) dτ If we take the upremum over all characteritic uch that f then we have P g (t) P g () + Utilizing (1) from Lemma 2.1 we find B(t) B + 2 B(τ) dτ P f () f() d.

10 1 JENNIFER RAE ANDERSON So, we conider t [, T 2 ] for ome T 2 > to be determined and etimate the integral above uing (5), (P2), and (P3) ince f G T ( τ ) B(τ) dτ B + 2 P f () f() d dτ ( ) T2 T2 ( B + 2 B + 2 T2 T2 P f () f() d (R 1 + 1) f d T 2 B + 2(T 2 ) 2 (R 1 + 1) f. We can then take T 2 mall enough uch that T 2 B + 2T 2 (R 1 + 1) f < 1, and becaue g() = f, we have P g (t) P g () + R B(τ) dτ Latly, let T 3 < 1 R. Since 1+1 d X(,, x, v) = V (,, x, v), d we take a characteritic X(, t, x, v) along which g and find X(t,, x, v) = X(,, x, v) + x + T3 (R 1 + 1)d = x + T 3 (R 1 + 1) R + 1. V (,, x, v)d Finally, if we conider T1 < mint 1, T 2, T 3 } then all of the above tatement mut hold on [, T1 ]. Thu, condition (P2) and (P3) hold for g = φ(f) and for any T (, T1 ), we have φ : G T G T. Next, we how that the mapping we have contructed i indeed a contraction on G T for T > ufficiently mall. Lemma 3.2. There exit T 2 > uch that for any T (, T 2 ), φ i a contraction on G T. Proof. Let f 1, f 2 G T, Let B i be the olution of t B i + x B i = f i dv B i (, x) = B (x) and define g i = φ(f i ) for i = 1, 2. Then, a in Lemma 2.1, we can write B 1 (t, x) B 2 (t, x) = B 1 (, x t) B 2 (, x t) R1+1 R 1 1 ) dτ dτ (f 1 (, t + x, v) f 2 (, t + x, v)) dvd.

11 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD 11 Taking the abolute value and applying the triangle inequality, B 1 (t, x) B 2 (t, x) B (x t) B (x t) + R1+1 R 1 1 2(R 1 + 1) and we conclude the uniform bound R1+1 (f 1 f 2 )() dvd (f 1 f 2 )() d R 1 1 (7) (B 1 B 2 )(t) 2(R 1 + 1)T up (f 1 f 2 )() [,T ] for every t [, T ]. Subtracting the g equation we ee g 1 g 2 atifie, Adding and ubtracting B 2 v g 1, t (g 1 g 2 ) + v x (g 1 g 2 ) + B 1 v g 1 B 2 v g 2 =. t (g 1 g 2 ) + v x (g 1 g 2 ) + B 1 v g 1 B 2 v g 1 + B 2 v g 1 B 2 v g 2 =. Rearranging we now have, t (g 1 g 2 ) + v x (g 1 g 2 ) + B 2 v (g 1 g 2 ) = (B 1 B 2 ) v g 1. Conider, the characteritic X 2 (, t, x, v), V 2 (, t, x, v) that atify X 2(, t, x, v) = V 2 (, t, x, v) V 2(, t, x, v) = B 2 (, X 2 (, t, x, v)) X 2 (t, t, x, v) = x V 2 (t, t, x, v) = v. Then we write the equation for g 1 g 2 a f 1 (, t + x, v) f 2 (, t + x, v) dvd d d (g 1 g 2 )(, X 2 (, t, x, v), V 2 (, t, x, v)) = (B 1 B 2 ) v g 1 (, X 2 (, t, x, v), V 2 (, t, x, v)) Integrating over [, t], thi become (g 1 g 2 )(t, X 2 (t, t, x, v), V 2 (t, t, x, v)) = (g 1 g 2 )(, X 2 (, t, x, v), V 2 (, t, x, v)) (B 1 B 2 ) v g 1 (, X 2 (, t, x, v), V 2 (, t, x, v))d But g 1 (, x, v) = g 2 (, x, v) = f (x, v) o (g 1 g 2 )(, x, v) =. Uing thi, the bound on g 1 L from (P3), and (7) we have (g 1 g 2 )(t, x, v) T (B 1 B 2 )(, X 2 (, t, x, v)) g 1 () Lip d (B 1 B 2 )() g 1 L d 2T 2 (R 1 + 1) g 1 L up [,T ] 8T 2 (R 1 + 1) f Lip f 1 f 2 GT (f 1 f 2 )()

