Counterexamples in PDE - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:25:15Z http://mathoverflow.net/feeds/question/68680 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68680/counterexamples-in-pde Counterexamples in PDE timur 2011-06-24T00:56:42Z 2013-01-08T01:42:57Z <p>Let us compile a list of counterexamples in PDE, similar in spirit to the books <em>Counterexamples in topology</em> and <em>Counterexamples in analysis</em>. Eventually I plan to type up the examples with their detailed derivations.</p> <p>Please give one example per answer, preferably with clear descriptions and pointers to literature.</p> <p><a href="http://mathoverflow.net/questions/4994/fundamental-examples" rel="nofollow">A related question</a>:</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68681#68681 Answer by Yakov Shlapentokh-Rothman for Counterexamples in PDE Yakov Shlapentokh-Rothman 2011-06-24T01:04:22Z 2011-06-24T01:04:22Z <p>Lewy's Example gives a PDE where local solvability fails.</p> <p><a href="http://en.wikipedia.org/wiki/Lewy%27s_example" rel="nofollow">http://en.wikipedia.org/wiki/Lewy%27s_example</a></p> <p>Fritz John's PDE book has a detailed discussion.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68682#68682 Answer by timur for Counterexamples in PDE timur 2011-06-24T01:14:03Z 2011-06-24T01:14:03Z <p>in 1955 Ennio De Giorgi constructed an example of parabolic-type linear equation, whose Cauchy problem has non-unique solution. An English translation of this paper appears in De Giorgi's collected works.</p> <p>To be more specific, he constructs 4 smooth functions $a(x,t)$, $b(x,t)$, $c(x,t)$, and $u(x,t)$ defined on the strip $\mathbb{R}\times[0,1]$, such that</p> <p>$$\frac{\partial^8u}{\partial t^8} = a\frac{\partial^4u}{\partial x^4} + b\frac{\partial^2u}{\partial x^2}+cu,$$</p> <p>on the whole strip, </p> <p>$$\frac{\partial^nu}{\partial t^n} = 0,$$</p> <p>identically on the line $t=0$ for $n=0,\ldots,7$, and $u$ not identically zero.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68696#68696 Answer by Anatoly Kochubei for Counterexamples in PDE Anatoly Kochubei 2011-06-24T05:18:12Z 2011-06-24T05:18:12Z <p>There is an example of non-uniqueness of solutions of the Cauchy problem for the heat equation in a class of functions with possible rapid growth at infinity. The example is constructed using the theory of quasi-analytic classes. See, for example, Section 1.9 in A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68699#68699 Answer by Andrew for Counterexamples in PDE Andrew 2011-06-24T06:26:55Z 2011-06-24T06:59:46Z <p>There is well-known <a href="http://www.math.unl.edu/~scohn1/8423/wellposed.pdf" rel="nofollow">example</a> of not <a href="http://en.wikipedia.org/wiki/Well-posed_problem" rel="nofollow">well-posed problem</a> by Hadamard.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68705#68705 Answer by Andrey Rekalo for Counterexamples in PDE Andrey Rekalo 2011-06-24T07:41:28Z 2011-06-24T07:41:28Z <p>Scheffer has shown that there is a nontrivial weak solution $u(x,t)\in L^2(\mathbb R^2\times\mathbb R)$ to the incompressible <a href="http://en.wikipedia.org/wiki/Euler_equations_%28fluid_dynamics%29" rel="nofollow">Euler equations</a> in 2D </p> <p>$$\left\{\begin{eqnarray} \frac{\partial u}{\partial t}+\nabla\cdot(u\otimes u) +\nabla p=0 \\ \nabla\cdot u=0\qquad\qquad\qquad\qquad\qquad\ \end{eqnarray}\right.$$ such that $u(x,t)\equiv 0$ for $|x|^2+|t|^2>1$. In other words, the solution is identically zero for $t&lt;-1$, then "something happens" and the solution becomes non-zero, and for all $t>1$ the solution vanishes again. In the real world, this would look like if the water suddenly started to move in a cup that stands firmly on a table. </p> <p>See V. Scheffer, <a href="http://www.springerlink.com/content/51283358j88mj175/" rel="nofollow">"An inviscid flow with compact support in space-time"</a>, <em>Journal of Geometric Analysis</em>, vol. 3 (1993), pp. 343-401. </p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68731#68731 Answer by Denis Serre for Counterexamples in PDE Denis Serre 2011-06-24T13:08:12Z 2011-06-24T16:01:38Z <p>It seems that non-uniqueness is the main source of counter-example, at least in the above answers. So, one more:</p> <p>Consider the heat equation for harmonic <em>maps</em>: $$u_t-\Delta u+|\nabla u|^2u=0,\qquad |u(x,t)|\equiv1,\qquad(1)$$ with prescribed boundary data $u=g$. A steady solution is a $\phi$ (a <em>harmonic map</em>) such that $$-\Delta \phi+|\nabla u|^2\phi=0,\qquad |\phi(x,t)|\equiv1$$ and $\phi=g$ on the boundary. It is a critical point of the functional $$I[z]:=\int_\Omega|\nabla z|^2dx$$ under the constraints that $|z|\equiv1$ in $\Omega$ and $z=g$ on the boundary. </p> <p>One may choose $g$ such that there exists a harmonic map $\phi$ that does not minimize locally $I[z]$. In this case, the Cauchy problem for (1), with initial data $\phi$, has <strong>two solutions</strong>. One is $\phi$, and the other one is time-dependent, with $I[u(t)]$ non-constant (it decays).</p> <p>This result was due to <a href="http://www.ams.org/mathscinet-getitem?mr=1167832" rel="nofollow">Bethuel, Coron, Ghidaglia, and Soyeur</a>. See also the work of <a href="http://www.numdam.org/item?id=AIHPC_1990__7_4_335_0" rel="nofollow">Coron</a> and later <a href="http://www.springerlink.com/content/e4jcb1qb77gdg9db/" rel="nofollow">Bertsch, Dal Passo, and van der Hout</a>.