==== Front SN Partial Differ Equ Appl SN Partial Differ Equ Appl Sn Partial Differential Equations and Applications 2662-2963 2662-2971 Springer International Publishing Cham 37398934 241 10.1007/s42985-023-00241-3 Original Paper Existence and non-uniqueness of stationary states for the Vlasov–Poisson equation on R3 subject to attractive background charges http://orcid.org/0000-0002-5385-4670 Winter Raphael raphael.elias.winter@univie.ac.at grid.10420.37 0000 0001 2286 1424 University of Vienna, Vienna, Austria 29 6 2023 29 6 2023 2023 4 4 302 8 2022 13 5 2023 © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge. Mathematics Subject Classification 35Q83 Austrian Science Fund (FWF)Open access funding provided by Austrian Science Fund (FWF). issue-copyright-statement© Springer Nature Switzerland AG 2023 ==== Body pmcIntroduction We consider the response of a spatially homogeneous plasma to a given background distribution of charges. This phenomenon can be modeled by the following nonlinear stationary Vlasov–Poisson equation for the plasma electron density f(x, v), on the three dimensional phase space R3×R31.1 v·∇xf-∇xQ·∇vf=0, 1.2 -ΔQ=ρ[f]-1+μ, 1.3 lim|x|→∞f(x,v)=f0(v). Here μ describes the distribution of background charges and f0 the electron distribution far away from the perturbation. As usual, ρ[f] denotes the spatial density of electrons given byρ[f](x)=∫R3f(x,v)dv. The system (1.1)–(1.3) is often considered in plasma physics, since the onset of Debye screening can quickly be derived for the linearized system. A detailed discussion can be found in the plasma physics textbooks [5, 6]. The model (1.1)–(1.3) is also used in other contexts in plasma physics, for instance for characterizing plasma waves (cf. [10]). For a plasma interacting with a repulsive point charge, screening has been proved rigorously in [1]. A number of results have been shown for the nonlinear Vlasov–Poisson equation in the case of finite mass and finite energy. In this case, the conserved quantities can be used to study existence and stability of stationary solutions (cf. [9]). We also refer to recent results on the stability of a point charge interacting with a plasma of finite mass [7, 8]. A closely related problem is the existence and stability of stationary solutions to the Vlasov–Poisson Boltzmann equation. In the presence of collisions, the natural boundary condition f0 are Maxwellian distributions. We refer to [2–4] for details. The key point of this paper is to show that stationary solutions to  (1.1)–(1.3) exist for general background measures μ, and infinitely many stationary states f exist as soon as the total charge of the background measure is attractive. The main result is contained in the following theorem. Theorem 1.1 Let f0(v)=F0(12|v|2) for some function F0∈C1(R) satisfying Assumption 1.2 below, and μ∈M(R3) be a measure with finite total variation, i.e.1.4 ∫R3|μ|(dx)<∞. Then there exist a solution f∈Wloc1,1(R3×R3)∩Lloc1(R3;L1(R3)), Q∈W1,1(R3)∩L2(R3) to the stationary Vlasov–Poisson problem (1.1)–(1.3). Here the equation for Q is understood in the weak sense, and the boundary condition (1.3) as‖f(·,v)-f0(v)‖L2(R3)<∞,v∈R3a.e. If the total charge θ∈R given by1.5 θ=∫R3μ(dx) is negative, then there exist infinitely many different solutions to the stationary problem (1.1)–(1.3). We split the proof of Theorem 1.1 in parts. The existence of solutions is shown in Proposition 3.2, using the extensions F of F0 constructed in Sect. 2. The non-uniqueness of solutions is given by Proposition 4.1. In [1], the existence of stationary solutions and their screening properties have been investigated forμ(dx)=θδ0(dx),θ>0. The existence proof in [1] relies on radial symmetry of the constructed solution f and the compact embedding Hr1(R3)⊂Lrp(R3), 20. The remainder of the paper is devoted to the proof of Theorem 1.1. Some parts of the proof follow similar to [1]. New ideas are needed to deal