Skip to main content
SpringerLink
Log in

From the Relativistic Mixture of Gases to the Relativistic Cucker–Smale Flocking

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We present a relativistic model for a mixture of Euler gases with multiple temperatures. For the proposed relativistic model, we explicitly determine production terms resulting from the interchange of energy–momentum between the constituents via the entropy principle. We use the analogy with the homogeneous solutions of a mixture of gases and the thermomechanical Cucker–Smale (in short TCS) flocking model in a classical setting (Ha and Ruggeri in Arch Ration Mech Anal 223:1397–1425, 2017) to derive a relativistic counterpart of the TCS model. Moreover, we employ the theory of a principal subsystem to derive the relativistic Cucker–Smale (in short CS) model. For the derived relativistic CS model, we provide a sufficient framework leading to the exponential flocking in terms of communication weights and also show that the relativistic CS model reduces to the classical CS model, as the speed of light tends to infinity in any finite-time interval.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arzeliés, H.: Relativistic transformation of temperature and some other thermodynamical quantities (in French). N. Cim. 35, 792–804, 1965

    ADS  Google Scholar 

  2. Boillat, G.: Recent mathematical methods in nonlinear wave propagation in CIME Course. In: Ruggeri, T. (ed.) Lecture Notes in Mathematics, vol. 1640, pp. 103–152. Springer, Berlin 1995

    Google Scholar 

  3. Boillat, G.: Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systémes hyperboliques. C. R. Acad. Sci. Paris A278, 909, 1974

    MathSciNet  MATH  Google Scholar 

  4. Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Ration. Mech. Anal. 137, 305–320, 1997

    MathSciNet  MATH  Google Scholar 

  5. Boillat, G., Ruggeri, T.: Maximum wave velocity in the moments system of a relativistic gas. Contin. Mech. Thermodyn. 11, 107, 1999

    ADS  MathSciNet  MATH  Google Scholar 

  6. Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Contin. Mech. Thermodyn. 9, 205, 1997

    ADS  MathSciNet  MATH  Google Scholar 

  7. Boillat, G., Ruggeri, T.: Relativistic gas: moment equations and maximum wave velocity. J. Math. Phys. 40, 6399, 1999

    ADS  MathSciNet  MATH  Google Scholar 

  8. Cercignani, C., Kremer, G.M.: The Relativistic Boltzmann Equation: Theory and Applications. Birkhäuser, Basel 2002

    MATH  Google Scholar 

  9. Choi, Y.-P., Ha, S.-Y., Jung, J., Kim, J.: Global dynamics of the thermomechanical Cucker–Smale ensemble immersed in incompressible viscous fluids. Nonlinearity32, 1597–1640, 2019

    ADS  MathSciNet  MATH  Google Scholar 

  10. Choi, Y.-P., Ha, S.-Y., Jung, J., Kim, J.: On the coupling of kinetic thermomechanical Cucker–Smale equation and compressible viscous fluid system. To appear in J. Math. Fluid Mech.

  11. Choi, Y.-P., Ha, S.-Y., Li, Z.: Emergent dynamics of the Cucker–Smale flocking model and its variants. Active particles. Vol. 1. Advances in theory, models, and applications, 299–331, Model. Simul. Sci. Eng. Technol., Birkhauser/Springer, Cham, 2017

  12. Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control52, 852–862, 2007

    MathSciNet  MATH  Google Scholar 

  13. Degond, P., Motsch, S.: Large-scale dynamics of the Persistent Turing Walker model of fish behavior. J. Stat. Phys. 131, 989–1022, 2008

    ADS  MathSciNet  MATH  Google Scholar 

  14. De Groot, S.R., Van Leeuwen, W.A., Van Weert, C.G.: Relativistic Kinetic Theory. Principles and Applications. North-holland, Amsterdam 1980

    Google Scholar 

  15. Freistühler, H.: Relativistic barotropic fluids: a godunov-boillat formulation for their dynamics and a discussion of two special classes. Arch. Ration. Mech. Anal. 232, 473–488, 2019

    MathSciNet  MATH  Google Scholar 

  16. Godunov, S.K.: An interesting class of quasilinear systems. Sov. Math. 2, 947, 1961

    MATH  Google Scholar 

  17. Gouin, H., Ruggeri, T.: Identification of an average temperature and a dynamical pressure in a multitemperature mixture of fluids. Phys. Rev. E78, 016303, 2008

