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Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance
A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains
Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea |
$ L_{p} $ |
$ p>1 $ |
$ \begin{equation*} \partial_{t}^{\alpha}u(t,x) = a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x), \quad t>0, x\in \Omega, \end{equation*} $ |
$ \alpha\in (0,2) $ |
$ \partial_{t}^{\alpha} $ |
$ \alpha $ |
$ \Omega $ |
$ C^1 $ |
$ \mathbb{R}^d $ |
References:
[1] |
D. Beleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012.
doi: 10.1142/9789814355216. |
[2] |
H. Dong and D. Kim,
$L_p$-estimates for time fractional parabolic equations in divergence form with measurable coefficients, J. Funct. Anal., 278 (2020), 108338.
doi: 10.1016/j.jfa.2019.108338. |
[3] |
H. Dong and D. Kim,
$L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.
doi: 10.1016/j.aim.2019.01.016. |
[4] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. |
[5] |
B.-S. Han, K.-H. Kim and D. Park,
Weighed $L_{q}(L_{p})$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives, J. Differ. Equ., 269 (2020), 3515-3550.
doi: 10.1016/j.jde.2020.03.005. |
[6] |
D. Kim, K.-H. Kim and K. Lee, Parabolic systems with measurable coefficients in weighted Sobolev spaces, arXiv: 1809.01325, (2018). |
[7] |
I. Kim, K.-H. Kim and S. Lim,
A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives, Ann. Probab., 47 (2019), 2087-2139.
doi: 10.1214/18-AOP1303. |
[8] |
I. Kim, K.-H. Kim and S. Lim,
An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.
doi: 10.1016/j.aim.2016.08.046. |
[9] |
K.-H. Kim,
On $L_{p}$-theory of stochastic partial differential equations of divergence form in $C^{1}$ domains, Probab. Theory. Rel., 130 (2004), 473-492.
doi: 10.1007/s00440-004-0368-5. |
[10] |
K.-H. Kim,
On stochastic partial differential equations with variable coefficients in $C^1$ domains, Stoch. Proc. Appl., 112 (2004), 261-283.
doi: 10.1016/j.spa.2004.02.006. |
[11] |
K.-H. Kim and N. V. Krylov,
On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal., 36 (2004), 618-642.
doi: 10.1137/S0036141003421145. |
[12] |
K.-H. Kim and N. V. Krylov,
On SPDEs with variable coefficients in one space dimension, Potential Anal., 21 (2004), 209-239.
doi: 10.1023/B:POTA.0000033334.06990.9d. |
[13] |
K.-H. Kim and S. Lim,
Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.
doi: 10.4134/JKMS.j150343. |
[14] |
N. V. Krylov and S. V. Lototsky,
A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298-325.
doi: 10.1137/S0036141097326908. |
[15] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, 2008.
doi: 10.1090/gsm/096. |
[16] |
N. V. Krylov,
Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Comm. Partial Differential Equations, 24 (1999), 1611-1653.
doi: 10.1080/03605309908821478. |
[17] |
G. M. Lieberman,
Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.
doi: 10.2140/pjm.1985.117.329. |
[18] |
S. V. Lototsky,
Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods. Appl. Anal., 7 (2000), 195-204.
doi: 10.4310/MAA.2000.v7.n1.a9. |
[19] |
F. Mainardi,
Fractional diffusive waves in viscoelastic solids, Nonlinear waves in solids, 137 (1995), 93-97.
|
[20] |
F. Mainardi and P. Paradisi,
Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.
doi: 10.1142/S0218396X01000826. |
[21] |
R. Metzler, E. Barkai and J. Kalfter,
Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82 (1999), 35-63.
|
[22] |
R. Metzler and J. Klafter,
The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[23] |
I. Podludni, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999. |
[24] |
H. Richard, Fractional Calculus: An Introduction for Physicists, , World Scientific, 2014. |
[25] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, , CRC Press, 1993. |
[26] |
H. Ye, J. Gao and Y. Ding,
A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.
doi: 10.1016/j.jmaa.2006.05.061. |
[27] |
R. Zacher,
Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.
