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]

Arch. Ration. Mech. Anal., 162 (2002), 1-23. doi: doi:10.1007/s002050100176.  Google Scholar

[2]

Adv. Differential Equations, 14 (2009), 1-25.  Google Scholar

[3]

Comm. Partial Differential Equations, 28 (2003), 1085-1111. doi: doi:10.1081/PDE-120021187.  Google Scholar

[4]

in "Advances in Kinetic Theory and Computing," Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, (1994), 171-190.  Google Scholar

[5]

Comm. Partial Differential Equations, 24 (1999), 2173-2189. doi: doi:10.1080/03605309908821498.  Google Scholar

[6]

SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: doi:10.1137/S0036142997317353.  Google Scholar

[7]

Arch. Rational Mech. Anal., 140 (1997), 197-223. doi: doi:10.1007/s002050050064.  Google Scholar

[8]

Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, MA, 2004.  Google Scholar

[9]

SIAM J. Math. Anal., 34 (2003), 925-938. doi: doi:10.1137/S0036141001399350.  Google Scholar

[10]

Commun. Math. Sci., 1 (2003), 593-621.  Google Scholar

[11]

Commun. Comput. Phys., 5 (2009), 565-581.  Google Scholar

[12]

SIAM J. Numer. Anal., 45 (2007), 2408-2441. doi: doi:10.1137/050644124.  Google Scholar

[13]

Acta Appl. Math., 99 (2007), 161-183. doi: doi:10.1007/s10440-007-9161-7.  Google Scholar

[14]

J. Math. Anal. Appl., 18 (1967), 238-251. doi: doi:10.1016/0022-247X(67)90054-6.  Google Scholar

[15]

Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  Google Scholar

[16]

Indiana Univ. Math. J., 26 (1977), 1097-1119. doi: doi:10.1512/iumj.1977.26.26088.  Google Scholar

[17]

Mat. Sb. (N.S.) 51 (1960), 99-128.  Google Scholar

[18]

J. Differential Equations, 5 (1969), 515-530. doi: doi:10.1016/0022-0396(69)90091-6.  Google Scholar

[19]

Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993.  Google Scholar

[20]

Phys. D, 152/153 (2001), 620-635. doi: doi:10.1016/S0167-2789(01)00195-6.  Google Scholar

[21]

Numer. Math., 90 (2002), 721-753. doi: doi:10.1007/s002110100309.  Google Scholar

[22]

C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171-174.  Google Scholar

[23]

Mathématiques & Applications, 26, Springer-Verlag, Berlin, 1997.  Google Scholar

[24]

Comm. Math. Phys., 222 (2001), 117-146. doi: doi:10.1007/s002200100506.  Google Scholar

[25]

J. Differential Equations, 176 (2001), 1-28. doi: doi:10.1006/jdeq.2000.3980.  Google Scholar

[26]

Graduate Studies in Mathematics, 12, American Mathematical Society, Providence, RI, 1996.  Google Scholar

[27]

Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967.  Google Scholar

[28]

Research Notes in Mathematics 69, Pitman (Advanced Publishing Program), Boston-London, 1982.  Google Scholar

[29]

Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[30]

Comm. Partial Differential Equations, 22 (1997), 337-358. doi: doi:10.1080/03605309708821265.  Google Scholar

[31]

J. Differential Equations, 56 (1985), 345-390. doi: doi:10.1016/0022-0396(85)90084-1.  Google Scholar

[32]

J. Evol. Equ., 7 (2007), 669-700.  Google Scholar

[33]

Nonlinear Anal., 73 (2010), 1802-1811.  Google Scholar

[34]

Arch. Math. (Basel), 94 (2010), 579-589.  Google Scholar

[35]

Trans. Amer. Math. Soc., 350 (1998), 119-133. doi: doi:10.1090/S0002-9947-98-01648-1.  Google Scholar

[36]

Comm. Math. Phys., 177 (1996), 349-380. doi: doi:10.1007/BF02101897.  Google Scholar

[37]

Lecture Notes in Mathematics, 1700, Springer-Verlag, Berlin, 1998.  Google Scholar

[38]

Astron. Astrophys., 5 (1970), 84-89. Google Scholar

show all references

References:
[1]