    ADS  Google Scholar 

  18. Ha, S.-Y.: Lyapunov functional approach and collective dynamics of some interacting many-body systems. Proceedings of the International Congress of Mathematicians: Seoul 2014. Vol. III, 1123–1140, Kyung Moon Sa, Seoul, 2014

  19. Ha, S.-Y., Kim, J., Min, C., Ruggeri, T., Zhang, X.: A global existence of classical solution to the hydrodynamic Cucker–Smale model in presence of temperature field. Anal. Appl. 16, 757–805, 2018

    MathSciNet  MATH  Google Scholar 

  20. Ha, S.-Y., Kim, J., Min, C., Ruggeri, T., Zhang, X.: Uniform stability and mean-field limit of thermodynamic Cucker–Smale model. Quart. Appl. Math. 77, 131–176, 2019

    MathSciNet  MATH  Google Scholar 

  21. Ha, S.-Y., Kim, J., Ruggeri, T.: Emergent behaviors of thermodynamic Cucker–Smale particles. SIAM J. Math. Anal. 50, 3092–3121, 2018

    MathSciNet  MATH  Google Scholar 

  22. Ha, S.-Y., Ko, D., Park, J., Zhang, X.: Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3, 209–267, 2016

    ADS  MathSciNet  MATH  Google Scholar 

  23. Ha, S.-Y., Ruggeri, T.: Emergent dynamics of a thermodynamically consistent particle model. Arch. Ration. Mech. Anal. 223, 1397–1425, 2017

    MathSciNet  MATH  Google Scholar 

  24. Hutter, K., Müller, I.: On mixtures of relativistic fluids. Helvetica Physica Acta48, 675, 1975

    Google Scholar 

  25. Ikenberry, E., Truesdell, C.: On the pressure and the flux of energy in a gas according to Maxwell’s kinetic theory. J. Ration. Mech. Anal. 5, 1, 1956

    MathSciNet  MATH  Google Scholar 

  26. Kang, M.-J., Ha, S.-Y., Kim, J., Shim, W.J.: Hydrodynamic limit of the kinetic thermomechanical Cucker–Smale model in a strong local alignment regime. To appear in Commun. Pure Appl. Anal.

  27. Ko, C.M., Li, Q.: Relativistic Vlasov–Uehling–Uhlenbeck model for heavy-ion collisions. Phys. Rev. C37, 2270, 1988

    ADS  Google Scholar 

  28. Kremer, G.M., Marques Jr., W.: Grad’s moment method for relativistic gas mixtures of Maxwellian particles. Phys. Fluids25, 017102, 2013

    ADS  Google Scholar 

  29. Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420, 1975

    ADS  MathSciNet  Google Scholar 

  30. Leonard, N.E., Paley, D.A., Lekien, F., Sepulchre, R., Fratantoni, D.M., Davis, R.E.: Collective motion, sensor networks and ocean sampling. Proc. IEEE95, 48–74, 2007

    Google Scholar 

  31. Moratto, V., Kremer, G.M.: Mixtures of relativistic gases in gravitational fields: combined Chapman–Enskog and Grad method and the Onsager relations. Phys. Rev. E91, 052139, 2015

    ADS  MathSciNet  Google Scholar 

  32. Müller, I.: A thermodynamic theory of mixtures of fluids. Arch. Ration. Mech. Anal. 28, 1, 1968

    MathSciNet  MATH  Google Scholar 

  33. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, New York 1998

    MATH  Google Scholar 

  34. Muschik, W.: Entropy Identity inducing Non-Equilibrium Thermodynamics of Relativistic Multi-Component Systems and their Newtonian Limits. arXiv:1806.01648v1, 2018

  35. Nielsen, M., Providência, C., da Providência, J.: Collective modes in hot and dense nuclear matter. Phys. Rev. C47, 200, 1993

    ADS  Google Scholar 

  36. Ott, H.: Lorentz–Transformation der Wiirme und der Temperatur. Zs. f. Phys. 175, 70–104, 1963

    ADS  MATH  Google Scholar 

  37. Paley, D.A., Leonard, N.E., Sepulchre, R., Grunbaum, D., Parrish, J.K.: Oscillator models and collective motion. IEEE Control Syst. 27, 89–105, 2007

    Google Scholar 

  38. Perea, L., Elosegui, P., Gómez, G.: Extension of the Cucker–Smale control law to space flight formation. J. Guidance Control Dyn. 32, 526–536, 2009

