Minsuk Yang

Yonsei University · Department of Mathematics

We deal with a weak solution \({{\textbf {v}}}\) to the Navier–Stokes initial value problem in \({{\mathbb {R}}}^3\times (0,T)\), that satisfies the strong energy inequality. We impose conditions on certain spectral projections of \({\varvec{\omega }}:={\textbf {curl}}\, {{\textbf {v}}}\) or just \({{\textbf {v}}}\), and we prove the regularity of...

We present new regularity criteria in terms of the negative part of the pressure p or the positive part of the extended Bernoulli pressure B≔p+12|u|2+12|b|2, where u is the velocity, and b is the magnetic field. The criteria extend the previously known results, and the extension is enabled by the use of an appropriate Orlicz norm.

We assume that Ω is either the whole space R3 or a half-space or a smooth bounded or exterior domain in R3, T>0 and (u,b,p) is a suitable weak solution of the MHD equations in Ω×(0,T). We show that (x0,t0)∈Ω×(0,T) is a regular point of the solution (u,b,p) if the limit inferior (for t→t0−) of the sum of the L3–norms of u and b over an arbitrarily s...

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pre...

We assume that \(\Omega \) is either a smooth bounded domain in \({{\mathbb {R}}}^3\) or \(\Omega ={{\mathbb {R}}}^3\), and \(\Omega '\) is a sub-domain of \(\Omega \). We prove that if \(0\le T_1<T_2\le T\le \infty \), \(({\mathbf {u}},{\mathbf {b}},p)\) is a suitable weak solution of the initial–boundary value problem for the MHD equations in \(\...

We study the initial-boundary value problem of the stochastic Navier--Stokes equations in the half space. We prove the existence of weak solutions in the standard Besov space valued random processes when the initial data belong to the critical Besov space.

We present a new regularity criterion for suitable weak solutions of the one-dimensional surface growth initial-value problem in terms of mixed Lebesgue norms.

We prove the existence of a mild solution to the three dimensional incompressible stochastic magnetohydrodynamic equations in the whole space with the initial data which belong to the Sobolev spaces.

We prove that if $0<T_0<T\leq\infty$, $(\mathbf{u},\mathbf{b},p)$ is a suitable weak solution of the MHD equations in $\mathbb{R}^3\times(0,T)$ and either $\mathcal{F}_{\gamma}(p_-)\in L^{\infty}(0,T_0;\, L^{3/2}(\mathbb{R}^3))$ or $\mathcal{F}_{\gamma}((|\mathbf{u}|^2+ |\mathbf{b}|^2+2p)_+)\in L^{\infty}(0,T_0;\, L^{3/2}(\mathbb{R}^3))$ for some $...

We study the partial regularity problem of the three-dimensional incompressible Navier–Stokes equations. We present a new boundary regularity criterion for boundary suitable weak solutions. As an application, a bound for the parabolic Minkowski dimension of possible singular points on the boundary is obtained.

In this paper, we study the potential singular points of interior and boundary suitable weak solutions to the 3D Navier--Stokes equations. It is shown that upper box dimension of interior singular points and boundary singular points are bounded by $7/6$ and $10/9$, respectively. Both proofs rely on recent progress of $\varepsilon$-regularity criter...

We prove the existence of a weak solution to the compressible Navier–Stokes system with hard sphere possibly non-monotone pressure law involving, in particular, the Carnahan–Starling model [2] largely employed in various physical and industrial applications. We take into account large velocities prescribed at the boundary of a bounded piecewise C²...

We study Besov and Triebel–Lizorkin space estimates for fractional diffusion. We measure the smoothing effect of the fractional heat flow in terms of the Besov and Triebel–Lizorkin scale. These estimates have many applications to various partial differential equations.

We study the partial regularity problem of the three-dimensional incompressible Navier--Stokes equations. We present a new boundary regularity criterion for boundary suitable weak solutions. As an application, a bound for the parabolic Minkowski dimension of possible singular points on the boundary is obtained.

