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October  2018, 38(10): 4837-4873. doi: 10.3934/dcds.2018212

On continuity equations in space-time domains

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90025, USA

* Corresponding author: Yuming Zhang

Received  May 2017 Revised  May 2018 Published  July 2018

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.

Citation: Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008.  Google Scholar

[2]

J.A. CarrilloM. DiFrancescoA. FigalliT. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Mathematical Journal, 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.  Google Scholar

[3]

J.A. CarrilloS. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials, Nonlinear Analysis: Theory, Methods & Applications, 100 (2014), 122-147.  doi: 10.1016/j.na.2014.01.010.  Google Scholar

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J.A. CarrilloD. Slepcev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 1209-1247.   Google Scholar

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F.H. ClarkeR. Stern and P. Wolenski, Proximal smoothness and the lower-c2 property, J. Convex Anal, 2 (1995), 117-144.   Google Scholar

[6]

E. CozziG.-M. Gie and J.P. Kelliher, The aggregation equation with newtonian potential: The vanishing viscosity limit, Journal of Mathematical Analysis and Applications, 453 (2017), 841-893.  doi: 10.1016/j.jmaa.2017.04.009.  Google Scholar

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K. Craig, The exponential formula for the wasserstein metric, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 169-187.  doi: 10.1051/cocv/2014069.  Google Scholar

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M. DelPino and J. Dolbeault, The optimal euclidean l p-sobolev logarithmic inequality, Journal of Functional Analysis, 197 (2003), 151-161.  doi: 10.1016/S0022-1236(02)00070-8.  Google Scholar

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S. DiMarinoB. Maury and F. Santambrogio, Measure sweeping processes, Journal of Convex Analysis, 23 (2016), 567-601.   Google Scholar

[10]

S. DiMarino and A.R. Mészáros, Uniqueness issues for evolution equations with density constraints, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1761-1783.  doi: 10.1142/S0218202516500445.  Google Scholar

[11]

J.F. Edmond and L. Thibault, Bv solutions of nonconvex sweeping process differential inclusion with perturbation, Journal of Differential Equations, 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar

[12]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: L^ p Spaces, Springer Science & Business Media, 2007.  Google Scholar

[13]

L. Gross, Logarithmic sobolev inequalities, American Journal of Mathematics, 97 (1975), 1061-1083.  doi: 10.2307/2373688.  Google Scholar

[14]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the fokker--planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[15]

R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Mathematical Journal, 80 (1995), 309-324.  doi: 10.1215/S0012-7094-95-08013-2.  Google Scholar

[16]

L. Petrelli and A. Tudorascu, Variational principle for general diffusion problems, Applied Mathematics and Optimization, 50 (2004), 229-257.  doi: 10.1007/s00245-004-0801-2.  Google Scholar

[17]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Communications in Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008.  Google Scholar

[2]

J.A. CarrilloM. DiFrancescoA. FigalliT. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Mathematical Journal, 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.  Google Scholar

[3]

J.A. CarrilloS. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials, Nonlinear Analysis: Theory, Methods & Applications, 100 (2014), 122-147.  doi: 10.1016/j.na.2014.01.010.  Google Scholar

[4]

J.A. CarrilloD. Slepcev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 1209-1247.   Google Scholar

[5]

F.H. ClarkeR. Stern and P. Wolenski, Proximal smoothness and the lower-c2 property, J. Convex Anal, 2 (1995), 117-144.   Google Scholar

[6]

E. CozziG.-M. Gie and J.P. Kelliher, The aggregation equation with newtonian potential: The vanishing viscosity limit, Journal of Mathematical Analysis and Applications, 453 (2017), 841-893.  doi: 10.1016/j.jmaa.2017.04.009.  Google Scholar

[7]

K. Craig, The exponential formula for the wasserstein metric, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 169-187.  doi: 10.1051/cocv/2014069.  Google Scholar

[8]

M. DelPino and J. Dolbeault, The optimal euclidean l p-sobolev logarithmic inequality, Journal of Functional Analysis, 197 (2003), 151-161.  doi: 10.1016/S0022-1236(02)00070-8.  Google Scholar

[9]

S. DiMarinoB. Maury and F. Santambrogio, Measure sweeping processes, Journal of Convex Analysis, 23 (2016), 567-601.   Google Scholar

[10]

S. DiMarino and A.R. Mészáros, Uniqueness issues for evolution equations with density constraints, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1761-1783.  doi: 10.1142/S0218202516500445.  Google Scholar

[11]

J.F. Edmond and L. Thibault, Bv solutions of nonconvex sweeping process differential inclusion with perturbation, Journal of Differential Equations, 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar

[12]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: L^ p Spaces, Springer Science & Business Media, 2007.  Google Scholar

[13]

L. Gross, Logarithmic sobolev inequalities, American Journal of Mathematics, 97 (1975), 1061-1083.  doi: 10.2307/2373688.  Google Scholar

[14]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the fokker--planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[15]

R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Mathematical Journal, 80 (1995), 309-324.  doi: 10.1215/S0012-7094-95-08013-2.  Google Scholar

[16]

L. Petrelli and A. Tudorascu, Variational principle for general diffusion problems, Applied Mathematics and Optimization, 50 (2004), 229-257.  doi: 10.1007/s00245-004-0801-2.  Google Scholar

[17]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Communications in Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033.  Google Scholar

Figure 1.  the particle system and the Px,t operator
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