# American Institute of Mathematical Sciences

March  2011, 10(2): 479-506. doi: 10.3934/cpaa.2011.10.479

## A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit

 1 Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden

Received  January 2010 Revised  July 2010 Published  December 2010

We study the following system of the viscous Hamilton--Jacobi and the continuity equations in the limit as $\varepsilon \downarrow 0$:

$S^\varepsilon_t+\frac{1}{2}|D S^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon =0$ in $Q_T$, $S^\varepsilon(0,x)=S_0(x)$ in $R^n;$

$\rho^\varepsilon_t+$ Div$(\rho^\varepsilon D S^\varepsilon)=0$ in $Q_T$, $\rho^\varepsilon(0,x)=\rho_0(x)$ in $R^n$.

Here $Q_T=(0,T]\times R^n$. The potential $V$ and the initial function $S_0$ are allowed to grow quadratically while $\rho_0$ is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity--measure solution $(S,\rho)$.

Citation: Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure and Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479
##### References:
 [1] P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23. doi: doi:10.1007/s002050100176. [2] S. Benachour, M. Ben-Artzi and P. Laurençot, Sharp decay estimates and vanishing viscosity for diffusive Hamilton-Jacobi equations, Adv. Differential Equations, 14 (2009), 1-25. [3] B. Ben Moussa and G.T. Kossioris, On the system of Hamilton-Jacobi and transport equations arising in geometrical optics, Comm. Partial Differential Equations, 28 (2003), 1085-1111. doi: doi:10.1081/PDE-120021187. [4] F. Bouchut, On zero pressure gas dynamics, in "Advances in Kinetic Theory and Computing," Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, (1994), 171-190. [5] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 2173-2189. doi: doi:10.1080/03605309908821498. [6] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: doi:10.1137/S0036142997317353. [7] P. Cannarsa, A. Mennucci and C. Sinestrari, Regularity results for solutions of a class of Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 140 (1997), 197-223. doi: doi:10.1007/s002050050064. [8] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'' Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, MA, 2004. [9] G.-Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938. doi: doi:10.1137/S0036141001399350. [10] L.-T. Cheng, H. Liu and S. Osher, Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations, Commun. Math. Sci., 1 (2003), 593-621. [11] A. Chertock and A. Kurganov, Computing multivalued solutions of pressureless gas dynamics by deterministic particle methods, Commun. Comput. Phys., 5 (2009), 565-581. [12] A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. Numer. Anal., 45 (2007), 2408-2441. doi: doi:10.1137/050644124. [13] Ph. Choquard and T. Strömberg, A one-dimensional inviscid and compressible fluid in a harmonic potential well, Acta Appl. Math., 99 (2007), 161-183. doi: doi:10.1007/s10440-007-9161-7. [14] E. Conway, Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. Math. Anal. Appl., 18 (1967), 238-251. doi: doi:10.1016/0022-247X(67)90054-6. [15] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. [16] C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26 (1977), 1097-1119. doi: doi:10.1512/iumj.1977.26.26088. [17] A. F. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.) 51 (1960), 99-128. [18] W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations, 5 (1969), 515-530. doi: doi:10.1016/0022-0396(69)90091-6. [19] W. H. Fleming and S. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993. [20] U. Frisch, J. Bec and B. Villone, Singularities and the distribution of density in the Burgers/adhesion model, Phys. D, 152/153 (2001), 620-635. doi: doi:10.1016/S0167-2789(01)00195-6. [21] L. Gosse and F. James, Convergence results for an inhomogeneous system arising in various high frequency approximations, Numer. Math., 90 (2002), 721-753. doi: doi:10.1007/s002110100309. [22] E. Grenier, Existence globale pour le système des gaz sans pression, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171-174. [23] L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations," Mathématiques & Applications, 26, Springer-Verlag, Berlin, 1997. [24] F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: doi:10.1007/s002200100506. [25] K. Ito, Existence of solutions to the Hamilton-Jacobi-Bellman equation under quadratic growth conditions, J. Differential Equations, 176 (2001), 1-28. doi: doi:10.1006/jdeq.2000.3980. [26] N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces,'' Graduate Studies in Mathematics, 12, American Mathematical Society, Providence, RI, 1996. [27] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967. [28] P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,'' Research Notes in Mathematics 69, Pitman (Advanced Publishing Program), Boston-London, 1982. [29] W. Pauli, "General Principles of Quantum Mechanics,'' Springer-Verlag, Berlin-New York, 1980. [30] F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equations with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358. doi: doi:10.1080/03605309708821265. [31] P. E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations, 56 (1985), 345-390. doi: doi:10.1016/0022-0396(85)90084-1. [32] T. Strömberg, Well-posedness for the system of the Hamilton-Jacobi and the continuity equations, J. Evol. Equ., 7 (2007), 669-700. [33] T. Strömberg, On a viscous Hamilton-Jacobi equation with an unbounded potential term, Nonlinear Anal., 73 (2010), 1802-1811. [34] T. Strömberg, Semiconcavity estimates for viscous Hamilton-Jacobi equations, Arch. Math. (Basel), 94 (2010), 579-589. [35] M. A. Sychev and V. J. Mizel, A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems, Trans. Amer. Math. Soc., 350 (1998), 119-133. doi: doi:10.1090/S0002-9947-98-01648-1. [36] E. Weinan, Y. G. Rykov and Y. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. doi: doi:10.1007/BF02101897. [37] W. A. Woyczyński, "Burgers-KPZ turbulence. Göttingen lectures,'' Lecture Notes in Mathematics, 1700, Springer-Verlag, Berlin, 1998. [38] Y. B. Zeldovich, Gravitational instability: an approximate theory for large density perturbations, Astron. Astrophys., 5 (1970), 84-89.