    ADS  Google Scholar 

  39. Ruggeri, T.: Can constitutive relations be represented by non-local equations? Quart. Appl. Math. 70, 597–611, 2012

    MathSciNet  MATH  Google Scholar 

  40. Ruggeri, T., Simić, S.: Average temperature and Maxwellian iteration in multitemperature mixtures of fluids. Phys. Rev. E80, 026317, 2009

    ADS  Google Scholar 

  41. Ruggeri, T., Simić, S.: On the hyperbolic system of a mixture of eulerian fluids: a comparison between single and multi-temperature models. Math. Methods Appl. Sci. 30, 827, 2007

    ADS  MathSciNet  MATH  Google Scholar 

  42. Ruggeri, T., Strumia, A.: Main field and convex covariant density for quasi-linear hyperbolic systems: relativistic fluid dynamics. Ann. Inst. Henri Poincaré34, 65–84, 1981

    MathSciNet  MATH  Google Scholar 

  43. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Cham 2015

    MATH  Google Scholar 

  44. Saito, K., Maruyama, T., Soutome, K.: Collective modes in hot and dense matter. Phys. Rev. C40, 407, 1989

    ADS  Google Scholar 

  45. Toner, J., Tu, Y.: Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E58, 4828–4858, 1998

    ADS  MathSciNet  Google Scholar 

  46. Topaz, C.M., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65, 152–174, 2004

    MathSciNet  MATH  Google Scholar 

  47. Truesdell, C.: Rational Thermodynamics. McGraw-Hill, New York 1969

    MATH  Google Scholar 

  48. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Schochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229, 1995

    ADS  MathSciNet  Google Scholar 

  49. Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42, 1967

    Google Scholar 

Download references

Acknowledgements

The work of S.-Y. Ha is partially supported by National Research Foundation of Korea Grant (NRF-2017R1A2B2001864) funded by the Korea Government, and the work of T. Ruggeri is supported by the National Group of Mathematical Physics GNFM-INdAM.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea

    Seung-Yeal Ha

  2. School of Mathematics, Korea Institute for Advanced Studies, Seoul, 02455, Republic of Korea

    Seung-Yeal Ha

  3. Institute of New Media and Communications, Seoul National University, Seoul, 08826, Republic of Korea

    Jeongho Kim

  4. Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM², University of Bologna, Bologna, Italy

    Tommaso Ruggeri

Authors
  1. Seung-Yeal Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeongho Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tommaso Ruggeri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seung-Yeal Ha.

Additional information

Communicated by E. Virga

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Proof of Lemma 6.6

Appendix A: Proof of Lemma 6.6

In this appendix, we provide a detailed proof of Lemma 6.6. Since we have two vectors \({\mathbf{x}}\) and \({\mathbf{y}}\), we only need to consider when \({\mathbf{x}},{\mathbf{y}}\in {\mathbb {R}}^2\) by changing the basis. Moreover, it is possible to choose \({\mathbf{x}}\) as a first direction. Therefore, we set

$$\begin{aligned} {\mathbf{x}}:=(x_1,0) \quad \text{ and } \quad {\mathbf{y}}:=(y_1,y_2). \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned}&({\mathbf{x}}-{\mathbf{y}})\cdot \left( \frac{{\mathbf{x}}}{F(x^2)}-\frac{{\mathbf{y}}}{F(y^2)}\right) \\&\quad =(x_1-y_1)\left( \frac{x_1}{F(x^2)}-\frac{y_1}{F(y^2)}\right) +\frac{y_2^2}{F(y^2)}\\&\quad =(x_1-y_1)\left( \frac{x_1F(y^2)-y_1F(x^2)}{F(x^2)F(y^2)}\right) +\frac{y_2^2}{F(y^2)}\\&\quad =(x_1-y_1)\left( \frac{(x_1-y_1)F(y^2)+y_1(F(y^2)-F(x^2))}{F(x^2)F(y^2)}\right) +\frac{y_2^2}{F(y^2)}\\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)}\left( F(y^2)+y_1\frac{F(y^2)-F(x^2)}{x_1-y_1}\right) +\frac{y_2^2}{F(y^2)}\\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)}\left( F(y^2)-y_1\frac{x^2-y^2}{x_1-y_1}\frac{F(y^2)-F(x^2)}{y^2-x^2}\right) +\frac{y_2^2}{F(y^2)}. \end{aligned} \end{aligned}$$