doi: 10.1007/s00028-004-0161-z. |
[28] |
R. Zacher,
Global strong solvability of a quasilinear subdiffusion problem, J. Evol. Equ., 12 (2012), 813-831.
doi: 10.1007/s00028-012-0156-0. |
show all references
References:
[1] |
D. Beleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012.
doi: 10.1142/9789814355216. |
[2] |
H. Dong and D. Kim,
$L_p$-estimates for time fractional parabolic equations in divergence form with measurable coefficients, J. Funct. Anal., 278 (2020), 108338.
doi: 10.1016/j.jfa.2019.108338. |
[3] |
H. Dong and D. Kim,
$L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.
doi: 10.1016/j.aim.2019.01.016. |
[4] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. |
[5] |
B.-S. Han, K.-H. Kim and D. Park,
Weighed $L_{q}(L_{p})$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives, J. Differ. Equ., 269 (2020), 3515-3550.
doi: 10.1016/j.jde.2020.03.005. |
[6] |
D. Kim, K.-H. Kim and K. Lee, Parabolic systems with measurable coefficients in weighted Sobolev spaces, arXiv: 1809.01325, (2018). |
[7] |
I. Kim, K.-H. Kim and S. Lim,
A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives, Ann. Probab., 47 (2019), 2087-2139.
doi: 10.1214/18-AOP1303. |
[8] |
I. Kim, K.-H. Kim and S. Lim,
An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.
doi: 10.1016/j.aim.2016.08.046. |
[9] |
K.-H. Kim,
On $L_{p}$-theory of stochastic partial differential equations of divergence form in $C^{1}$ domains, Probab. Theory. Rel., 130 (2004), 473-492.
doi: 10.1007/s00440-004-0368-5. |
[10] |
K.-H. Kim,
On stochastic partial differential equations with variable coefficients in $C^1$ domains, Stoch. Proc. Appl., 112 (2004), 261-283.
doi: 10.1016/j.spa.2004.02.006. |
[11] |
K.-H. Kim and N. V. Krylov,
On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal., 36 (2004), 618-642.
doi: 10.1137/S0036141003421145. |
[12] |
K.-H. Kim and N. V. Krylov,
On SPDEs with variable coefficients in one space dimension, Potential Anal., 21 (2004), 209-239.
doi: 10.1023/B:POTA.0000033334.06990.9d. |
[13] |
K.-H. Kim and S. Lim,
Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.
doi: 10.4134/JKMS.j150343. |
[14] |
N. V. Krylov and S. V. Lototsky,
A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298-325.
doi: 10.1137/S0036141097326908. |
[15] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, 2008.
doi: 10.1090/gsm/096. |
[16] |
N. V. Krylov,
Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Comm. Partial Differential Equations, 24 (1999), 1611-1653.
doi: 10.1080/03605309908821478. |
[17] |
G. M. Lieberman,
Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.
doi: 10.2140/pjm.1985.117.329. |
[18] |
S. V. Lototsky,
Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods. Appl. Anal., 7 (2000), 195-204.
doi: 10.4310/MAA.2000.v7.n1.a9. |
[19] |
F. Mainardi,
Fractional diffusive waves in viscoelastic solids, Nonlinear waves in solids, 137 (1995), 93-97.
|
[20] |
F. Mainardi and P. Paradisi,
Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.
doi: 10.1142/S0218396X01000826. |
[21] |
R. Metzler, E. Barkai and J. Kalfter,
Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82 (1999), 35-63.
|
[22] |
R. Metzler and J. Klafter,
The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[23] |
I. Podludni, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999. |
[24] |
H. Richard, Fractional Calculus: An Introduction for Physicists, , World Scientific, 2014. |
[25] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, , CRC Press, 1993. |
[26] |
H. Ye, J. Gao and Y. Ding,
A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.
doi: 10.1016/j.jmaa.2006.05.061. |
[27] |
R. Zacher,
Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.
doi: 10.1007/s00028-004-0161-z. |
[28] |
R. Zacher,
Global strong solvability of a quasilinear subdiffusion problem, J. Evol. Equ., 12 (2012), 813-831.
doi: 10.1007/s00028-012-0156-0. |
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