Figure 2.
The vorticity distribution at time $ t = 1 $ , for a solution to (1.2) with initial vorticity $ \overline \omega_\varepsilon $
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Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations
January
2021, 41(1): 113-130.
doi: 10.3934/dcds.2020168
A posteriori error estimates for self-similar solutions to the Euler equations
Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA |
The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.
Keywords:
A posterior error estimate,
Euler equations,
incompressible fluid,
self-similar solutions.
Mathematics Subject Classification:
Primary: 35L60, 35Q31; Secondary: 35Q35.
Citation:
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems,
2021, 41
(1)
: 113-130.
doi: 10.3934/dcds.2020168
![]()
References:
| [1] |
M. S. Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems in Mathematical Analysis, Pure and Applied Mathematics, Academic Press, New York, 1977.
|
| [2] |
F. Bernicot and T. Hmidi,
On the global well-posedness for Euler equations with unbounded vorticity, Dyn. Partial Differ. Equ., 12 (2015), 127-155.
doi: 10.4310/DPDE.2015.v12.n2.a3. |
| [3] |
S. C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
| [4] |
A. Bressan, Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations, Graduate Studies in Mathematics, 143, American Mathematical Society, Providence, RI, 2013. |
| [5] |
A. Bressan and R. Murray, On self-similar solutions to the Euler equations, submitted. |
| [6] |
P. Ciarlet and C. Mardare,
On the Newton-Kantorovich theorem, Anal. Appl. (Singap.), 10 (2012), 249-269.
doi: 10.1142/S0219530512500121. |
| [7] |
P. Clément,
Approximation by finite element functions using local regularization, Rev. Francaise Automat. Informat. Rech. Opérationnelle Sér., 9 (1975), 77-84.
doi: 10.1051/m2an/197509R200771. |
| [8] |
S. Daneri,
Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations, Comm. Math. Phys., 329 (2014), 745-786.
doi: 10.1007/s00220-014-1973-5. |
| [9] |
S. Daneri, E. Runa and L. Székelyhidi, Non-uniqueness for the Euler equations up to Onsager's critical exponent, work in progress. |
| [10] |
S. Daneri and L. Székelyhidi,
Non-uniqueness and $h$-principle for Hölder-continuous weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 224 (2017), 471-514.
doi: 10.1007/s00205-017-1081-8. |
| [11] |
C. De Lellis and L. Székelyhidi, The Euler equations as a differential inclusion, Ann. of Math. (2), 170 (2009), 1417-1436.
doi: 10.4007/annals.2009.170.1417. |
| [12] |
C. De Lellis and L. Székelyhidi,
On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260.
doi: 10.1007/s00205-008-0201-x. |
| [13] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
| [14] |
V. Elling,
Algebraic spiral solutions of 2d incompressible Euler, J. Differential Equations, 255 (2013), 3749-3787.
doi: 10.1016/j.jde.2013.07.021. |
| [15] |
V. Elling,
Self-Similar 2d Euler solutions with mixed-sign vorticity, Comm. Math. Phys., 348 (2016), 27-68.
doi: 10.1007/s00220-016-2755-z. |
| [16] |
V. Elling,
Algebraic spiral solutions of the 2d incompressible Euler equations, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 323-334.
doi: 10.1007/s00574-016-0141-2. |
| [17] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
| [18] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
| [19] |
J. R. Kuttler and V. G. Sigillito,
Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26 (1984), 163-193.
doi: 10.1137/1026033. |
| [20] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4284-0. |
| [21] |
L. R. Scott and S. Zhang,
Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483-493.
doi: 10.1090/S0025-5718-1990-1011446-7. |
| [22] |
V. Scheffer,
An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401.
doi: 10.1007/BF02921318. |
| [23] |
W. Shen, Matlab codes for the numerical simulation of the incompressible Euler equations., Available from: http://www.personal.psu.edu/wxs27/SimEuler/. |
| [24] |
A. Shnirelman,
On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., 50 (1997), 1261-1286.
doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6. |
| [25] |
M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part Ⅰ and Ⅱ, preprint, arXiv: 1805.09426 and arXiv: 1805.09440. |
| [26] |
V. Yudovich,
Non-stationary flow of an ideal incompressible liquid, Comp. Math. Math. Phys., 3 (1963), 1407-1457.
doi: 10.1016/0041-5553(63)90247-7. |
show all references
References:
| [1] |
M. S. Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems in Mathematical Analysis, Pure and Applied Mathematics, Academic Press, New York, 1977.