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68740#68740 Answer by Hendrik Vogt for Counterexamples in PDE Hendrik Vogt 2011-06-24T14:41:23Z 2011-06-24T14:41:23Z <p><a href="http://www.ams.org/mathscinet-getitem?mr=1906035" rel="nofollow">In a paper of 2001</a>, N. Filonov has constructed a second order uniformly elliptic operator in divergence form on $\mathbb R^n$ (where $n\ge3$) with Hölder continuous coefficients and <em>compactly supported eigenfunctions</em>. (It is known that this can't happen for Lipschitz continuous coefficients since then one has the unique continuation property.)</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68783#68783 Answer by Yuri Bakhtin for Counterexamples in PDE Yuri Bakhtin 2011-06-25T11:39:46Z 2011-06-25T11:39:46Z <p>In his <a href="http://www.mathnet.ru/php/getFT.phtml?jrnid=sm&amp;paperid=6410&amp;volume=42&amp;year=1935&amp;issue=2&amp;fpage=199&amp;what=fullt&amp;option_lang=eng" rel="nofollow"> classic 1935 work</a> Tikhonov showed that the Cauchy problem for the heat equation with 0 initial data has nonzero solutions. He also identified uniqueness classes, of course.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/68837#68837 Answer by Kristal Cantwell for Counterexamples in PDE Kristal Cantwell 2011-06-26T05:27:14Z 2011-06-26T05:27:14Z <p>In the paper "One cannot hear the shape of a drum", Carolyn Gordon, David L. Webb and Scott Wolpert give an example of two simply connected regions which are isospectral but not isometric. The article is available <a href="http://www.ams.org/journals/bull/1992-27-01/S0273-0979-1992-00289-6/S0273-0979-1992-00289-6.pdf" rel="nofollow">here</a>. It was in answer to the article "Can you hear the shape of a drum by Mark Kac. Also see <a href="http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum" rel="nofollow">this article</a> for more information about this problem.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/97250#97250 Answer by Uday for Counterexamples in PDE Uday 2012-05-17T18:49:45Z 2012-05-18T05:36:14Z <p>Lewy's example and most other allied examples of non-solvable operators have only complex coefficients. Leaving open the question of unsolvable operators with real coefficient. F.Treves in 1962 has constructed a fourth order operator iterating Lewy operator. </p> <p>If $P$ is Lewy operator then $PP\bar{P}\bar{P}$ is a real coefficient fourth order operator which is not local solvable. </p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/118326#118326 Answer by Luis Silvestre for Counterexamples in PDE Luis Silvestre 2013-01-08T01:22:10Z 2013-01-08T01:29:37Z <p>Here are three nontrivial examples with uniformly elliptic equations.</p> <p><strong>1</strong> Nadirashvilli and Valduts example of a solution to a fully nonlinear uniformly elliptic PDE with constant coefficients which is not $C^2$.</p> <p><strong>Nadirashvili, N. and Vlăduţ, S.</strong> <em>Nonclassical solutions of fully nonlinear elliptic equations.</em> Geom. Funct. Anal. 17 (2007), no. 4, 1283–1296. </p> <p>The original example was in dimension 12. In a later work (joint with Vladimir Tkachev) they brought the dimension down to 5. The example is known to be impossible in dimension 2. Dimensions 3 and 4 are still open.</p> <p><strong>2</strong> Safonov's example of a uniformly elliptic equation in 3D whose solution cannot be Lipschitz continuous.</p> <p><strong>Safonov, M. V.</strong> <em>Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients.</em> (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; translation in Math. USSR-Sb. 60 (1988), no. 1, 269–281 </p> <p>Such example is known to be impossible in dimension 2.</p> <p><strong>3.</strong> Plis' example of a uniformly elliptic equation with Holder coefficients for which the unique continuation property does not hold.</p> <p><strong>Pliś, A.</strong> <em>On non-uniqueness in Cauchy problem for an elliptic second order differential equation.</em> Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 1963 95–100. 35.42 </p> <p>In particular, he proves that there exists a non zero solution to some uniformly elliptic PDE in 3D with Holder coefficients which is identically zero outside of a ball.</p> <p>It is related to the example that Hendrik Vogt suggested. Again, in dimension 2 it would not be possible.</p> http://mathoverflow.net/questions/68680/counterexamples-in-pde/118327#118327 Answer by soulphysics for Counterexamples in PDE soulphysics 2013-01-08T01:42:57Z 2013-01-08T01:42:57Z <p>A particularly simple example is Norton's dome, with height given as a function of radial distance on the surface of the dome by</p> <p>$h = \frac{2}{3g}r^{3/2}$</p> <p>where $g$ is the gravitational constant near the surface of the earth. The dome has a curvature singularity at the apex. And, if we model a mass at the ($r=0$) apex of this dome with zero velocity, we find that Newton's equation does not have a unique solution; the mass can "fall" at any arbitrary time $t$ for no reason at all.</p> <p>Norton's paper about the dome: <a href="http://philsci-archive.pitt.edu/2943/" rel="nofollow">http://philsci-archive.pitt.edu/2943/</a></p> <p>A helpful reply: <a href="http://philsci-archive.pitt.edu/3195/" rel="nofollow">http://philsci-archive.pitt.edu/3195/</a></p>