with general, non-radial solutions, the attractive case θ<0 and non-uniqueness of solutions. As in [1], we look for solutions f of the form1.6 f(x,v)=F(12|v|2+Q(x)). Hence, if Q(x)→0 as |x|→∞, then the boundary condition in (1.3) yieldsf0(v)=F(12|v|2). In the case of a repulsive point charge, we can show Q≥0. Therefore, the function F in (1.6) is completely determined by the boundary condition f0 in (1.3). In the presence of an attractive test charge, Q also attains negative values and the function F is not uniquely determined by f0. Physically, this can be explained by the presence of electrons trapped on orbits around the attractive background charge. This phonomenon is also discussed in [10]. In order to prove that infinitely many solutions exist for θ<0 in (1.5), we construct a family of admissible extensions F of F0F(r)=F0(r)ifr≥0,F~(r)ifr<0. The set of admissible extensions F is characterized by the function1.7 g(r)=4π2∫0∞sF(r+s)ds,r∈R. We make the following assumption on the distribution f0 of the plasma at |x|→∞, which is also assumed in [1]. Assumption 1.2 (Velocity distribution at infinity) The boundary condition f0 can be represented as f0(v)=F0(12|v|2), where F0∈C1(R+;R+) satisfies (i) normalization 1.8 4π2∫0∞rF0(r)dr=1. (ii) decay condition 1.9 |F0(r)|+|F0′(r)|≤C1+r3. (iii) stability condition: F0 satisfies: 1.10 F0′(r)<0,r≥0. For future reference, we define1.11 σ=-g′(0)>0. Extension to the negative half-line We show the existence of solutions if the function g defined in (1.7) satisfies the following four properties: (i) normalization: 2.1 g(0)=1. (ii) differentiability and monotonicity: g∈C2(R) and 2.2 g′(r)<0,r∈R. (iii) sub-differential at zero: 2.3 g(r)≥(g(0)+g′(0)r). (iv) growth condition: for some 1<α<32 we have 2.4 g(r)-(g(0)+g′(0)r)≤C1|r|α,|g′(r)-g′(0)|≤C2|r|α-1. Before we establish the existence of solutions under the conditions  (2.1)–(2.4), we demonstrate that it is possible to extend the function F0 to the negative half-line such that these conditions are met. This is the content of the following lemma. Lemma 2.1 Let F0 satisfy Assumption 1.2. For β∈(0,12) and cβ>0 consider the functionF~β,cβ(r)=cβr2⟨r⟩β+2+e-r2(F0(0)+F0′(0)r)r≤0, and let Fβ,cβ be the extension of F0 by F~β,cβ. Here ⟨r⟩=1+|r|2 is the Japanese bracket. Then Fβ,cβ∈Cb1(R;R+) and for cβ>0 large enough, the function gβ,cβ defined by (1.7) satisfies the conditions (2.1), (2.2), (2.3) and (2.4) with α=32-β. Proof Step 1. By construction we have Fβ,cβ∈Cb1(R). Moreover, Fβ,cβ>0 is positive since F0′>0 (cf. (1.10)). Step 2. The normalization condition (2.1) follows since F is an extension of F0 and F0 satisfies (1.8). Furthermore, the derivative of gβ,cβ can be represented asgβ,cβ′(y)=-4π2∫0∞Fβ,cβ(r+y)rdr. Since Fβ,cβ is positive, (2.2) follows. Step 3. For the proof of (2.3), we first remark that the condition holds for r≥0. As observed in [1] this follows since F0 satisfies (1.10), and therefore gβ,cβ isgβ,cβ′′(y)=-4π2∫0∞Fβ,cβ′(r+y)rdr>0,y≥0, convex on the positive half-line. Step 4. We prove that (2.3) holds for r<0 if cβ>0 large enough. To this end, we decompose gβ,cβ for r≤0 into2.5 gβ,cβ(r)=gβ,0(r)+cβ4π2∫0|r|s|r+s|2(1+|r+s|2)β+22ds. We now observe that there exists c′>0 small enough such that (2.3) is satisfied for r∈[-c′,0], independent of β, cβ. This follows from gβ,0′′(0)>0, and the other contribution in (2.5) being positive. On the other hand, for r≤-c′, we can estimate the second term in (2.5) below by4π2∫0|r|s|r+s|2(1+|r+s|2)β+22ds≥c(c′)cβ|r|32-β. Since β∈(0,12), we can choose cβ>0 large enough such that (2.3) holds. Step 4. Since gβ,cβ∈C2(R), the condition (2.4) holds locally. It remains to check the asymptotics for r→-∞. We again use (2.5). From the decay condition on F0 (1.9) we easily obtain|gβ,0(r)|≤C|r|α,|gβ,0′(r)|≤C|r|α-1. Similarly, the estimate follows for the integral term in (2.5) by a straightforward computation. □ Existence of solutions Lemma 3.1 Let F∈Cb1(R) be