A regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces

By establishing a new trilinear estimate, we show a regularity criterion for the three dimensional micropolar fluid system via the velocity in homogeneous Besov spaces. This improves [B. Q. Dong, Z. L. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal. 73(2010), 2334-2341] in some sense

국소 및 비국소 측도 데이터 문제의 정칙성 이론

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2023. 2. 변순식.In this thesis, we establish various regularity results for nonlinear measure data problems. The results obtained are part of a program devoted to nonlinear Calderón-Zygmund theory and nonlinear potential theory. Firstly, we obtain maximal integrability and fractional differentiability results for elliptic measure data problems with Orlicz growth and borderline double phase growth, respectively. We also obtain fractional differentiability results for parabolic measure data problems under a minimal assumption on the coefficients. Secondly, we obtain gradient potential estimates and fractional differentiability results for elliptic obstacle problems with measure data, by using linearization techniques. In particular, we develop a new method to obtain potential estimates for irregular obstacle problems. For the case of single obstacle problems with L¹-data, we further obtain uniqueness results and comparison principles in order to improve such regularity results. Lastly, we show existence, regularity and potential estimates for mixed local and nonlocal equations with measure data. Also, as a first step to the regularity theory for anisotropic nonlocal problems with nonstandard growth, we establish Hölder regularity for nonlocal double phase problems by identifying sharp assumptions analogous to those for local double phase problems.이 학위논문에서는 비선형 측도 데이터 문제들에 대하여 다양한 정칙성 결과들을 얻는다. 해당 결과들은 비선형 칼데론-지그문트 이론 및 비선형 퍼텐셜 이론을 다루는 과정의 일부이다. 첫 번째로, 오를리츠 성장조건 및 경계선 이중위상 성장조건을 가지는 타원형 측도 데이터 문제에 대하여 각각 최대 적분성 및 분수차수 미분성 결과를 얻는다. 또한 포물형 측도 데이터 문제에 대하여 분수차수 미분성을 계수에 대한 최소한의 가정 하에서 증명한다. 두 번째로, 측도 데이터를 가지는 타원형 장애물 문제에 대하여 선형화 기법을 이용함으로써 그레이디언트 퍼텐셜 가늠 및 분수차수 미분성을 증명한다. 특히 비정칙 장애물 문제에 대해 퍼텐셜 가늠을 얻기 위한 새로운 방법을 개발한다. 더 나아가, L¹ 데이터를 가지는 단일 장애물 문제에 대하여는 해의 유일성 및 비교 원리를 증명하여 이러한 정칙성 결과들을 개선한다. 마지막으로, 측도 데이터를 가지는 국소 및 비국소 혼합 방정식에 대하여 해의 존재성, 정칙성 및 퍼텐셜 가늠을 증명한다. 또한, 비표준 성장조건을 가지는 비등방적 비국소 문제에 대한 정칙성 이론의 첫걸음으로서, 비국소 이중위상 문제에 대한 횔더 정칙성을 국소 이중위상 문제의 경우과 유사한 최적의 조건 하에서 증명한다.1 Introduction 1 1.1 Measure data problems 1 1.1.1 Nonlinear Calderón-Zygmund theory 2 1.1.2 Nonlinear potential theory 4 1.2 Elliptic measure data problems with nonstandard growth 7 1.3 Elliptic obstacle problems with measure data 8 1.4 Nonlocal equations, mixed local and nonlocal equations 9 1.5 Nonlocal operators and measure data 10 1.6 Nonlocal operators with nonstandard growth 11 2 Preliminaries 13 2.1 General notations 13 2.2 Function spaces 15 2.2.1 Musielak-Orlicz spaces 15 2.2.2 Fractional Sobolev spaces 18 2.2.3 Lorentz spaces, Marcinkiewicz spaces 21 2.3 Auxiliary results 22 2.3.1 Basic properties of the vector fields V(·) and A(·) 22 2.3.2 Regularity for homogeneous equations 24 2.3.3 Technical lemmas 34 3 Elliptic and parabolic equations with measure data 35 3.1 Maximal integrability for elliptic measure data problems with Orlicz growth 35 3.1.1 Main results 35 3.1.2 Some technical results 37 3.1.3 Proof of Theorem 3.1.2 43 3.2 Fractional differentiability for elliptic measure data problems with double phase in the borderline case 53 3.2.1 Main results 53 3.2.2 Preliminaries 55 3.2.3 Regularity for homogeneous problems 56 3.2.4 Comparison estimates 61 3.2.5 Proof of Theorem 3.2.2 66 3.3 Fractional differentiability for parabolic measure data problems 71 3.3.1 Main results 71 3.3.2 Preliminaries 73 3.3.3 Some technical results 75 3.3.4 Proof of Theorem 3.3.3 79 4 Elliptic obstacle problems with measure data 83 4.1 Potential estimates for obstacle problems with measure data 84 4.1.1 Main results 85 4.1.2 Reverse Hölders inequalities for homogeneous obstacle problems 88 4.1.3 Basic comparison estimates 93 4.1.4 Linearized comparison estimates 109 4.1.5 The two-scales degenerate alternative 109 4.1.6 The two-scales non-degenerate alternative 111 4.1.7 Combining the two alternatives 126 4.1.8 Proof of Theorem 4.1.2 128 4.1.9 Proof of Theorem 4.1.3 132 4.2 Fractional differentiability for double obstacle problems with measure data 138 4.2.1 Main results 139 4.2.2 Comparison estimates 141 4.2.3 Proof of Theorem 4.2.2 156 4.2.4 Proof of Theorem 4.2.4 158 4.3 Comparison principle for obstacle problems with L¹-data 162 4.3.1 Comparison principles 163 4.3.2 Applications to regularity results 166 5 Mixed local and nonlocal equations with measure data 171 5.1 Main results 171 5.2 Preliminaries 177 5.3 Regularity for homogeneous equations 178 5.4 Comparison estimates 184 5.5 Existence of SOLA 189 5.6 Potential estimates 194 5.6.1 Proof of Theorems 5.1.4 and 5.1.7 194 5.6.2 Proof of Theorem 5.1.5 197 5.7 Continuity criteria for SOLA 204 5.7.1 Proof of Theorem 5.1.8 204 5.7.2 Proof of Theorem 5.1.10 205 6 Nonlocal double phase problems 207 6.1 Main results 208 6.2 Preliminaries 211 6.2.1 Function spaces 211 6.2.2 Inequalities 212 6.3 Existence of weak solutions 215 6.4 Caccioppoli estimates and local boundedness 217 6.5 Hölder continuity 225 6.5.1 Logarithmic estimates 225 6.5.2 Proof of Theorem 6.1.2 235 Abstract (in Korean) 261박

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement