# American Institute of Mathematical Sciences

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

* Corresponding author

Received  January 2020 Published  March 2020

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.

Citation: Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. Bressan and R. Murray, On self-similar solutions to the Euler equations, submitted. Google Scholar [6] P. Ciarlet and C. Mardare, On the Newton-Kantorovich theorem, Anal. Appl. (Singap.), 10 (2012), 249-269.  doi: 10.1142/S0219530512500121.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] S. Daneri, E. Runa and L. Székelyhidi, Non-uniqueness for the Euler equations up to Onsager's critical exponent, work in progress. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401.  doi: 10.1007/BF02921318.  Google Scholar [23] W. Shen, Matlab codes for the numerical simulation of the incompressible Euler equations., Available from: http://www.personal.psu.edu/wxs27/SimEuler/. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar

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##### 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. Bressan and R. Murray, On self-similar solutions to the Euler equations, submitted. Google Scholar [6] P. Ciarlet and C. Mardare, On the Newton-Kantorovich theorem, Anal. Appl. (Singap.), 10 (2012), 249-269.  doi: 10.1142/S0219530512500121.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] S. Daneri, E. Runa and L. Székelyhidi, Non-uniqueness for the Euler equations up to Onsager's critical exponent, work in progress. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401.  doi: 10.1007/BF02921318.  Google Scholar [23] W. Shen, Matlab codes for the numerical simulation of the incompressible Euler equations., Available from: http://www.personal.psu.edu/wxs27/SimEuler/. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar
The vorticity distribution at time $t = 1$, for a solution to (1.2) with initial vorticity $\overline \omega_\varepsilon$
The vorticity distribution at time $t = 1$, for a solution to (1.2) with initial vorticity $\overline\omega_\varepsilon^\dagger$
The supports of the initial vorticity considered in (1.3)-(1.5)
Decomposing the plane ${\mathbb R}^2 = {\mathcal D}^\sharp\cup{\mathcal D}^\natural\cup{\mathcal D}^\flat$ into an outer, a middle, and an inner domain. Left: the case of a single spiraling vortex, as in Fig. 2. Right: the case of two spiraling vortices, as in Fig. 3
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)
Computing the perturbed solution $\Omega^\varepsilon(y)$ in (3.4), by estimating the change in the characteristic through $y$
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