12 12 JENNIFER RAE ANDERSON If we take T2 1 <, then we ee for T < T 8(R1+1) f 2, t [, T ], and for all x, v R, Lip where A < 1. Thu it follow that (g 1 g 2 )(t, x, v) A f 1 f 2 GT g 1 g 2 GT A f 1 f 2 GT and φ : G T G T i a contraction for all T (, T 2 ). The lat lemma i devoted to howing that no olution of (SVM) other than the one that we contruct can exit in Lip([, T ] R 2 ). Lemma 3.3. There exit at mot one olution to (SVM), for (f, B) Lip([, T ] R 2 ) Lip([, T ] R) which atify the given initial condition (IC). Proof. Suppoe that (f 1, B 1 ), (f 2, B 2 ) Lip([, T ] R 2 ) Lip([, T ] R) are two olution to our ytem. Define f(t, x, v) := (f 1 f 2 )(t, x, v) and B(t, x, v) := (B 1 B 2 )(t, x, v). Then we ee = t (f 1 f 2 ) + v x (f 1 f 2 ) + B 1 v f 1 B 2 v f 2 = t f + v x f + B 1 v f 1 B 2 v f 1 + B 2 v f 2 = t f + v x f + B v f 1 + B 2 v f If we let X 2, V 2 be characteritic atifying X 2(, t, x, v) = V 2 (, t, x, v) V 2(, t, x, v) = B 2 (, X 2 (, t, x, v)) X 2 (t, t, x, v) = x V 2 (t, t, x, v) = v, then we have d d (f(, X 2(, t, x, v), V 2 (, t, x, v)) = (B v f 1 )(, X 2 (, t, x, v), V 2 (, t, x, v)). Integrating and uing f(, x, v) = f 1 (, x, v) f 2 (, x, v) = for every x, v R, we have f(t, x, v) = f(, X 2 (, t, x, v), V 2 (, t, x, v)) B(, X 2 (, t, x, v)) v f 1 (, X 2 (, t,, v), V 2 (, t, x, v))d f 1 L B() d Taking the upremum of both ide we ee, (8) f(t) f L B() d. Subtracting the B equation we oberve, t B + x B = fdv

13 WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD 13 and thu Integrating yield, d B(, t + x) = d B(t, x) = B(, x t) Pf () f(, t + x, v)dv. P f () f(, t + x, v)dvd. Becaue B(, x) = B 1 (, x) B 2 (, x) = B (x) B (x) = we find (9) B(t) 2 Adding (8) and (9) we have, f(t) + B(t) P f () f() d. max f L, 2P f ()} ( f() + B() ) d Since f Lip([, T ] R 2 ) i uniformly bounded and both f 1 and f 2 inherit the compact velocity upport of f, there i C > uch that f(t) + B(t) C Applying Grönwall inequality (cf. [3]), we arrive at f(t) + B(t). ( f() + B() ) d. So it mut follow that f and B, and thu f 1 = f 2 with B 1 = B 2. Hence, we ee that there can be at mot one olution to (SVM) within the pace of Lipchitz function. We are now at the point where we can apply Lemma 3.1, 3.2, and 3.3 in order to conclude by the contraction mapping principal that Theorem 1.1 i true. Proof of Theorem 1.1. Let T = mint1, T2 } and define G = G T. Then, we know from Lemma 3.1 and 3.2 that φ i a contraction on G. Becaue G i a complete, cloed ubet of a Banach pace, we may apply the contraction mapping principal and conclude that there exit a unique f = f(t, x, v) G uch that φ(f) = f. Let B = B(t, x) Lip([, T ] R be the unique olution to t B + x B = fdv (1) B(, x) = B (x). Notice that by Lemma 2.2, ince f Lip([, T ] R 2 ), it follow that B i alo Lipchitz. Then, becaue g = φ(f) = f atifie t g + v x g + B v g = g(, x, v) = f (x, v), we ee that (f, B) atifie (SVM) and (IC) t f + v x f + B v f = t B + x B = f(t, x, v)dv f(, x, v) = f (x, v) B(, x) = B (x)

14 14 JENNIFER RAE ANDERSON where f i the fixed point and B i defined by (1). Further from Lemma 3.3 we know that (f, B) i the only pair of Lipchitz function that atify (SVM) and (IC). Thu, our contructed olution i unique in thi pace. 4. Concluion Future work for thi topic include jutifying the exitence and uniquene of claical olution, f, B C 1. Alo of interet, i the quetion of global-in-time olution. Finally, we are alo intereted in howing that there are no teady tate olution except f = B and determining the tability of the zero olution in different L p norm. Reference [1] Bouchut, F., Gole, F., and Pallard, C. Claical olution and the Glaey-Strau theorem for the 3D Vlaov-Maxwell ytem. Arch. Ration. Mech. Anal. 17, 1 (23), [2] DiPerna, R. J., and Lion, P.-L. Global weak olution of Vlaov-Maxwell ytem. Comm. Pure Appl. Math. 42, 6 (1989), [3] Evan, L. C. Partial differential equation, vol. 19 of Graduate Studie in Mathematic. American Mathematical Society, Providence, RI, [4] Glaey, R. T. The Cauchy problem in kinetic theory. Society for Indutrial and Applied Mathematic (SIAM), Philadelphia, PA, [5] Glaey, R. T., and Strau, W. A. Singularity formation in a colliionle plama could occur only at high velocitie. Arch. Rational Mech. Anal. 92, 1 (1986), [6] Klainerman, S., and Staffilani, G. A new approach to tudy the Vlaov-Maxwell ytem. Commun. Pure Appl. Anal. 1, 1 (22), [7] Rudin, W. Principle of mathematical analyi, third ed. McGraw-Hill Book Co., New York, International Serie in Pure and Applied Mathematic. [8] van Kampen, N., and Felderhof, B. Theoretical Method in Plama Phyic. Wiley, New York, NY, 1967.