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 & 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.  doi: doi:10.1007/s002050100176.  Google Scholar

[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.   Google Scholar

[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.  doi: doi:10.1081/PDE-120021187.  Google Scholar

[4]

F. Bouchut, On zero pressure gas dynamics,, in, (1994), 171.   Google Scholar

[5]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Equations, 24 (1999), 2173.  doi: doi:10.1080/03605309908821498.  Google Scholar

[6]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws,, SIAM J. Numer. Anal., 35 (1998), 2317.  doi: doi:10.1137/S0036142997317353.  Google Scholar

[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.  doi: doi:10.1007/s002050050064.  Google Scholar

[8]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'', Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[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.  doi: doi:10.1137/S0036141001399350.  Google Scholar

[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.   Google Scholar

[11]

A. Chertock and A. Kurganov, Computing multivalued solutions of pressureless gas dynamics by deterministic particle methods,, Commun. Comput. Phys., 5 (2009), 565.   Google Scholar

[12]

A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics,, SIAM J. Numer. Anal., 45 (2007), 2408.  doi: doi:10.1137/050644124.  Google Scholar

[13]

Ph. Choquard and T. Strömberg, A one-dimensional inviscid and compressible fluid in a harmonic potential well,, Acta Appl. Math., 99 (2007), 161.  doi: doi:10.1007/s10440-007-9161-7.  Google Scholar

[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.  doi: doi:10.1016/0022-247X(67)90054-6.  Google Scholar

[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.   Google Scholar

[16]

C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws,, Indiana Univ. Math. J., 26 (1977), 1097.  doi: doi:10.1512/iumj.1977.26.26088.  Google Scholar

[17]

A. F. Filippov, Differential equations with discontinuous right-hand side,, Mat. Sb. (N.S.) \textbf{51} (1960), 51 (1960), 99.   Google Scholar

[18]

W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation,, J. Differential Equations, 5 (1969), 515.  doi: doi:10.1016/0022-0396(69)90091-6.  Google Scholar

[19]

W. H. Fleming and S. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Applications of Mathematics (New York), (1993).   Google Scholar

[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.  doi: doi:10.1016/S0167-2789(01)00195-6.  Google Scholar

[21]

L. Gosse and F. James, Convergence results for an inhomogeneous system arising in various high frequency approximations,, Numer. Math., 90 (2002), 721.  doi: doi:10.1007/s002110100309.  Google Scholar

[22]

E. Grenier, Existence globale pour le système des gaz sans pression,, C. R. Acad. Sci. Paris S\'er. I Math., 321 (1995), 171.   Google Scholar

[23]

L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations,", Math\'ematiques & Applications, (1997).   Google Scholar

[24]

F. Huang and Z. Wang, Well posedness for pressureless flow,, Comm. Math. Phys., 222 (2001), 117.  doi: doi:10.1007/s002200100506.  Google Scholar

[25]

K. Ito, Existence of solutions to the Hamilton-Jacobi-Bellman equation under quadratic growth conditions,, J. Differential Equations, 176 (2001), 1.  doi: doi:10.1006/jdeq.2000.3980.  Google Scholar

[26]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces,'', Graduate Studies in Mathematics, (1996).   Google Scholar

[27]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', Translations of Mathematical Monographs, (1967).   Google Scholar

[28]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,'', Research Notes in Mathematics 69, (1982).   Google Scholar

[29]

W. Pauli, "General Principles of Quantum Mechanics,'', Springer-Verlag, (1980).   Google Scholar

[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.  doi: doi:10.1080/03605309708821265.  Google Scholar

[31]

P. E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations,, J. Differential Equations, 56 (1985), 345.  doi: doi:10.1016/0022-0396(85)90084-1.  Google Scholar

[32]

T. Strömberg, Well-posedness for the system of the Hamilton-Jacobi and the continuity equations,, J. Evol. Equ., 7 (2007), 669.   Google Scholar

[33]

T. Strömberg, On a viscous Hamilton-Jacobi equation with an unbounded potential term,, Nonlinear Anal., 73 (2010), 1802.   Google Scholar

[34]

T. Strömberg, Semiconcavity estimates for viscous Hamilton-Jacobi equations,, Arch. Math. (Basel), 94 (2010), 579.   Google Scholar

[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.  doi: doi:10.1090/S0002-9947-98-01648-1.  Google Scholar

[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.  doi: doi:10.1007/BF02101897.  Google Scholar

[37]

W. A. Woyczyński, "Burgers-KPZ turbulence. Göttingen lectures,'', Lecture Notes in Mathematics, (1700).   Google Scholar

[38]

Y. B. Zeldovich, Gravitational instability: an approximate theory for large density perturbations,, Astron. Astrophys., 5 (1970), 84.   Google Scholar

show all references

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.  doi: doi:10.1007/s002050100176.  Google Scholar

