Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

By Zhiwu Lin, C. Zeng. Memoirs of the American Mathematical Society, 2017

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<p>Consider a general linear Hamiltonian system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u equals upper J upper L u"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _{t}u=JLu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We assume that<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L colon upper X right-arrow upper X Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mtext> </mml:mtext> <mml:mi>L</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\ L:X\rightarrow X^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induces a bounded and symmetric bi-linear form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper L dot comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>L</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \langle L\cdot ,\cdot \right \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which has only finitely many negative dimensions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Superscript minus Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n^{-}(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. There is no restriction on the anti-self-dual operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J colon upper X Superscript asterisk Baseline superset-of upper D left-parenthesis upper J right-parenthesis right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:mo>⊃<!-- ⊃ --></mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">J:X^{\ast }\supset D(J)\rightarrow X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We first obtain a structural decomposition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into the direct sum of several closed subspaces so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is blockwise diagonalized and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper L"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">JL</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript t upper J upper L"> <mml:semantics> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">e^{tJL}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript t upper J upper L"> <mml:semantics> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">e^{tJL}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Superscript minus Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">n^{-}\left ( L\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the dimensions of generalized eigenspaces of eigenvalues of<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper L"> <mml:semantics> <mml:mrow> <mml:mtext> </mml:mtext> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\ JL</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.</p>

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8de6197058fe74e8aacc24819585bf2b789d491e