|
| [2] |
F. Bernicot and T. Hmidi,
On the global well-posedness for Euler equations with unbounded vorticity, Dyn. Partial Differ. Equ., 12 (2015), 127-155.
doi: 10.4310/DPDE.2015.v12.n2.a3. |
| [3] |
S. C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
| [4] |
A. Bressan, Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations, Graduate Studies in Mathematics, 143, American Mathematical Society, Providence, RI, 2013. |
| [5] |
A. Bressan and R. Murray, On self-similar solutions to the Euler equations, submitted. |
| [6] |
P. Ciarlet and C. Mardare,
On the Newton-Kantorovich theorem, Anal. Appl. (Singap.), 10 (2012), 249-269.
doi: 10.1142/S0219530512500121. |
| [7] |
P. Clément,
Approximation by finite element functions using local regularization, Rev. Francaise Automat. Informat. Rech. Opérationnelle Sér., 9 (1975), 77-84.
doi: 10.1051/m2an/197509R200771. |
| [8] |
S. Daneri,
Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations, Comm. Math. Phys., 329 (2014), 745-786.
doi: 10.1007/s00220-014-1973-5. |
| [9] |
S. Daneri, E. Runa and L. Székelyhidi, Non-uniqueness for the Euler equations up to Onsager's critical exponent, work in progress. |
| [10] |
S. Daneri and L. Székelyhidi,
Non-uniqueness and $h$-principle for Hölder-continuous weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 224 (2017), 471-514.
doi: 10.1007/s00205-017-1081-8. |
| [11] |
C. De Lellis and L. Székelyhidi, The Euler equations as a differential inclusion, Ann. of Math. (2), 170 (2009), 1417-1436.
doi: 10.4007/annals.2009.170.1417. |
| [12] |
C. De Lellis and L. Székelyhidi,
On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260.
doi: 10.1007/s00205-008-0201-x. |
| [13] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
| [14] |
V. Elling,
Algebraic spiral solutions of 2d incompressible Euler, J. Differential Equations, 255 (2013), 3749-3787.
doi: 10.1016/j.jde.2013.07.021. |
| [15] |
V. Elling,
Self-Similar 2d Euler solutions with mixed-sign vorticity, Comm. Math. Phys., 348 (2016), 27-68.
doi: 10.1007/s00220-016-2755-z. |
| [16] |
V. Elling,
Algebraic spiral solutions of the 2d incompressible Euler equations, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 323-334.
doi: 10.1007/s00574-016-0141-2. |
| [17] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
| [18] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
| [19] |
J. R. Kuttler and V. G. Sigillito,
Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26 (1984), 163-193.
doi: 10.1137/1026033. |
| [20] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4284-0. |
| [21] |
L. R. Scott and S. Zhang,
Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483-493.
doi: 10.1090/S0025-5718-1990-1011446-7. |
| [22] |
V. Scheffer,
An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401.
doi: 10.1007/BF02921318. |
| [23] |
W. Shen, Matlab codes for the numerical simulation of the incompressible Euler equations., Available from: http://www.personal.psu.edu/wxs27/SimEuler/. |
| [24] |
A. Shnirelman,
On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., 50 (1997), 1261-1286.
doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6. |
| [25] |
M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part Ⅰ and Ⅱ, preprint, arXiv: 1805.09426 and arXiv: 1805.09440. |
| [26] |
V. Yudovich,
Non-stationary flow of an ideal incompressible liquid, Comp. Math. Math. Phys., 3 (1963), 1407-1457.
doi: 10.1016/0041-5553(63)90247-7. |
Figure 3.
The vorticity distribution at time $ t = 1 $ , for a solution to (1.2) with initial vorticity $ \overline\omega_\varepsilon^\dagger $
Figure 1.
The supports of the initial vorticity considered in (1.3)-(1.5)
Figure 5.
According to (A1), every characteristic starting at a point $ y\in \Sigma_1\cap {\rm Supp }(h) $ exits from the domain $ {\mathcal D} $ at some boundary point $ z = z(y)\in \Sigma_2 $ , at a time $ T(y)\leq T^* $ . The shaded region represents the subdomain $ {\mathcal D}^* $ in (2.10)
Figure 6.
Computing the perturbed solution $ \Omega^\varepsilon(y) $ in (3.4), by estimating the change in the characteristic through $ y $
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