确定性与随机非线性发展方程在金融、物理、生物、流体力学以及大气海洋科学等领域具有广泛应用。本书以泛函分析、随机分析与无穷维动力系统及偏微分方程理论为主线,系统阐述若干类非自治确定性与随机发展方程的整体适定性、吸引子理论与不变测度等长期动力学行为,并融入了作者在该方向的系列研究成果。书中通过提出新的吸引子理论框架与紧性分析方法,建立研究此类系统长期行为的相关普适性工具,可应用于流体力学方程、非线性波动方程与非线性抛物方程等模型,进一步深化对确定性与随机无穷维动力系统理论及其应用的理解。

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1.2011.09-2015.07，海南师范大学，数学与统计学院，攻读理学学士学位 2.2015.09-2020.07，西南大学，数学与统计学院，攻读理学博士学位(硕博连读） 3.2018.09-2019.09,美国New Mexico Institute of Mining and Technology,数学系(联合培养博士）1.2020.06-2022.06，北京应用物理与计算数学研究所，邓稼先创新研究中心，博士后(合作导师:郭柏灵院士) 2.2022.06-至今，贵州师范大学，数学科学学院，校聘教授 3.2024.01-2024.07, 中国科学院数学与系统科学研究院，访问学者 4.2024.09-2025.06, 厦门大学数学科学学院，访问学者偏微分方程和随机动力系统王仁海，男，2020年博士毕业于西南大学，2020-2022北京应用物理与计算数学研究所博士后，现贵州师范大学校聘教授。王仁海教授主要从事无穷维动力系统方面的研究工作，其主要研究成果发表在Math. Ann.、Int. Math. Res. No.、Nonlinearity、SIAM J. Math. Anal.、JDE、JDDE、PAMS等学术期刊上。1.2023-2028，《Letters on Applied and Pure Mathematics》，主编 2.2024-2029，《Electronic Journal of Applied Mathematics》，副主编 3.2023-2028，《Applied Mathematics andStatistics-Numerical Analysis and ScicntificComputation》，副主编 4.2025-2028，《Journal of Inequalities and Applications》,编委 5.2022-2027，《Advances in Mathematical Physics》，编委 6.2022-2027，《Global Journals: Natural Science》，编委 7.2023-2024，《Electronic Research Archive》，客座编辑 8. 2022-2023，《Discrete and Continuous Dynamical Systems，Series S》，客座编辑

Contents
“博士后文库”序言
Preface
Chapter 1 Enhanced Pullback Attractors for 3D Primitive Equations 1
1.1 Introduction and main results 1
1.1.1 Statement of problems 1
1.1.2 Motivations and theoretical results 2
1.1.3 Applications of theoretical results 4
1.2 Theoretical results of enhanced pullback attractors 9
1.2.1 Enhanced pullback attractors for non-autonomous dynamical systems 9
1.2.2 Necessary and sufficient criterions for existence of enhanced pullback attractors 10
1.2.3 Alternative criteria for existence of enhanced pullback attractors 15
1.2.4 Relations, structures and topology properties of enhanced pullback attractors 16
1.2.5 A necessary and sufficient criterion for asymptotic stability of sets 18
1.3 Main results to non-autonomous viscous primitive equation 21
1.3.1 First result: existence and uniqueness of enhanced pullback attractors in V×V 27
1.3.2 Second result: asymptotic stability of enhanced pullback attractors in V×V 29
1.3.3 Third result: existence of enhanced pullback attractors in H2×H2 30
1.3.4 Fourth result: asymptotic stability of enhanced pullback attractors in H2×H2 31
1.4 Long-time behavior and stability analysis of solutions 31
1.4.1 Large-time uniform estimates in H1×H1 31
1.4.2 Uniform “Flattening Effects” in V×V 42
1.4.3 Asymptotic stability of solutions in V×V 46
1.4.4 Large-time uniform estimates in H2×H2 50
1.4.5 Uniform 1/2-H？lder continuity of solutions from V×V to H2×H2 57
1.5 Proof of main results 65
1.5.1 Proof of Theorem 3.6 65
1.5.2 Proof of Theorem 3.7 66
1.5.3 Proof of Theorem 3.8 66
1.5.4 Proof of Theorem 3.9 67
Chapter 2 Fractal Dimensions of Random Invariant Sets and Attractors 68
2.1 Introduction and main results 68
2.2 Improved bound of fractal dimension of random invariant sets 71
2.3 Random attractors of stochastic hydrodynamical systems 78
2.3.1 Description of stochastic hydrodynamical systems 78
2.3.2 Non-autonomous random dynamical systems 80
2.3.3 Long term D-uniform estimates in H 82
2.3.4 Pullback asymptotic compactness of solutions in H 84
2.3.5 Existence and uniqueness of pullback random attractors 92
2.4 (H,V)-regularity of pullback random attractors 94
2.4.1 Definition of bi-spatial D-pullback random (H,V)-attractor 94
2.4.2 Regularity of solutions to pathwise random system 94
2.4.3 Long term D-uniform estimates in V 95
2.4.4 Long time D-uniform “flattening effect” in V 98
2.4.5 Existence of bi-spatial D-pullback random (H,V)-attractor 101
2.5 Upper bound of fractal dimension of pullback random attractors 102
2.5.1 Lipschitz property of solutions in H 103
2.5.2 Lipschitz property of solutions in V 104
2.5.3 Lipschitz projection in H 105
2.5.4 Lipschitz projection in V 107
2.5.5 Finite fractal dimension of random attractor in H and V 110
Chapter 3 Multivalued Random Dynamics of BBM Equations Driven by Colored Noise 113