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##### References:
 [1] P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23. doi: doi:10.1007/s002050100176. [2] S. Benachour, M. Ben-Artzi and P. Laurençot, Sharp decay estimates and vanishing viscosity for diffusive Hamilton-Jacobi equations, Adv. Differential Equations, 14 (2009), 1-25. [3] B. Ben Moussa and G.T. Kossioris, On the system of Hamilton-Jacobi and transport equations arising in geometrical optics, Comm. Partial Differential Equations, 28 (2003), 1085-1111. doi: doi:10.1081/PDE-120021187. [4] F. Bouchut, On zero pressure gas dynamics, in "Advances in Kinetic Theory and Computing," Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, (1994), 171-190. [5] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 2173-2189. doi: doi:10.1080/03605309908821498. [6] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: doi:10.1137/S0036142997317353. [7] P. Cannarsa, A. Mennucci and C. Sinestrari, Regularity results for solutions of a class of Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 140 (1997), 197-223. doi: doi:10.1007/s002050050064. [8] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'' Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, MA, 2004. [9] G.-Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938. doi: doi:10.1137/S0036141001399350. [10] L.-T. Cheng, H. Liu and S. Osher, Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations, Commun. Math. Sci., 1 (2003), 593-621. [11] A. Chertock and A. Kurganov, Computing multivalued solutions of pressureless gas dynamics by deterministic particle methods, Commun. Comput. Phys., 5 (2009), 565-581. [12] A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. Numer. Anal., 45 (2007), 2408-2441. doi: doi:10.1137/050644124. [13] Ph. Choquard and T. Strömberg, A one-dimensional inviscid and compressible fluid in a harmonic potential well, Acta Appl. Math., 99 (2007), 161-183. doi: doi:10.1007/s10440-007-9161-7. [14] E. Conway, Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. Math. Anal. Appl., 18 (1967), 238-251. doi: doi:10.1016/0022-247X(67)90054-6. [15] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. [16] C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26 (1977), 1097-1119. doi: doi:10.1512/iumj.1977.26.26088. [17] A. F. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.) 51 (1960), 99-128. [18] W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations, 5 (1969), 515-530. doi: doi:10.1016/0022-0396(69)90091-6. [19] W. H. Fleming and S. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993. [20] U. Frisch, J. Bec and B. Villone, Singularities and the distribution of density in the Burgers/adhesion model, Phys. D, 152/153 (2001), 620-635. doi: doi:10.1016/S0167-2789(01)00195-6. [21] L. Gosse and F. James, Convergence results for an inhomogeneous system arising in various high frequency approximations, Numer. Math., 90 (2002), 721-753. doi: doi:10.1007/s002110100309. [22] E. Grenier, Existence globale pour le système des gaz sans pression, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171-174. [23] L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations," Mathématiques & Applications, 26, Springer-Verlag, Berlin, 1997. [24] F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: doi:10.1007/s002200100506. [25] K. Ito, Existence of solutions to the Hamilton-Jacobi-Bellman equation under quadratic growth conditions, J. Differential Equations, 176 (2001), 1-28. doi: doi:10.1006/jdeq.2000.3980. [26] N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces,'' Graduate Studies in Mathematics, 12, American Mathematical Society, Providence, RI, 1996. [27] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967. [28] P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,'' Research Notes in Mathematics 69, Pitman (Advanced Publishing Program), Boston-London, 1982. [29] W. Pauli, "General Principles of Quantum Mechanics,'' Springer-Verlag, Berlin-New York, 1980. [30] F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equations with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358. doi: doi:10.1080/03605309708821265. [31] P. E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations, 56 (1985), 345-390. doi: doi:10.1016/0022-0396(85)90084-1. [32] T. Strömberg, Well-posedness for the system of the Hamilton-Jacobi and the continuity equations, J. Evol. Equ., 7 (2007), 669-700. [33] T. Strömberg, On a viscous Hamilton-Jacobi equation with an unbounded potential term, Nonlinear Anal., 73 (2010), 1802-1811. [34] T. Strömberg, Semiconcavity estimates for viscous Hamilton-Jacobi equations, Arch. Math. (Basel), 94 (2010), 579-589. [35] M. A. Sychev and V. J. Mizel, A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems, Trans. Amer. Math. Soc., 350 (1998), 119-133. doi: doi:10.1090/S0002-9947-98-01648-1. [36] E. Weinan, Y. G. Rykov and Y. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. doi: doi:10.1007/BF02101897. [37] W. A. Woyczyński, "Burgers-KPZ turbulence. Göttingen lectures,'' Lecture Notes in Mathematics, 1700, Springer-Verlag, Berlin, 1998. [38] Y. B. Zeldovich, Gravitational instability: an approximate theory for large density perturbations, Astron. Astrophys., 5 (1970), 84-89.
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