<p>We construct a two-parameter continuum of type II blow up solutions for the energy-critical nonlinear Schrödinger equation in dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( d = 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. The solutions collapse to a single energy bubble in finite time and have the form <Equation ID="Equ505"> <EquationSource Format="TEX">\(\begin{aligned} u(t,x) = e^{i \alpha (t)}\lambda (t)^{\frac{1}{2}}W(\lambda (t) x) + \eta (t, x ),~ ~ t \in [0, T),~ x \in {{\,\mathrm{\mathbb {R}}\,}}^3, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>λ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>η</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>t</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mrow> <mspace width="0.166667em" /> <mi mathvariant="double-struck">R</mi> <mspace width="0.166667em" /> </mrow> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( W( x) = \big ( 1 + \frac{|x|^2}{3}\big )^{-\frac{1}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mn>1</mn> <mo>+</mo> <mfrac> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mn>3</mn> </mfrac> <msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is the ground state solution, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\lambda (t) = (T -t)^{- \frac{1}{2} - \nu } \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mi>ν</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for suitable <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( \nu &gt; 0 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, &#xa0;<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( \alpha (t) = \alpha _0 \log (T - t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>log</mo> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( T= T(\nu , \alpha _0) &gt; 0 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Further there holds <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( \Vert \eta (t) - \eta _T\Vert _{\dot{H}^1 \cap \dot{H}^2} = o(1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> <mi>η</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> </mrow> <msub> <mi>η</mi> <mi>T</mi> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>1</mn> </msup> <mo>∩</mo> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mrow> </msub> <mo>=</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\( t \rightarrow T^-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mi>T</mi> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\( \eta _T \in \dot{H}^{1} \cap \dot{H}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>η</mi> <mi>T</mi> </msub> <mo>∈</mo> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>1</mn> </msup> <mo>∩</mo> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Blow up dynamics for the 3D energy-critical nonlinear Schrödinger equation

  • Tobias Schmid

摘要

We construct a two-parameter continuum of type II blow up solutions for the energy-critical nonlinear Schrödinger equation in dimension \( d = 3\) d = 3 . The solutions collapse to a single energy bubble in finite time and have the form \(\begin{aligned} u(t,x) = e^{i \alpha (t)}\lambda (t)^{\frac{1}{2}}W(\lambda (t) x) + \eta (t, x ),~ ~ t \in [0, T),~ x \in {{\,\mathrm{\mathbb {R}}\,}}^3, \end{aligned}\) u ( t , x ) = e i α ( t ) λ ( t ) 1 2 W ( λ ( t ) x ) + η ( t , x ) , t [ 0 , T ) , x R 3 , where \( W( x) = \big ( 1 + \frac{|x|^2}{3}\big )^{-\frac{1}{2}}\) W ( x ) = ( 1 + | x | 2 3 ) - 1 2 is the ground state solution, \(\lambda (t) = (T -t)^{- \frac{1}{2} - \nu } \) λ ( t ) = ( T - t ) - 1 2 - ν for suitable \( \nu > 0 \) ν > 0 ,   \( \alpha (t) = \alpha _0 \log (T - t)\) α ( t ) = α 0 log ( T - t ) and \( T= T(\nu , \alpha _0) > 0 \) T = T ( ν , α 0 ) > 0 . Further there holds \( \Vert \eta (t) - \eta _T\Vert _{\dot{H}^1 \cap \dot{H}^2} = o(1)\) η ( t ) - η T H ˙ 1 H ˙ 2 = o ( 1 ) as \( t \rightarrow T^-\) t T - for some \( \eta _T \in \dot{H}^{1} \cap \dot{H}^2\) η T H ˙ 1 H ˙ 2 .