3.1 Introduction and main results 113
3.2 Preliminaries 115
3.2.1 Pullback attractors for multivalued non-autonomous random dynamical systems 115
3.2.2 Colored noise 117
3.3 Measurability of multivalued solutions to the BBM equation 118
3.3.1 Definition and existence of solutions 118
3.3.2 Uniform estimates on compact time intervals 120
3.3.3 Weak and strong continuity of solutions 122
3.4 Multivalued random dynamical system for the BBM equation 127
3.5 Pullback asymptotic compactness of the BBM equation 128
3.5.1 Uniform estimates in bounded domains 128
3.5.2 Uniform estimates outside bounded domains 129
3.5.3 Uniform estimates on spectral projections of solutions in bounded domains 136
3.5.4 D-pullback asymptotic compactness of solutions 139
3.6 Existence and uniqueness of pullback random attractors 140
Chapter 4 Dynamics of Weakly Dissipative Fractional Hyperbolic Equations Driven by Colored Noise 142
4.1 Introduction and main results 142
4.2 Random attractors of fractional wave equations with superlinear colored noise 146
4.2.1 Assumptions 146
4.2.2 Fractional Laplacian and Sobolev inequalities 148
4.2.3 Cocycle of solutions 150
4.2.4 D-Uniform estimates 151
4.2.5 D-Uniform tail-estimates 154
4.2.6 D-Uniform estimates on bounded domains 163
4.2.7 Existence and uniqueness of pullback random attractors 168
4.3 Random attractors of fractional wave equations with additive white noise 171
4.3.1 Cocycle of solutions 172
4.3.2 D-Uniform estimates 173
4.3.3 Existence and uniqueness of pullback random attractors 175
4.4 Robustness in probability of random attractors from colored noise to white noise 176
4.4.1 Random attractors of fractional wave equation with additive colored noise 176
4.4.2 Convergence of absorbing sets from colored noise to white noise 177
4.4.3 Convergence of solutions from colored noise to white noise 180
4.4.4 Eventual compactness of random attractors with respect to correlation time 182
4.4.5 Convergence in probability of random attractors 194
Chapter 5 Mean Attractors and Invariant Measures of SPDEs Driven by Superlinear Noise 196
5.1 Introduction and main results 196
5.1.1 Statement of problems 196
5.1.2 Global well-posedness of stochastic evolution equation 198
5.1.3 Mean random attractors stochastic evolution equation 199
5.1.4 Invariant measures of stochastic evolution equation: autonomous case 200
5.1.5 Invariant measures of stochastic evolution equation: non-autonomous case 203
5.1.6 Remarks on models 204
5.1.7 Applications of abstract results 205
5.1.8 Outline of chapter 206
5.2 Well-posedness of stochastic evolution equation in high-order Bochner spaces 206
5.2.1 Approximate systems and functional spaces 206
5.2.2 A priori estimates 207
5.2.3 Proof of Theorem 1.2 211
5.3 Mean random attractors of stochastic evolution equation: existence and uniqueness 216
5.4 Invariant measures of stochastic evolution equation: autonomous case 219
5.4.1 Notations related to measures 219
5.4.2 Feller properties, Markov process and process laws 222
5.4.3 Proof of Theorem 1.4 225
5.4.4 Proof of Theorem 1.5 227
5.4.5 Proof of Theorem 1.6 230
5.4.6 Proof of Theorem 1.7 233
5.5 Evolution systems of measures of stochastic evolution equation: non-autonomous case 238
5.5.1 Proof of Theorem 1.8 238
5.6 Applications to models 240
5.6.1 Fractional (s,p)-Laplacian equations 241
5.6.2 Convective Brinkman-Forchheimer equations 247
Chapter 6 Global Well-Posdness, Mean Attractors and Invariant Measures of FitzHugh-Nagumo Systems 257
6.1 Introduction and main results 257
6.2 Preliminaries 260
6.3 Existence of solutions under different noise conditions 264
6.3.1 Existence of solutions for regular additive noise 264
6.3.2 The existence of solutions for general additive noise 274
6.3.3 Existence of solutions for globally Lipschitz noise 282
6.3.4 Existence of solutions for locally Lipschitz noise 292
6.4 Weak pullback mean random attractors 299
6.5 Invariant measures of stochastic FitzHugh-Nagumo systems 303
6.5.1 Uniform estimates of solutions to stochastic FitzHugh-Nagumo system 304
6.5.2 Existence of invariant measures for stochastic FitzHugh-Nagumo system 314
References 320
编后记 335