On optimal regularity estimates for finite-entropy solutions of scalar conservation laws by Lamy, Xavier, Lorent, Andrew, Peng, Guanying

“…We consider finite-entropy solutions of scalar conservation laws $u_t +a(u)_x =0$, that is, bounded weak solutions whose entropy productions are locally finite Radon measures. …”
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The Double-Logarithmic Regularity Criterion of Pressure for the 3D Navier–Stokes Equations in the Besov Space by Min Liu, Juan Song, Tian-Li Li

“…In the paper, we consider the regularity of the weak solutions for the incompressible 3D Navier–Stokes (N–S) equations. …”
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The Regularity Criteria and the A Priori Estimate on the 3D Incompressible Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems by Fan Geng, Shu Wang, Yongxin Wang

“…We establish one regularity criteria of the weak solutions involving only in a vorticity component ω3 and one a priori estimate on the solution that H3u3L∞0,T;Lpℝ3 is bounded for 1≤p≤∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. …”
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Initial and Boundary Value Problems for a Class of Nonlinear Metaparabolic Equations by Huafei Di, Zefang Song

“…At low initial energy level (Ju0<d), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. …”
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Existence of Solutions for Nonhomogeneous A-Harmonic Equations with Variable Growth by Yongqiang Fu, Lifeng Guo

“…We obtain the existence of weak solutions for the above equations in subspace 𝔎01,p(x)(Ω,Λl-1) of W01,p(x)(Ω,Λl-1).…”
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Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity by Keimer, Alexander, Pflug, Lukas

“…We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf {L}^{\infty }(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf {C}\big ([0,T]; \mathsf {L}^{1}_{\mathrm{loc}}\big )$, as well as related maximum principles. …”
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A Generalization of Exponential Class and Its Applications by Hongya Gao, Chao Liu, Hong Tian

“…As an application, we obtain weak monotonicity property for very weak solutions of 𝒜-harmonic equation with variable coefficients under some suitable conditions related to Lθ,∞)(Ω), which provides a generalization of a known result due to Moscariello. …”
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On the critical behavior for a Sobolev-type inequality with Hardy potential by Jleli, Mohamed, Samet, Bessem

“…We investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality $ -\partial _t(\Delta u)-\Delta u+ \frac{\sigma }{|x|^2}u \ge |x|^{\mu }|u|^p$ in $(0,\infty )\times B$, under the inhomogeneous Dirichlet-type boundary condition $u(t,x)=f(x)$ on $(0,\infty )\times \partial B$, where $B$ is the unit open ball of $\mathbb{R}^N$, $N\ge 2$, $\sigma >-\bigl (\frac{N-2}{2}\bigr )^2$, $\mu \in \mathbb{R}$ and $p>1$. …”
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Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films by Matthew Michal, Marina Chugunova, Roman Taranets

“…For the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. …”
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Bifurcation analysis of fractional Kirchhoff–Schrödinger–Poisson systems in $\mathbb R^3$ by Linlin Wang, Yuming Xing

“…We show that the existence of components of (weak) solutions of the above equation bifurcates out from the first eigenvalue $\lambda_1$ of the problem $$(-\Delta)^s u+V(x)u=\lambda g(x)u\quad\mbox{in }\mathbb R^3.$$ The main feature of this paper is the inclusion of a potentially degenerate Kirchhoff model, combined with the critical nonlinearity.…”
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Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises by Arcady Ponosov, Lev Idels

“…Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. …”
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Global Existence of Solutions for a Nonstrictly Hyperbolic System by De-yin Zheng, Yun-guang Lu, Guo-qiang Song, Xue-zhou Lu

“…We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. …”
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Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms by Salvatore D’Asero

“…This paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mo>|</mo><mi>α</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></munder></mstyle><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>α</mi><mo>|</mo></mrow></msup><msup><mrow><mi mathvariant="normal">D</mi></mrow><mi>α</mi></msup><mfenced separators="" open="[" close="]"><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><msup><mrow><mi mathvariant="normal">D</mi></mrow><mn>1</mn></msup><mi>u</mi><mo>,</mo><msup><mrow><mi mathvariant="normal">D</mi></mrow><mn>2</mn></msup><mi>u</mi><mo>)</mo></mrow><mo>−</mo><msub><mi>E</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo> </mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>λ</mi><mo>(</mo><msub><mi>p</mi><mi>α</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo> </mo><mo>sign</mo><mi>u</mi></mfenced><mo>=</mo><mi>f</mi></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>, where the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula> satisfy a strengthened degenerate coercivity condition.…”
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