On the other hand, note that

$$\begin{aligned} \begin{aligned} x^2-y^2&=({\mathbf{x}}-{\mathbf{y}})\cdot ({\mathbf{x}}+{\mathbf{y}})=(x_1-y_1)(x_1+y_1)-y^2_2, \\ F(r)-F(s)&=\sqrt{\frac{r}{c^2}+\frac{r}{g(r)}}-\sqrt{\frac{s}{c^2}+\frac{s}{g(s)}}=\frac{\frac{r}{c^2}-\frac{s}{c^2}+\frac{r}{g(r)}-\frac{s}{g(s)}}{F(r)+F(s)}\\&=:\frac{r-s}{F(r)+F(s)}\left( \frac{1}{c^2}+G(r,s)\right) , \end{aligned} \end{aligned}$$

since g(r) is an increasing function, and it follows from (6.7)\(_2\) that \(\frac{r}{g(r)}\) is also an increasing function and G(r, s) is nonnegative. Thus, we have

$$\begin{aligned}&({\mathbf{x}}-{\mathbf{y}})\cdot \left( \frac{{\mathbf{x}}}{F(x^2)}-\frac{{\mathbf{y}}}{F(y^2)}\right) \\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)}\left( F(y^2)-y_1\frac{x^2-y^2}{x_1-y_1}\frac{F(y^2)-F(x^2)}{y^2-x^2}\right) +\frac{y_2^2}{F(y^2)} \\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)}\left( F(y^2)-y_1(x_1+y_1)\frac{\frac{1}{c^2}+G(x^2,y^2)}{F(x^2)+F(y^2)}\right) \\&\qquad +\frac{y_1(x_1-y_1)y_2^2}{F(x^2)F(y^2)}\frac{\frac{1}{c^2}+G(x^2,y^2)}{F(x^2)+F(y^2)}+\frac{y_2^2}{F(y^2)}\\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\\&\qquad \times \left( F(x^2)F(y^2)+F(y^2)^2-y_1(x_1+y_1)\left( \frac{1}{c^2}+G(x^2,y^2)\right) \right) \\&\qquad +\frac{y_2^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\\&\qquad \times \left( F(x^2)^2+F(x^2)F(y^2)+y_1(x_1-y_1)\left( \frac{1}{c^2}+G(x^2,y^2)\right) \right) . \end{aligned}$$

However, one has

$$\begin{aligned} \begin{aligned}&F(y^2)^2 -y_1^2\left( \frac{1}{c^2}+G(x^2,y^2)\right) =y^2\left( \frac{1}{c^2}+\frac{1}{g(y^2)}\right) -y_1^2\left( \frac{1}{c^2}+G(x^2,y^2)\right) \\&\quad =y^2\left( \frac{1}{g(y^2)}-G(x^2,y^2)\right) +y_2^2\left( \frac{1}{c^2}+G(x^2,y^2)\right) , \\&y^2\left( \frac{1}{g(y^2)}-G(x^2,y^2)\right) =y^2\left( \frac{1}{g(y^2)}-\frac{\frac{y^2}{g(y^2)}-\frac{x^2}{g(x^2)}}{y^2-x^2}\right) \\&\quad =\frac{x^2y^2(g(y^2)-g(x^2))}{g(x^2)g(y^2)(y^2-x^2)}. \end{aligned} \end{aligned}$$

We now use the estimate of \(\frac{r}{g(r)}\) in Lemma 6.5 to obtain

$$\begin{aligned} \begin{aligned}&\frac{r}{g(r)}\ge 1+\frac{1}{c^4}, \quad \frac{g(r)-g(s)}{r-s}=g'(a),\quad \min \{r,s\}\leqq a\leqq \max \{r,s\},\\&\quad \text{ and } \text{ thus, } \quad y^2\left( \frac{1}{g(y^2)}-G(x^2,y^2)\right) \ge \left( 1+\frac{1}{c^4}\right) ^2\min _{0\leqq a\leqq R} g'(a)=:C(R) \end{aligned} \end{aligned}$$