We prove existence of weak solutions to the compressible Navier‐Stokes system in barotropic regime (adiabatic coefficient , in three dimensions, in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded Lipschitz piecewise regular domain, without any restriction neither on the...

We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. There have been several attempts to refine the Caffarelli--Kohn--Nirenberg criterion (1982). We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin's...

We study the strong solution to the 3-D compressible Navier– Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density ρ and the divergence of the velocity u without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the underlyin...

We consider suitable weak solutions to the Navier–Stokes equations in time varying domains. We develop Schauder theory for the approximate Stokes equations in time varying domains whose solutions satisfy a uniform localized energy estimate including boundary. Existence of suitable weak solutions in time varying domains follows from compactness in L...

We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $\rho$ and the divergence of the velocity $\bu$ without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the...

We study the partial regularity problem of the incompressible Navier--Stokes equations. In this paper, we show that a reverse H\"older inequality of velocity gradient with increasing support holds under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the...

We establish the existence and the pointwise bound of the fundamental solution for the stationary Stokes system with measurable coefficients in the whole space $\mathbb{R}^d$, $d \ge 3$, under the assumption that weak solutions of the system are locally H\"older continuous. We also discuss the existence and the pointwise bound of the Green function...

We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier-Stokes equations. We present a new regularity criterion for the weak solution $u$ satisfying the condition $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ without any smallness assumption on that scale, where $L^{3,w}(\mathbb{R}^3)$ denotes the stan...

In this paper, we study the convergence of the inverse Laplace transform for valuing American put options when the dynamics of the risky asset is governed by the constant elasticity of variance (CEV) model. The CEV model is one popular alternative of the Black–Scholes model to describe well the real financial market. We calculate various coefficien...

We study the possible interior singular points of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. We present an improved parabolic upper Minkowski dimension of the possible singular set. It is bounded by $95/63$. The result also continue to hold for the three dimensional incompressible magnetohydrodynamic e...

We study boundary singular points of suitable weak solutions to the three dimensional incompressible magnetohydrodynamic equations. By using the generalised Hausdorff measure we estimate the size of boundary singular points and present the improved range of powers of logarithmic factors.

We consider the Keller-Segel model coupled with the incompressible Navier-Stokes equations in dimension three. We prove the local in time existence of the solution for large initial data and the global in time existence of the solution for small initial data plus some smallness condition on the gravitational potential in the critical Besov spaces,...

In this paper we prove a parabolic Triebel-Lizorkin space estimate for the operator given by \[T^{\alpha}f(t,x) = \int_0^t \int_{{\mathbb R}^d} P^{\alpha}(t-s,x-y)f(s,y) dyds,\] where the kernel is \[P^{\alpha}(t,x) = \int_{{\mathbb R}^d} e^{2\pi ix\cdot\xi} e^{-t|\xi|^\alpha} d\xi.\] The operator $T^{\alpha}$ maps from $L^{p}F_{s}^{p,q}$ to $L^{p}...

We derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations. Combining Biot-Savart law, interpolation inequalities and the local energy inequality, we prove a partial regularity theorem for suitable weak solutions. Furthermore, we obtain an improved estimate for the logarithmic Hausdorff dimension o...

In this paper, we study the Cauchy problem of the Camassa-Holm equation. We present a well-posedness result and a blow-up criterion of solutions in the inhomogeneous Triebel-Lizorkin spaces.

We investigate the second moment of a random sampling ζ(1/2+iXt)ζ(1/2+iXt) of the Riemann zeta function on the critical line. Our main result states that if XtXt is an increasing random sampling with gamma distribution, then for all sufficiently large t, E|ζ(1/2+iXt)|2=logt+O(logtloglogt).

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional inte...

We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero cylinder condition. We prove a regularity result after finding suitable spaces for the solution and the pre-assigne...