[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.   Google Scholar

[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.  doi: doi:10.1081/PDE-120021187.  Google Scholar

[4]

F. Bouchut, On zero pressure gas dynamics,, in, (1994), 171.   Google Scholar

[5]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Equations, 24 (1999), 2173.  doi: doi:10.1080/03605309908821498.  Google Scholar

[6]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws,, SIAM J. Numer. Anal., 35 (1998), 2317.  doi: doi:10.1137/S0036142997317353.  Google Scholar

[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.  doi: doi:10.1007/s002050050064.  Google Scholar

[8]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'', Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[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.  doi: doi:10.1137/S0036141001399350.  Google Scholar

[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.   Google Scholar

[11]

A. Chertock and A. Kurganov, Computing multivalued solutions of pressureless gas dynamics by deterministic particle methods,, Commun. Comput. Phys., 5 (2009), 565.   Google Scholar

[12]

A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics,, SIAM J. Numer. Anal., 45 (2007), 2408.  doi: doi:10.1137/050644124.  Google Scholar

[13]

Ph. Choquard and T. Strömberg, A one-dimensional inviscid and compressible fluid in a harmonic potential well,, Acta Appl. Math., 99 (2007), 161.  doi: doi:10.1007/s10440-007-9161-7.  Google Scholar

[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.  doi: doi:10.1016/0022-247X(67)90054-6.  Google Scholar

[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.   Google Scholar

[16]

C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws,, Indiana Univ. Math. J., 26 (1977), 1097.  doi: doi:10.1512/iumj.1977.26.26088.  Google Scholar

[17]

A. F. Filippov, Differential equations with discontinuous right-hand side,, Mat. Sb. (N.S.) \textbf{51} (1960), 51 (1960), 99.   Google Scholar

[18]

W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation,, J. Differential Equations, 5 (1969), 515.  doi: doi:10.1016/0022-0396(69)90091-6.  Google Scholar

[19]

W. H. Fleming and S. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Applications of Mathematics (New York), (1993).   Google Scholar

[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.  doi: doi:10.1016/S0167-2789(01)00195-6.  Google Scholar

[21]

L. Gosse and F. James, Convergence results for an inhomogeneous system arising in various high frequency approximations,, Numer. Math., 90 (2002), 721.  doi: doi:10.1007/s002110100309.  Google Scholar

[22]

E. Grenier, Existence globale pour le système des gaz sans pression,, C. R. Acad. Sci. Paris S\'er. I Math., 321 (1995), 171.   Google Scholar

[23]

L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations,", Math\'ematiques & Applications, (1997).   Google Scholar

[24]

F. Huang and Z. Wang, Well posedness for pressureless flow,, Comm. Math. Phys., 222 (2001), 117.  doi: doi:10.1007/s002200100506.  Google Scholar

[25]

K. Ito, Existence of solutions to the Hamilton-Jacobi-Bellman equation under quadratic growth conditions,, J. Differential Equations, 176 (2001), 1.  doi: doi:10.1006/jdeq.2000.3980.  Google Scholar

[26]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces,'', Graduate Studies in Mathematics, (1996).   Google Scholar

[27]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', Translations of Mathematical Monographs, (1967).   Google Scholar

[28]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,'', Research Notes in Mathematics 69, (1982).   Google Scholar

[29]

W. Pauli, "General Principles of Quantum Mechanics,'', Springer-Verlag, (1980).   Google Scholar

[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.  doi: doi:10.1080/03605309708821265.  Google Scholar

[31]

P. E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations,, J. Differential Equations, 56 (1985), 345.  doi: doi:10.1016/0022-0396(85)90084-1.  Google Scholar

[32]

T. Strömberg, Well-posedness for the system of the Hamilton-Jacobi and the continuity equations,, J. Evol. Equ., 7 (2007), 669.   Google Scholar

[33]

T. Strömberg, On a viscous Hamilton-Jacobi equation with an unbounded potential term,, Nonlinear Anal., 73 (2010), 1802.   Google Scholar

[34]

T. Strömberg, Semiconcavity estimates for viscous Hamilton-Jacobi equations,, Arch. Math. (Basel), 94 (2010), 579.   Google Scholar

[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.  doi: doi:10.1090/S0002-9947-98-01648-1.  Google Scholar

[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.  doi: doi:10.1007/BF02101897.  Google Scholar

[37]

W. A. Woyczyński, "Burgers-KPZ turbulence. Göttingen lectures,'', Lecture Notes in Mathematics, (1700).   Google Scholar

[38]

Y. B. Zeldovich, Gravitational instability: an approximate theory for large density perturbations,, Astron. Astrophys., 5 (1970), 84.   Google Scholar

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