to see that

$$\begin{aligned}&({\mathbf{x}}-{\mathbf{y}})\cdot \left( \frac{{\mathbf{x}}}{F(x^2)}-\frac{{\mathbf{y}}}{F(y^2)}\right) \\&\quad \ge \frac{(x_1-y_1)^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\left( F(x^2)F(y^2) \right. \\&\qquad \left. -(x_1y_1)\left( \frac{1}{c^2}+G(x^2,y^2)\right) +C(R)\right) \\&\qquad +\frac{y_2^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\\&\qquad \times \left( ((x_1-y_1)^2+y_1(x_1-y_1))\left( \frac{1}{c^2}+G(x^2,y^2)\right) \right. \\&\qquad \left. +F(x^2)^2+F(x^2)F(y^2)\right) \\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\left( F(x^2)F(y^2) \right. \\&\qquad \left. -(x_1y_1)\left( \frac{1}{c^2}+G(x^2,y^2)\right) +C(R)\right) \\&\qquad +\frac{y_2^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\\&\qquad \times \left( (x_1-y_1)x_1\left( \frac{1}{c^2}+G(x^2,y^2)\right) +F(x^2)^2+F(x^2)F(y^2)\right) \\&\quad =\frac{(x_1-y_1)^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\left( F(x^2)F(y^2) \right. \\&\qquad \left. -(x_1y_1)\left( \frac{1}{c}+G(x^2,y^2)\right) +C(R)\right) \\&\qquad +\frac{y_2^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\\&\qquad \times \left( x_1^2\left( \frac{1}{c^2}+G(x^2,y^2)\right) +F(x^2)^2+F(x^2)F(y^2) \right. \\&\qquad \left. -x_1y_1\left( \frac{1}{c^2}+G(x^2,y^2)\right) \right) . \end{aligned}$$

On the other hand, since g(r) is an increasing function of r, we also have the following estimate:

$$\begin{aligned} \frac{1}{c^2}+G(r,s)=\frac{1}{c^2}+\frac{\frac{r}{g(r)}-\frac{s}{g(s)}}{r-s}=\frac{1}{c^2}+\frac{1}{g(r)}+s\frac{\frac{1}{g(r)}-\frac{1}{g(s)}}{r-s}\leqq \frac{1}{c^2}+\frac{1}{g(r)}. \end{aligned}$$

By the symmetry between r and s, we have

$$\begin{aligned} \frac{1}{c^2}+G(r,s)\leqq \min \left\{ \frac{1}{c^2}+\frac{1}{g(r)},\frac{1}{c^2}+\frac{1}{g(s)}\right\} \leqq \sqrt{\frac{1}{c^2}+\frac{1}{g(r)}}\sqrt{\frac{1}{c^2}+\frac{1}{g(s)}}. \end{aligned}$$

Thus, we use \(x_1=|{\mathbf{x}}|=x\) and \(y_1\leqq |{\mathbf{y}}|=y\) to obtain

$$\begin{aligned} x_1y_1\left( \frac{1}{c^2}+G(x^2,y^2)\right)&\leqq xy\left( \frac{1}{c^2}+G(x^2,y^2)\right) \\&\leqq \sqrt{\frac{x^2}{c^2}+\frac{x^2}{g(x^2)}}\sqrt{\frac{y^2}{c^2}+\frac{y^2}{g(x^2)}}=F(x^2)F(y^2). \end{aligned}$$

This implies that

$$\begin{aligned}&({\mathbf{x}}-{\mathbf{y}})\cdot \left( \frac{{\mathbf{x}}}{F(x^2)}-\frac{{\mathbf{y}}}{F(y^2)}\right) \\&\quad \ge \frac{(x_1-y_1)^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}C(R)\\&\qquad +\frac{y_2^2}{F(x^2)F(y^2)(F(x^2)+F(y^2))}\left( x_1^2\left( \frac{1}{c^2}+G(x^2,y^2)\right) +F(x^2)^2\right) \\&\quad \ge (x_1-y_1)^2 \frac{C(R)}{2F(R)^3}+y_2^2 \frac{F(x^2)}{F(y^2)(F(x^2)+F(y^2))}\quad (\because G(x^2,y^2)\ge 0)\\&\quad \ge (x_1-y_1)^2 \frac{C(R)}{2F(R)^3}+y_2^2 \frac{1}{2F(R)^2}\ge |{\mathbf{x}}-\mathbf{y}|^2\min \left\{ \frac{C(R)}{2F(R)^3},\frac{1}{2F(R)^2}\right\} , \end{aligned}$$

where we used \(F\ge 1\) in the last inequality. Finally, we choose

$$\begin{aligned} {\varLambda }(R):=\min \left\{ \frac{C(R)}{2F(R)^3},\frac{1}{2F(R)^2}\right\} \end{aligned}$$

to obtain the desired estimate. This completes the proof.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ha, SY., Kim, J. & Ruggeri, T. From the Relativistic Mixture of Gases to the Relativistic Cucker–Smale Flocking. Arch Rational Mech Anal 235, 1661–1706 (2020). https://doi.org/10.1007/s00205-019-01452-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-019-01452-y

Search

Navigation