Arch. Ration. Mech. Anal., 162 (2002), 1-23. doi: doi:10.1007/s002050100176.  Google Scholar

[2]

Adv. Differential Equations, 14 (2009), 1-25.  Google Scholar

[3]

Comm. Partial Differential Equations, 28 (2003), 1085-1111. doi: doi:10.1081/PDE-120021187.  Google Scholar

[4]

in "Advances in Kinetic Theory and Computing," Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, (1994), 171-190.  Google Scholar

[5]

Comm. Partial Differential Equations, 24 (1999), 2173-2189. doi: doi:10.1080/03605309908821498.  Google Scholar

[6]

SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: doi:10.1137/S0036142997317353.  Google Scholar

[7]

Arch. Rational Mech. Anal., 140 (1997), 197-223. doi: doi:10.1007/s002050050064.  Google Scholar

[8]

Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, MA, 2004.  Google Scholar

[9]

SIAM J. Math. Anal., 34 (2003), 925-938. doi: doi:10.1137/S0036141001399350.  Google Scholar

[10]

Commun. Math. Sci., 1 (2003), 593-621.  Google Scholar

[11]

Commun. Comput. Phys., 5 (2009), 565-581.  Google Scholar

[12]

SIAM J. Numer. Anal., 45 (2007), 2408-2441. doi: doi:10.1137/050644124.  Google Scholar

[13]

Acta Appl. Math., 99 (2007), 161-183. doi: doi:10.1007/s10440-007-9161-7.  Google Scholar

[14]

J. Math. Anal. Appl., 18 (1967), 238-251. doi: doi:10.1016/0022-247X(67)90054-6.  Google Scholar

[15]

Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  Google Scholar

[16]

Indiana Univ. Math. J., 26 (1977), 1097-1119. doi: doi:10.1512/iumj.1977.26.26088.  Google Scholar

[17]

Mat. Sb. (N.S.) 51 (1960), 99-128.  Google Scholar

[18]

J. Differential Equations, 5 (1969), 515-530. doi: doi:10.1016/0022-0396(69)90091-6.  Google Scholar

[19]

Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993.  Google Scholar

[20]

Phys. D, 152/153 (2001), 620-635. doi: doi:10.1016/S0167-2789(01)00195-6.  Google Scholar

[21]

Numer. Math., 90 (2002), 721-753. doi: doi:10.1007/s002110100309.  Google Scholar

[22]

C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171-174.  Google Scholar

[23]

Mathématiques & Applications, 26, Springer-Verlag, Berlin, 1997.  Google Scholar

[24]

Comm. Math. Phys., 222 (2001), 117-146. doi: doi:10.1007/s002200100506.  Google Scholar

[25]

J. Differential Equations, 176 (2001), 1-28. doi: doi:10.1006/jdeq.2000.3980.  Google Scholar

[26]

Graduate Studies in Mathematics, 12, American Mathematical Society, Providence, RI, 1996.  Google Scholar

[27]

Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967.  Google Scholar

[28]

Research Notes in Mathematics 69, Pitman (Advanced Publishing Program), Boston-London, 1982.  Google Scholar

[29]

Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[30]

Comm. Partial Differential Equations, 22 (1997), 337-358. doi: doi:10.1080/03605309708821265.  Google Scholar

[31]

J. Differential Equations, 56 (1985), 345-390. doi: doi:10.1016/0022-0396(85)90084-1.  Google Scholar

[32]

J. Evol. Equ., 7 (2007), 669-700.  Google Scholar

[33]

Nonlinear Anal., 73 (2010), 1802-1811.  Google Scholar

[34]

Arch. Math. (Basel), 94 (2010), 579-589.  Google Scholar

[35]

Trans. Amer. Math. Soc., 350 (1998), 119-133. doi: doi:10.1090/S0002-9947-98-01648-1.  Google Scholar

[36]

Comm. Math. Phys., 177 (1996), 349-380. doi: doi:10.1007/BF02101897.  Google Scholar

[37]

Lecture Notes in Mathematics, 1700, Springer-Verlag, Berlin, 1998.  Google Scholar

[38]

Astron. Astrophys., 5 (1970), 84-89. Google Scholar

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