Pub Date : 2023-10-15DOI: 10.1007/s40072-023-00317-6
Ilya Chevyrev
Abstract We consider a non-linear heat equation $$partial _t u = Delta u + B(u,Du)+P(u)$$ ∂tu=Δu+B(u,Du)+P(u) posed on the d -dimensional torus, where P is a polynomial of degree at most 3 and B is a bilinear map that is not a total derivative. We show that, if the initial condition $$u_0$$ u0 is taken from a sequence of smooth Gaussian fields with a specified covariance, then u exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck–Yang–Mills heat flow extends continuously, which complements recent well-posedness results of Cao–Chatterjee and the author with Chandra–Hairer–Shen. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $$B^{-1/2}_{infty ,infty }$$ B∞,∞-1/2 ; the space $$B^{-1/2}_{infty ,infty }$$ B∞,∞-1/2 is an endpoint since the equation is locally well-posed for $$B^{eta }_{infty ,infty }$$ B∞,∞η for every $$eta >-frac{1}{2}$$ η>-12 .
我们考虑一个非线性热方程$$partial _t u = Delta u + B(u,Du)+P(u)$$∂t u = Δ u + B (u, du) + P (u)，其中P是一个最多3次的多项式，B是一个非全导数的双线性映射。我们证明，如果初始条件$$u_0$$ u 0取自具有指定协方差的光滑高斯场序列，则u表现出高概率的范数膨胀。该结果的一个结果是，不存在在三维环面上携带Gaussian自由场且DeTurck-Yang-Mills热流连续延伸的分布的Banach空间，这补充了cho - chatterjee和作者最近的适定性结果与Chandra-Hairer-Shen。另一个结果是(确定性)非线性热方程在Besov空间$$B^{-1/2}_{infty ,infty }$$ B∞，∞- 1 / 2上的每一点都表现出范数膨胀，因此是局部不适定的;空间$$B^{-1/2}_{infty ,infty }$$ B∞，∞- 1 / 2是一个端点，因为方程对于$$B^{eta }_{infty ,infty }$$ B∞，∞η对于每个$$eta >-frac{1}{2}$$ η ＆gt是局部适定的;- 12。
{"title":"Norm inflation for a non-linear heat equation with gaussian initial conditions","authors":"Ilya Chevyrev","doi":"10.1007/s40072-023-00317-6","DOIUrl":"https://doi.org/10.1007/s40072-023-00317-6","url":null,"abstract":"Abstract We consider a non-linear heat equation $$partial _t u = Delta u + B(u,Du)+P(u)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> posed on the d -dimensional torus, where P is a polynomial of degree at most 3 and B is a bilinear map that is not a total derivative. We show that, if the initial condition $$u_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is taken from a sequence of smooth Gaussian fields with a specified covariance, then u exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck–Yang–Mills heat flow extends continuously, which complements recent well-posedness results of Cao–Chatterjee and the author with Chandra–Hairer–Shen. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $$B^{-1/2}_{infty ,infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> ; the space $$B^{-1/2}_{infty ,infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> is an endpoint since the equation is locally well-posed for $$B^{eta }_{infty ,infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mi>η</mml:mi> </mml:msubsup> </mml:math> for every $$eta >-frac{1}{2}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>></mml:mo> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> .","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136184881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-09DOI: 10.1007/s40072-023-00314-9
Yassine Tahraoui, Fernanda Cipriano
{"title":"Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions","authors":"Yassine Tahraoui, Fernanda Cipriano","doi":"10.1007/s40072-023-00314-9","DOIUrl":"https://doi.org/10.1007/s40072-023-00314-9","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-03DOI: 10.1007/s40072-023-00311-y
Alexander Davie, Fabian Germ, István Gyöngy
Abstract A partially observed jump diffusion $$Z=(X_t,Y_t)_{tin [0,T]}$$ Z=(Xt,Yt)t∈[0,T] given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $$X_t$$ Xt given the observations $$(Y_s)_{sin [0,t]}$$ (Ys)s∈[0,t] exists and belongs to $$L_p$$ Lp if the conditional density of $$X_0$$ X0 given $$Y_0$$ Y0 exists and belongs to $$L_p$$ Lp .
