We study the bubbling phenomena for focusing mass-critical nonlinear Schrödinger equations, driven by linear multiplicative noise in the sense of controlled rough paths. In both dimensions one and two, we give a pathwise construction of stochastic multi-bubble blow-up solutions, which concentrate at finitely many distinct points, and behave asymptotically like a sum of pseudo-conformal blow-up solutions. In particular, this provides the first examples of mass quantization phenomenon in the stochastic case. It applies to the canonical deterministic model as well and complements the classical work (Merle, Comm. Math. Phys. 129(2), 223–240 (1990)). The second main result is concerned with the uniqueness of multi-bubble solutions in the energy class with the rate <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1421_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$(T-t)^{3+\zeta }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo>+</mo> <mi>ζ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1421_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> can be any small positive constant. The uniqueness result is new in both the stochastic and deterministic cases. Via the pseudo-conformal symmetry, it also yields the corresponding uniqueness of deterministic multi-solitons. In the special single-bubble case, it provides the conditional uniqueness of stochastic critical-mass blow-up solutions, which improves the recent work (Su and Zhang, J. Funct. Anal. 284, 109796 (2023)).

On the multi-bubble blow-up solutions to focusing mass-critical stochastic nonlinear Schrödinger equations in dimensions one and two

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发表日期：

2025年9月12日

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Yiming Su,
Deng Zhang

摘要

We study the bubbling phenomena for focusing mass-critical nonlinear Schrödinger equations, driven by linear multiplicative noise in the sense of controlled rough paths. In both dimensions one and two, we give a pathwise construction of stochastic multi-bubble blow-up solutions, which concentrate at finitely many distinct points, and behave asymptotically like a sum of pseudo-conformal blow-up solutions. In particular, this provides the first examples of mass quantization phenomenon in the stochastic case. It applies to the canonical deterministic model as well and complements the classical work (Merle, Comm. Math. Phys. 129(2), 223–240 (1990)). The second main result is concerned with the uniqueness of multi-bubble solutions in the energy class with the rate $(T-t)^{3+\zeta }$ ${(T - t)}^{3 + ζ}$ , where $\zeta $ $ζ$ can be any small positive constant. The uniqueness result is new in both the stochastic and deterministic cases. Via the pseudo-conformal symmetry, it also yields the corresponding uniqueness of deterministic multi-solitons. In the special single-bubble case, it provides the conditional uniqueness of stochastic critical-mass blow-up solutions, which improves the recent work (Su and Zhang, J. Funct. Anal. 284, 109796 (2023)).