a non-negative function such that the function g (cf. (1.7)) satisfies the properties (2.1)–(2.4) for some 1<α<32. Recall σ>0 introduced in (1.11) and for P∈Lp(R3) define3.1 B[P](x)=g(P(x))-1+σP(x). Then we have3.2 0≤B[P](x)≤C0(|P(x)|α∧|P|2). Moreover, B is a continuous operator B:L2α(R3)→L2(R3). Proof The non-negativity of B follows from the subdifferential condition (2.3) on g. For the upper bound, we first recall that g∈C2, and σ is defined by (1.11). Since α<2 this yields0≤B[P]≤C(|P|α∧|P|2),|P|≤1. For |P|→∞ the inequality follows from the growth condition (2.4). Continuity of the operator B:L2α(R3)→L2(R3) follows from‖B[P]-B[Q]‖L2(R3)≤‖(|P|+|Q|)α-1|P-Q|‖L2(R3)≲‖P‖L2α(R3)+‖Q‖L2α(R3)‖P-Q‖L2α(R3), and finishes the proof. □ Proposition 3.2 (Existence of solutions) Let F∈Cb1(R) be an extension of F0 such that the function g (cf. (1.7)) satisfies the properties (2.1)–(2.4) for some 1<α<32. Then there exists a solution f∈Wloc1,1(R3×R3)∩Lloc1(R3;L1(R3)), Q∈W1,1(R3)∩L2(R3) to the stationary Vlasov–Poisson system (1.1)–(1.3) in the sense of Theorem 1.1. Proof Step 1. Recall σ>0 defined in (1.11), and let Φσ be the fundamental solution to (σ-Δ)-1, i.e.3.3 Φσ(x)=e-σ|x|4π|x|. Define S∈S′(R3) by3.4 S=Φσ∗μ. Since μ∈M(R3) has finite total variation (cf. (1.4)), we know‖S‖Lp(R3)<∞,p∈[1,3), and in light of (3.2), B[S] satisfies‖B[S]‖Lp(R3)<∞,p∈1,3α. By the Green’s function property of Φσ, S is a weak solution to3.5 (σ-Δ)S=μ. Let us further introduce the functions H1, H byH1(x)=ϕ∗B[S],H(x)=C0((|S|+|H1|)α∧(|S|+|H1|)2), where C0>0 is the constant appearing in (3.2) and3.6 ϕ(x)=14π|x|. Using that ϕ defined in (3.6) is a Riesz potential, we find that3.7 ‖H1‖Lp(R3)<∞,p∈(3,∞),‖H‖L2(R3)<∞. Step 2. We pick the coefficient q0 asq0=2α, and define the operatorK:Lq0(R3)→Lq0(R3)R↦Φσ∗B(R+S)∧H. We claim that K is a continuous compact operator. Continuity follows from the continuity of B shown in Lemma 3.1. It remains to prove that the image of K is precompact. To this end, consider a sequence Rk∈Lq0(R3), and Gk:=K[Rk]. Then for some M>0 we have3.8 0≤Gk≤H:=Φσ∗H, 3.9 ‖Gk‖W1,q0(R3)0 such that for P∈Lq0(R3) and λ∈[0,1] with P=λK(P) we have‖P‖Lq(R3)0 large enough, the functions constructed in Lemma 2.1. Let further f1,Q1 and f2,Q2 be associated solutions to the system (1.1)–(1.3) provided by Proposition 3.2. Then we havef1≠f2∈Wloc1,1(R3),Q1≠Q2∈W1,1(R3). Proof Step 1. We first prove Q1(x)<0 on a set of positive measure. Recall the equation for Q-ΔQ=g(Q)-1+μ∈M(R3). Assume Q≥0 a.e.. Then g(Q)-1≤0, and therefore0=∫R3-ΔQ(x)dx≤∫R3μ(dx)=θ<0, which yields a contradiction. Step 2. The functions Q1 and Q2 do not coincide, i.e. |Q1-Q2|≠0∈Lloc1(R3). Again we argue by contradiction. If Q1=Q2 a.e. theng1(Q1(x))=g2(Q1(x))x∈R3a.e.. However, we have g1(r)>g2(r) on the negative half-axis r<0 since β1<β2. Since Q1 takes negative values on a set of positive measure we reach a contradiction. Step 3. Finally |f1-f2|≠0∈Lloc1(R3). Assume the contrary, and let A⊂R3 the set on which Q1<0. By Step 1, A has positive measure. Then we estimate0=∫R3×R3|f1(x,v)-f2(x,v)|dxdv≥∫A∫R3|F1(12|v|2+Q1(x))-F2(12|v|2+Q2(x))|dvdx≥∫Ainfr∈R∫R3|F1(12|v|2+Q1(x))-F2(12|v|2+r)|dvdx. On the other hand, for any y<0 the integrandinfr∈R∫R3|F1(12|v|2+y)-F2(12|v|2+r)|dv>0 is positive. This leads to a contradiction to the assumption. □ Acknowledgements R.W. acknowledges financial support from the Austrian Science Fund (FWF) project F65. Furthermore, R.W. would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Frontiers in kinetic theory: connecting microscopic to macroscopic scales—KineCon 2022” when work on this paper was undertaken. This work was supported by EPSRC Grant number EP/R014604/1. Funding Information Open access funding provided by Austrian Science Fund (FWF). Data Availability Statement Data sharing is not applicable to this article as no new data were created or analysed in this study. 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