考虑由Wiener过程和泊松鞅测度驱动的随机微分方程给出的部分观测跳跃扩散$$Z=(X_t,Y_t)_{tin [0,T]}$$ Z = (X t, Y t) t∈[0,t]，当方程的系数满足适当的Lipschitz条件和生长条件时。在一般条件下，如果$$X_0$$ X 0在$$Y_0$$ Y 0下的条件密度存在并属于$$L_p$$ L p，则在观测值$$(Y_s)_{sin [0,t]}$$ (Y s) s∈[0,t]下的未观测分量$$X_t$$ X t的条件密度存在且属于$$L_p$$ L p。
{"title":"On partially observed jump diffusions II: the filtering density","authors":"Alexander Davie, Fabian Germ, István Gyöngy","doi":"10.1007/s40072-023-00311-y","DOIUrl":"https://doi.org/10.1007/s40072-023-00311-y","url":null,"abstract":"Abstract A partially observed jump diffusion $$Z=(X_t,Y_t)_{tin [0,T]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $$X_t$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> given the observations $$(Y_s)_{sin [0,t]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:math> exists and belongs to $$L_p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> if the conditional density of $$X_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> given $$Y_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> exists and belongs to $$L_p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> .","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135739399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1007/s40072-023-00313-w
Christoph Schwab, Andreas Stein
Abstract We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $${mathcal {D}}subset {mathbb {R}}^d$$ D⊂Rd , with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for (a) nonuniform pathwise upper and lower coefficient bounds, and for (b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (“error vs work”) analysis of the MLMC-FEM approximations is provided.
{"title":"Multilevel Monte Carlo FEM for elliptic PDEs with Besov random tree priors","authors":"Christoph Schwab, Andreas Stein","doi":"10.1007/s40072-023-00313-w","DOIUrl":"https://doi.org/10.1007/s40072-023-00313-w","url":null,"abstract":"Abstract We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $${mathcal {D}}subset {mathbb {R}}^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> , with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for (a) nonuniform pathwise upper and lower coefficient bounds, and for (b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (“error vs work”) analysis of the MLMC-FEM approximations is provided.","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136248793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1007/s40072-023-00310-z
David Candil, Le Chen, Cheuk Yin Lee
{"title":"Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds","authors":"David Candil, Le Chen, Cheuk Yin Lee","doi":"10.1007/s40072-023-00310-z","DOIUrl":"https://doi.org/10.1007/s40072-023-00310-z","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135246029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-25DOI: 10.1007/s40072-023-00309-6
Jaehoon Kang, Daehan Park
{"title":"An $$L_q(L_p)$$-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps","authors":"Jaehoon Kang, Daehan Park","doi":"10.1007/s40072-023-00309-6","DOIUrl":"https://doi.org/10.1007/s40072-023-00309-6","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86882898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1007/s40072-023-00306-9
Gavin Stewart
{"title":"On the wellposedness of periodic nonlinear Schrödinger equations with white noise dispersion","authors":"Gavin Stewart","doi":"10.1007/s40072-023-00306-9","DOIUrl":"https://doi.org/10.1007/s40072-023-00306-9","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91227135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-25DOI: 10.1007/s40072-023-00307-8
Fengling Wang, T. Caraballo, Yangrong Li, Renhai Wang
{"title":"Asymptotic stability of evolution systems of probability measures of stochastic discrete modified Swift–Hohenberg equations","authors":"Fengling Wang, T. Caraballo, Yangrong Li, Renhai Wang","doi":"10.1007/s40072-023-00307-8","DOIUrl":"https://doi.org/10.1007/s40072-023-00307-8","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87944234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-24DOI: 10.1007/s40072-023-00294-w
Hexiang Wan, Guangchen Wang, Jie Xiong
{"title":"A branching particle system approximation for solving partially observed stochastic optimal control problems via stochastic maximum principle","authors":"Hexiang Wan, Guangchen Wang, Jie Xiong","doi":"10.1007/s40072-023-00294-w","DOIUrl":"https://doi.org/10.1007/s40072-023-00294-w","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83414119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-01DOI: 10.1007/s40072-021-00223-9
V. Barbu, M. Röckner
{"title":"Correction to: Uniqueness for nonlinear Fokker–Planck equations and weak uniqueness for McKean-Vlasov SDEs","authors":"V. Barbu, M. Röckner","doi":"10.1007/s40072-021-00223-9","DOIUrl":"https://doi.org/10.1007/s40072-021-00223-9","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76965863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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